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1 ·9842bb6bd9e3f8249e6213b3265cce52·60668·editors·optional·elpa-macaulay2_1.24.11+ds-5_all.deb1 ·9842bb6bd9e3f8249e6213b3265cce52·60668·editors·optional·elpa-macaulay2_1.24.11+ds-5_all.deb
2 ·eb2bce0488ab6b90f274a85c7b1dd2a0·29932940·math·optional·macaulay2-common_1.24.11+ds-5_all.deb 
3 ·6927050bd3d312fd1c04ce954c2a8545·48895688·debug·optional·macaulay2-dbgsym_1.24.11+ds-5_amd64.deb 
4 ·e268fce2347ac27216d7d4323336114b·2580884·math·optional·macaulay2_1.24.11+ds-5_amd64.deb2 ·2553c344aa7ac9e6a6bbdd18fce662a9·29934912·math·optional·macaulay2-common_1.24.11+ds-5_all.deb
 3 ·8d9e165effe88124e07ae4b512fc357f·48895452·debug·optional·macaulay2-dbgsym_1.24.11+ds-5_amd64.deb
 4 ·22d32c10e0743c8b3359416ab11f4e3a·2579892·math·optional·macaulay2_1.24.11+ds-5_amd64.deb
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1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary
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1 Package:·macaulay2-common1 Package:·macaulay2-common
2 Source:·macaulay22 Source:·macaulay2
3 Version:·1.24.11+ds-53 Version:·1.24.11+ds-5
4 Architecture:·all4 Architecture:·all
5 Maintainer:·Debian·Math·Team·<team+math@tracker.debian.org>5 Maintainer:·Debian·Math·Team·<team+math@tracker.debian.org>
6 Installed-Size:·2874446 Installed-Size:·287462
7 Depends:·fonts-glyphicons-halflings·(>=·1.009~3.4.1+dfsg),·fonts-katex·(>=·0.16.10+~cs6.1.0),·libjs-bootsidemenu·(>=·1.0.0),·libjs-bootstrap·(>=·3.4.1+dfsg),·libjs-d3·(>=·3.5.17),·libjs-jquery·(>=·3.6.1+dfsg+~3.5.14),·libjs-katex·(>=·0.16.10+~cs6.1.0),·libjs-nouislider·(>=·15.8.1+ds),·libjs-three·(>=·111+dfsg1),·node-clipboard·(>=·2.0.11+ds+~cs9.6.11)7 Depends:·fonts-glyphicons-halflings·(>=·1.009~3.4.1+dfsg),·fonts-katex·(>=·0.16.10+~cs6.1.0),·libjs-bootsidemenu·(>=·1.0.0),·libjs-bootstrap·(>=·3.4.1+dfsg),·libjs-d3·(>=·3.5.17),·libjs-jquery·(>=·3.6.1+dfsg+~3.5.14),·libjs-katex·(>=·0.16.10+~cs6.1.0),·libjs-nouislider·(>=·15.8.1+ds),·libjs-three·(>=·111+dfsg1),·node-clipboard·(>=·2.0.11+ds+~cs9.6.11)
8 Section:·math8 Section:·math
9 Priority:·optional9 Priority:·optional
10 Multi-Arch:·foreign10 Multi-Arch:·foreign
11 Homepage:·http://macaulay2.com11 Homepage:·http://macaulay2.com
12 Description:·Software·system·for·algebraic·geometry·research·(common·files)12 Description:·Software·system·for·algebraic·geometry·research·(common·files)
13 ·Macaulay·2·is·a·software·system·for·algebraic·geometry·research,·written·by13 ·Macaulay·2·is·a·software·system·for·algebraic·geometry·research,·written·by
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10 o2·=·cokernel·|·a·b·c·|10 o2·=·cokernel·|·a·b·c·|
  
11 ····························111 ····························1
12 o2·:·R-module,·quotient·of·R12 o2·:·R-module,·quotient·of·R
  
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17 ·····························································17 ·····························································
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24 ·····························································24 ·····························································
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90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·burkeResolution(M,·7,·Check·=>·false)91 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·burkeResolution(M,·7,·Check·=>·false)
92 ·--·1.24218s·elapsed92 ·--·2.0146s·elapsed
  
93 ······1······3······9······27······81······243······729······218793 ······1······3······9······27······81······243······729······2187
94 o3·=·R··&lt;--·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R····&lt;--·R94 o3·=·R··&lt;--·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R····&lt;--·R
95 ·····························································95 ·····························································
96 ·····0······1······2······3·······4·······5········6········796 ·····0······1······2······3·······4·······5········6········7
  
97 o3·:·Complex</code></pre>97 o3·:·Complex</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·burkeResolution(M,·7,·Check·=>·true)100 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·burkeResolution(M,·7,·Check·=>·true)
101 ·--·1.79817s·elapsed101 ·--·3.57397s·elapsed
  
102 ······1······3······9······27······81······243······729······2187102 ······1······3······9······27······81······243······729······2187
103 o4·=·R··&lt;--·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R····&lt;--·R103 o4·=·R··&lt;--·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R····&lt;--·R
104 ·····························································104 ·····························································
105 ·····0······1······2······3·······4·······5········6········7105 ·····0······1······2······3·······4·······5········6········7
  
106 o4·:·Complex</code></pre>106 o4·:·Complex</code></pre>
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Offset 24, 24 lines modifiedOffset 24, 24 lines modified
24 i2·:·M·=·coker·vars·R24 i2·:·M·=·coker·vars·R
  
25 o2·=·cokernel·|·a·b·c·|25 o2·=·cokernel·|·a·b·c·|
  
26 ····························126 ····························1
27 o2·:·R-module,·quotient·of·R27 o2·:·R-module,·quotient·of·R
28 i3·:·elapsedTime·burkeResolution(M,·7,·Check·=>·false)28 i3·:·elapsedTime·burkeResolution(M,·7,·Check·=>·false)
29 ·--·1.24218s·elapsed29 ·--·2.0146s·elapsed
  
30 ······1······3······9······27······81······243······729······218730 ······1······3······9······27······81······243······729······2187
31 o3·=·R··<--·R··<--·R··<--·R···<--·R···<--·R····<--·R····<--·R31 o3·=·R··<--·R··<--·R··<--·R···<--·R···<--·R····<--·R····<--·R
  
32 ·····0······1······2······3·······4·······5········6········732 ·····0······1······2······3·······4·······5········6········7
  
33 o3·:·Complex33 o3·:·Complex
34 i4·:·elapsedTime·burkeResolution(M,·7,·Check·=>·true)34 i4·:·elapsedTime·burkeResolution(M,·7,·Check·=>·true)
35 ·--·1.79817s·elapsed35 ·--·3.57397s·elapsed
  
36 ······1······3······9······27······81······243······729······218736 ······1······3······9······27······81······243······729······2187
37 o4·=·R··<--·R··<--·R··<--·R···<--·R···<--·R····<--·R····<--·R37 o4·=·R··<--·R··<--·R··<--·R···<--·R···<--·R····<--·R····<--·R
  
38 ·····0······1······2······3·······4·······5········6········738 ·····0······1······2······3·······4·······5········6········7
  
39 o4·:·Complex39 o4·:·Complex
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./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/___Calculations_spwith_sprandom_spsimplicial_spcomplexes.out
    
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1 --·-*-·M2-comint·-*-·hash:·64566133369511003201 --·-*-·M2-comint·-*-·hash:·6456613336951100320
  
2 i1·:·setRandomSeed(currentTime());2 i1·:·setRandomSeed(currentTime());
  
3 i2·:·K·=·randomAbstractSimplicialComplex(4)3 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
5 ·······························0·=>·{{2},·{3}}5 ·······························0·=>·{{1}}
6 ·······························1·=>·{{2,·3}} 
  
7 o2·:·AbstractSimplicialComplex6 o2·:·AbstractSimplicialComplex
  
8 i3·:·prune·HH·simplicialChainComplex·K7 i3·:·prune·HH·simplicialChainComplex·K
  
9 ·······18 ·······1
10 o3·=·ZZ9 o3·=·ZZ
Offset 19, 17 lines modifiedOffset 18, 18 lines modified
  
19 o3·:·Complex18 o3·:·Complex
  
20 i4·:·setRandomSeed(currentTime());19 i4·:·setRandomSeed(currentTime());
  
21 i5·:·L·=·randomAbstractSimplicialComplex(6,3)20 i5·:·L·=·randomAbstractSimplicialComplex(6,3)
  
22 o5·=·AbstractSimplicialComplex{-1·=>·{{}}·····}21 o5·=·AbstractSimplicialComplex{-1·=>·{{}}···················}
23 ·······························0·=>·{{2},·{3}}22 ·······························0·=>·{{1},·{5},·{6}}
 23 ·······························1·=>·{{1,·5},·{1,·6},·{5,·6}}
24 ·······························1·=>·{{2,·3}}24 ·······························2·=>·{{1,·5,·6}}
  
25 o5·:·AbstractSimplicialComplex25 o5·:·AbstractSimplicialComplex
  
26 i6·:·prune·HH·simplicialChainComplex·L26 i6·:·prune·HH·simplicialChainComplex·L
  
27 ·······127 ·······1
28 o6·=·ZZ28 o6·=·ZZ
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38 o6·:·Complex38 o6·:·Complex
  
39 i7·:·setRandomSeed(currentTime());39 i7·:·setRandomSeed(currentTime());
  
40 i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)40 i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
41 o8·=·AbstractSimplicialComplex{-1·=>·{{}}···················································}41 o8·=·AbstractSimplicialComplex{-1·=>·{{}}···········································}
42 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}42 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}
43 ·······························1·=>·{{2,·3},·{2,·5},·{2,·6},·{3,·4},·{3,·5},·{4,·5},·{5,·6}}43 ·······························1·=>·{{2,·3},·{2,·4},·{2,·5},·{2,·6},·{3,·6},·{4,·5}}
44 ·······························2·=>·{{2,·3,·5},·{2,·5,·6},·{3,·4,·5}}44 ·······························2·=>·{{2,·3,·6},·{2,·4,·5}}
  
45 o8·:·AbstractSimplicialComplex45 o8·:·AbstractSimplicialComplex
  
46 i9·:·prune·HH·simplicialChainComplex·M46 i9·:·prune·HH·simplicialChainComplex·M
  
47 ·······147 ·······1
48 o9·=·ZZ48 o9·=·ZZ
Offset 59, 24 lines modifiedOffset 59, 24 lines modified
59 o9·:·Complex59 o9·:·Complex
  
60 i10·:·setRandomSeed(currentTime());60 i10·:·setRandomSeed(currentTime());
  
61 i11·:·K·=·randomAbstractSimplicialComplex(4)61 i11·:·K·=·randomAbstractSimplicialComplex(4)
  
62 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}62 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
63 ································0·=>·{{2},·{3}}63 ································0·=>·{{1},·{3}}
64 ································1·=>·{{2,·3}}64 ································1·=>·{{1,·3}}
  
65 o11·:·AbstractSimplicialComplex65 o11·:·AbstractSimplicialComplex
  
66 i12·:·J·=·randomSubSimplicialComplex(K)66 i12·:·J·=·randomSubSimplicialComplex(K)
  
67 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}67 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
68 ································0·=>·{{2},·{3}}68 ································0·=>·{{1},·{3}}
69 ································1·=>·{{2,·3}}69 ································1·=>·{{1,·3}}
  
70 o12·:·AbstractSimplicialComplex70 o12·:·AbstractSimplicialComplex
  
71 i13·:·inducedSimplicialChainComplexMap(K,J)71 i13·:·inducedSimplicialChainComplexMap(K,J)
  
72 ············2················272 ············2················2
73 o13·=·0·:·ZZ··<-----------·ZZ··:·073 o13·=·0·:·ZZ··<-----------·ZZ··:·0
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./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Abstract__Simplicial__Complex.out
    
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1 --·-*-·M2-comint·-*-·hash:·92224415997616292451 --·-*-·M2-comint·-*-·hash:·9222441599761629245
  
2 i1·:·setRandomSeed(currentTime());2 i1·:·setRandomSeed(currentTime());
  
3 i2·:·K·=·randomAbstractSimplicialComplex(4)3 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}···················}4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
5 ·······························0·=>·{{1},·{3},·{4}} 
6 ·······························1·=>·{{1,·3},·{1,·4},·{3,·4}} 
7 ·······························2·=>·{{1,·3,·4}}5 ·······························0·=>·{{2}}
  
8 o2·:·AbstractSimplicialComplex6 o2·:·AbstractSimplicialComplex
  
9 i3·:·setRandomSeed(currentTime());7 i3·:·setRandomSeed(currentTime());
  
10 i4·:·L·=·randomAbstractSimplicialComplex(6,3)8 i4·:·L·=·randomAbstractSimplicialComplex(6,3)
  
11 o4·=·AbstractSimplicialComplex{-1·=>·{{}}·····}9 o4·=·AbstractSimplicialComplex{-1·=>·{{}}···························}
12 ·······························0·=>·{{1},·{4}}10 ·······························0·=>·{{1},·{4},·{5},·{6}}
 11 ·······························1·=>·{{1,·4},·{1,·5},·{4,·5},·{5,·6}}
13 ·······························1·=>·{{1,·4}}12 ·······························2·=>·{{1,·4,·5}}
  
14 o4·:·AbstractSimplicialComplex13 o4·:·AbstractSimplicialComplex
  
15 i5·:·setRandomSeed(currentTime());14 i5·:·setRandomSeed(currentTime());
  
16 i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)15 i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
17 o6·=·AbstractSimplicialComplex{-1·=>·{{}}···························································}16 o6·=·AbstractSimplicialComplex{-1·=>·{{}}···························································}
18 ·······························0·=>·{{1},·{2},·{3},·{4},·{5},·{6}}17 ·······························0·=>·{{1},·{2},·{3},·{4},·{6}}
19 ·······························1·=>·{{1,·5},·{1,·6},·{2,·3},·{2,·4},·{2,·6},·{3,·4},·{4,·6},·{5,·6}}18 ·······························1·=>·{{1,·2},·{1,·3},·{1,·6},·{2,·3},·{2,·4},·{2,·6},·{3,·6},·{4,·6}}
20 ·······························2·=>·{{1,·5,·6},·{2,·3,·4},·{2,·4,·6}}19 ·······························2·=>·{{1,·2,·3},·{1,·3,·6},·{2,·4,·6}}
  
21 o6·:·AbstractSimplicialComplex20 o6·:·AbstractSimplicialComplex
  
22 i7·:·21 i7·:·
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./usr/share/doc/Macaulay2/AbstractSimplicialComplexes/example-output/_random__Sub__Simplicial__Complex.out
    
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1 --·-*-·M2-comint·-*-·hash:·134731048092355422971 --·-*-·M2-comint·-*-·hash:·13473104809235542297
  
2 i1·:·setRandomSeed(currentTime());2 i1·:·setRandomSeed(currentTime());
  
3 i2·:·K·=·randomAbstractSimplicialComplex(4)3 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}4 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
5 ·······························0·=>·{{2}}5 ·······························0·=>·{{2},·{3}}
 6 ·······························1·=>·{{2,·3}}
  
6 o2·:·AbstractSimplicialComplex7 o2·:·AbstractSimplicialComplex
  
7 i3·:·J·=·randomSubSimplicialComplex(K)8 i3·:·J·=·randomSubSimplicialComplex(K)
  
8 o3·=·AbstractSimplicialComplex{-1·=>·{{}}}9 o3·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
9 ·······························0·=>·{{2}}10 ·······························0·=>·{{2},·{3}}
 11 ·······························1·=>·{{2,·3}}
  
10 o3·:·AbstractSimplicialComplex12 o3·:·AbstractSimplicialComplex
  
11 i4·:·13 i4·:·
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52 ········<table·class="examples">52 ········<table·class="examples">
53 ··········<tr>53 ··········<tr>
54 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>54 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>
55 </td>··········</tr>55 </td>··········</tr>
56 ··········<tr>56 ··········<tr>
57 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)57 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)
  
58 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}58 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
59 ·······························0·=>·{{2},·{3}}59 ·······························0·=>·{{1}}
60 ·······························1·=>·{{2,·3}} 
  
61 o2·:·AbstractSimplicialComplex</code></pre>60 o2·:·AbstractSimplicialComplex</code></pre>
62 </td>··········</tr>61 </td>··········</tr>
63 ··········<tr>62 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i3·:·prune·HH·simplicialChainComplex·K63 <td>··············<pre><code·class="language-macaulay2">i3·:·prune·HH·simplicialChainComplex·K
  
65 ·······164 ·······1
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79 ········<table·class="examples">78 ········<table·class="examples">
80 ··········<tr>79 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i4·:·setRandomSeed(currentTime());</code></pre>80 <td>··············<pre><code·class="language-macaulay2">i4·:·setRandomSeed(currentTime());</code></pre>
82 </td>··········</tr>81 </td>··········</tr>
83 ··········<tr>82 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i5·:·L·=·randomAbstractSimplicialComplex(6,3)83 <td>··············<pre><code·class="language-macaulay2">i5·:·L·=·randomAbstractSimplicialComplex(6,3)
  
85 o5·=·AbstractSimplicialComplex{-1·=>·{{}}·····}84 o5·=·AbstractSimplicialComplex{-1·=>·{{}}···················}
86 ·······························0·=>·{{2},·{3}}85 ·······························0·=>·{{1},·{5},·{6}}
 86 ·······························1·=>·{{1,·5},·{1,·6},·{5,·6}}
87 ·······························1·=>·{{2,·3}}87 ·······························2·=>·{{1,·5,·6}}
  
88 o5·:·AbstractSimplicialComplex</code></pre>88 o5·:·AbstractSimplicialComplex</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i6·:·prune·HH·simplicialChainComplex·L91 <td>··············<pre><code·class="language-macaulay2">i6·:·prune·HH·simplicialChainComplex·L
  
92 ·······192 ·······1
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106 ········<table·class="examples">106 ········<table·class="examples">
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i7·:·setRandomSeed(currentTime());</code></pre>108 <td>··············<pre><code·class="language-macaulay2">i7·:·setRandomSeed(currentTime());</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)111 <td>··············<pre><code·class="language-macaulay2">i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
112 o8·=·AbstractSimplicialComplex{-1·=>·{{}}···················································}112 o8·=·AbstractSimplicialComplex{-1·=>·{{}}···········································}
113 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}113 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}
114 ·······························1·=>·{{2,·3},·{2,·5},·{2,·6},·{3,·4},·{3,·5},·{4,·5},·{5,·6}}114 ·······························1·=>·{{2,·3},·{2,·4},·{2,·5},·{2,·6},·{3,·6},·{4,·5}}
115 ·······························2·=>·{{2,·3,·5},·{2,·5,·6},·{3,·4,·5}}115 ·······························2·=>·{{2,·3,·6},·{2,·4,·5}}
  
116 o8·:·AbstractSimplicialComplex</code></pre>116 o8·:·AbstractSimplicialComplex</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i9·:·prune·HH·simplicialChainComplex·M119 <td>··············<pre><code·class="language-macaulay2">i9·:·prune·HH·simplicialChainComplex·M
  
120 ·······1120 ·······1
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135 ··········<tr>135 ··········<tr>
136 <td>··············<pre><code·class="language-macaulay2">i10·:·setRandomSeed(currentTime());</code></pre>136 <td>··············<pre><code·class="language-macaulay2">i10·:·setRandomSeed(currentTime());</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i11·:·K·=·randomAbstractSimplicialComplex(4)139 <td>··············<pre><code·class="language-macaulay2">i11·:·K·=·randomAbstractSimplicialComplex(4)
  
140 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}140 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
141 ································0·=>·{{2},·{3}}141 ································0·=>·{{1},·{3}}
142 ································1·=>·{{2,·3}}142 ································1·=>·{{1,·3}}
  
143 o11·:·AbstractSimplicialComplex</code></pre>143 o11·:·AbstractSimplicialComplex</code></pre>
144 </td>··········</tr>144 </td>··········</tr>
145 ··········<tr>145 ··········<tr>
146 <td>··············<pre><code·class="language-macaulay2">i12·:·J·=·randomSubSimplicialComplex(K)146 <td>··············<pre><code·class="language-macaulay2">i12·:·J·=·randomSubSimplicialComplex(K)
  
147 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}147 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
148 ································0·=>·{{2},·{3}}148 ································0·=>·{{1},·{3}}
149 ································1·=>·{{2,·3}}149 ································1·=>·{{1,·3}}
  
150 o12·:·AbstractSimplicialComplex</code></pre>150 o12·:·AbstractSimplicialComplex</code></pre>
151 </td>··········</tr>151 </td>··········</tr>
152 ··········<tr>152 ··········<tr>
153 <td>··············<pre><code·class="language-macaulay2">i13·:·inducedSimplicialChainComplexMap(K,J)153 <td>··············<pre><code·class="language-macaulay2">i13·:·inducedSimplicialChainComplexMap(K,J)
  
154 ············2················2154 ············2················2
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9 In·what·follows·we·illustrate·a·collection·of·homological·calculations·that·can9 In·what·follows·we·illustrate·a·collection·of·homological·calculations·that·can
10 be·performed·on·random·simplicial·complexes.10 be·performed·on·random·simplicial·complexes.
11 Create·a·random·abstract·simplicial·complex·with·vertices·supported·on·a·subset11 Create·a·random·abstract·simplicial·complex·with·vertices·supported·on·a·subset
12 of·[n]·=·{1,...,n}.12 of·[n]·=·{1,...,n}.
13 i1·:·setRandomSeed(currentTime());13 i1·:·setRandomSeed(currentTime());
14 i2·:·K·=·randomAbstractSimplicialComplex(4)14 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
15 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}15 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
16 ·······························0·=>·{{2},·{3}}16 ·······························0·=>·{{1}}
17 ·······························1·=>·{{2,·3}} 
  
18 o2·:·AbstractSimplicialComplex17 o2·:·AbstractSimplicialComplex
19 i3·:·prune·HH·simplicialChainComplex·K18 i3·:·prune·HH·simplicialChainComplex·K
  
20 ·······119 ·······1
21 o3·=·ZZ20 o3·=·ZZ
  
22 ·····021 ·····0
  
23 o3·:·Complex22 o3·:·Complex
24 Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.23 Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.
25 i4·:·setRandomSeed(currentTime());24 i4·:·setRandomSeed(currentTime());
26 i5·:·L·=·randomAbstractSimplicialComplex(6,3)25 i5·:·L·=·randomAbstractSimplicialComplex(6,3)
  
27 o5·=·AbstractSimplicialComplex{-1·=>·{{}}·····}26 o5·=·AbstractSimplicialComplex{-1·=>·{{}}···················}
28 ·······························0·=>·{{2},·{3}}27 ·······························0·=>·{{1},·{5},·{6}}
 28 ·······························1·=>·{{1,·5},·{1,·6},·{5,·6}}
29 ·······························1·=>·{{2,·3}}29 ·······························2·=>·{{1,·5,·6}}
  
30 o5·:·AbstractSimplicialComplex30 o5·:·AbstractSimplicialComplex
31 i6·:·prune·HH·simplicialChainComplex·L31 i6·:·prune·HH·simplicialChainComplex·L
  
32 ·······132 ·······1
33 o6·=·ZZ33 o6·=·ZZ
  
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48 binomial(binomial(n,d+1),m)·possibilities.48 binomial(binomial(n,d+1),m)·possibilities.
49 i7·:·setRandomSeed(currentTime());49 i7·:·setRandomSeed(currentTime());
50 i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)50 i8·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
51 o8·=·AbstractSimplicialComplex{-1·=>·{{}}51 o8·=·AbstractSimplicialComplex{-1·=>·{{}}
52 }52 }
53 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}53 ·······························0·=>·{{2},·{3},·{4},·{5},·{6}}
54 ·······························1·=>·{{2,·3},·{2,·5},·{2,·6},·{3,·4},·{3,·5},54 ·······························1·=>·{{2,·3},·{2,·4},·{2,·5},·{2,·6},·{3,·6},
55 {4,·5},·{5,·6}}55 {4,·5}}
56 ·······························2·=>·{{2,·3,·5},·{2,·5,·6},·{3,·4,·5}}56 ·······························2·=>·{{2,·3,·6},·{2,·4,·5}}
  
57 o8·:·AbstractSimplicialComplex57 o8·:·AbstractSimplicialComplex
58 i9·:·prune·HH·simplicialChainComplex·M58 i9·:·prune·HH·simplicialChainComplex·M
  
59 ·······159 ·······1
60 o9·=·ZZ60 o9·=·ZZ
  
Offset 66, 23 lines modifiedOffset 66, 23 lines modified
  
66 o9·:·Complex66 o9·:·Complex
67 Creates·a·random·sub-simplicial·complex·of·a·given·simplicial·complex.67 Creates·a·random·sub-simplicial·complex·of·a·given·simplicial·complex.
68 i10·:·setRandomSeed(currentTime());68 i10·:·setRandomSeed(currentTime());
69 i11·:·K·=·randomAbstractSimplicialComplex(4)69 i11·:·K·=·randomAbstractSimplicialComplex(4)
  
70 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}70 o11·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
71 ································0·=>·{{2},·{3}}71 ································0·=>·{{1},·{3}}
72 ································1·=>·{{2,·3}}72 ································1·=>·{{1,·3}}
  
73 o11·:·AbstractSimplicialComplex73 o11·:·AbstractSimplicialComplex
74 i12·:·J·=·randomSubSimplicialComplex(K)74 i12·:·J·=·randomSubSimplicialComplex(K)
  
75 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}75 o12·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
76 ································0·=>·{{2},·{3}}76 ································0·=>·{{1},·{3}}
77 ································1·=>·{{2,·3}}77 ································1·=>·{{1,·3}}
  
78 o12·:·AbstractSimplicialComplex78 o12·:·AbstractSimplicialComplex
79 i13·:·inducedSimplicialChainComplexMap(K,J)79 i13·:·inducedSimplicialChainComplexMap(K,J)
  
80 ············2················280 ············2················2
81 o13·=·0·:·ZZ··<-----------·ZZ··:·081 o13·=·0·:·ZZ··<-----------·ZZ··:·0
82 ·················|·1·0·|82 ·················|·1·0·|
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50 ········<table·class="examples">50 ········<table·class="examples">
51 ··········<tr>51 ··········<tr>
52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>
53 </td>··········</tr>53 </td>··········</tr>
54 ··········<tr>54 ··········<tr>
55 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)55 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)
  
56 o2·=·AbstractSimplicialComplex{-1·=>·{{}}···················}56 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
57 ·······························0·=>·{{1},·{3},·{4}} 
58 ·······························1·=>·{{1,·3},·{1,·4},·{3,·4}} 
59 ·······························2·=>·{{1,·3,·4}}57 ·······························0·=>·{{2}}
  
60 o2·:·AbstractSimplicialComplex</code></pre>58 o2·:·AbstractSimplicialComplex</code></pre>
61 </td>··········</tr>59 </td>··········</tr>
62 ········</table>60 ········</table>
63 ········<div>61 ········<div>
64 ··········<p>Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.</p>62 ··········<p>Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.</p>
65 ········</div>63 ········</div>
66 ········<table·class="examples">64 ········<table·class="examples">
67 ··········<tr>65 ··········<tr>
68 <td>··············<pre><code·class="language-macaulay2">i3·:·setRandomSeed(currentTime());</code></pre>66 <td>··············<pre><code·class="language-macaulay2">i3·:·setRandomSeed(currentTime());</code></pre>
69 </td>··········</tr>67 </td>··········</tr>
70 ··········<tr>68 ··········<tr>
71 <td>··············<pre><code·class="language-macaulay2">i4·:·L·=·randomAbstractSimplicialComplex(6,3)69 <td>··············<pre><code·class="language-macaulay2">i4·:·L·=·randomAbstractSimplicialComplex(6,3)
  
72 o4·=·AbstractSimplicialComplex{-1·=>·{{}}·····}70 o4·=·AbstractSimplicialComplex{-1·=>·{{}}···························}
73 ·······························0·=>·{{1},·{4}}71 ·······························0·=>·{{1},·{4},·{5},·{6}}
 72 ·······························1·=>·{{1,·4},·{1,·5},·{4,·5},·{5,·6}}
74 ·······························1·=>·{{1,·4}}73 ·······························2·=>·{{1,·4,·5}}
  
75 o4·:·AbstractSimplicialComplex</code></pre>74 o4·:·AbstractSimplicialComplex</code></pre>
76 </td>··········</tr>75 </td>··········</tr>
77 ········</table>76 ········</table>
78 ········<div>77 ········<div>
79 ··········<p>Create·the·random·complex·Y_d(n,m)·which·has·vertex·set·[n]·and·complete·(d·−·1)-skeleton,·and·has·exactly·m·d-dimensional·faces,·chosen·at·random·from·all·binomial(binomial(n,d+1),m)·possibilities.</p>78 ··········<p>Create·the·random·complex·Y_d(n,m)·which·has·vertex·set·[n]·and·complete·(d·−·1)-skeleton,·and·has·exactly·m·d-dimensional·faces,·chosen·at·random·from·all·binomial(binomial(n,d+1),m)·possibilities.</p>
80 ········</div>79 ········</div>
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86 ··········<tr>85 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i5·:·setRandomSeed(currentTime());</code></pre>86 <td>··············<pre><code·class="language-macaulay2">i5·:·setRandomSeed(currentTime());</code></pre>
88 </td>··········</tr>87 </td>··········</tr>
89 ··········<tr>88 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)89 <td>··············<pre><code·class="language-macaulay2">i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
91 o6·=·AbstractSimplicialComplex{-1·=>·{{}}···························································}90 o6·=·AbstractSimplicialComplex{-1·=>·{{}}···························································}
92 ·······························0·=>·{{1},·{2},·{3},·{4},·{5},·{6}}91 ·······························0·=>·{{1},·{2},·{3},·{4},·{6}}
93 ·······························1·=>·{{1,·5},·{1,·6},·{2,·3},·{2,·4},·{2,·6},·{3,·4},·{4,·6},·{5,·6}}92 ·······························1·=>·{{1,·2},·{1,·3},·{1,·6},·{2,·3},·{2,·4},·{2,·6},·{3,·6},·{4,·6}}
94 ·······························2·=>·{{1,·5,·6},·{2,·3,·4},·{2,·4,·6}}93 ·······························2·=>·{{1,·2,·3},·{1,·3,·6},·{2,·4,·6}}
  
95 o6·:·AbstractSimplicialComplex</code></pre>94 o6·:·AbstractSimplicialComplex</code></pre>
96 </td>··········</tr>95 </td>··········</tr>
97 ········</table>96 ········</table>
98 ······</div>97 ······</div>
99 ······<div>98 ······<div>
100 ········<h2>See·also</h2>99 ········<h2>See·also</h2>
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6 *\x8**\x8**\x8**\x8**\x8**\x8*·r\x8ra\x8an\x8nd\x8do\x8om\x8mA\x8Ab\x8bs\x8st\x8tr\x8ra\x8ac\x8ct\x8tS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x·-\x8--\x8-·C\x8Cr\x8re\x8ea\x8at\x8te\x8e·a\x8a·r\x8ra\x8an\x8nd\x8do\x8om\x8m·s\x8si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8l·s\x8se\x8et\x8t·*\x8**\x8**\x8**\x8**\x8**\x8*6 *\x8**\x8**\x8**\x8**\x8**\x8*·r\x8ra\x8an\x8nd\x8do\x8om\x8mA\x8Ab\x8bs\x8st\x8tr\x8ra\x8ac\x8ct\x8tS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x·-\x8--\x8-·C\x8Cr\x8re\x8ea\x8at\x8te\x8e·a\x8a·r\x8ra\x8an\x8nd\x8do\x8om\x8m·s\x8si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8l·s\x8se\x8et\x8t·*\x8**\x8**\x8**\x8**\x8**\x8*
7 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*7 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
8 Create·a·random·abstract·simplicial·complex·with·vertices·supported·on·a·subset8 Create·a·random·abstract·simplicial·complex·with·vertices·supported·on·a·subset
9 of·[n]·=·{1,...,n}.9 of·[n]·=·{1,...,n}.
10 i1·:·setRandomSeed(currentTime());10 i1·:·setRandomSeed(currentTime());
11 i2·:·K·=·randomAbstractSimplicialComplex(4)11 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
12 o2·=·AbstractSimplicialComplex{-1·=>·{{}}···················}12 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}
13 ·······························0·=>·{{1},·{3},·{4}} 
14 ·······························1·=>·{{1,·3},·{1,·4},·{3,·4}} 
15 ·······························2·=>·{{1,·3,·4}}13 ·······························0·=>·{{2}}
  
16 o2·:·AbstractSimplicialComplex14 o2·:·AbstractSimplicialComplex
17 Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.15 Create·a·random·simplicial·complex·on·[n]·with·dimension·at·most·equal·to·r.
18 i3·:·setRandomSeed(currentTime());16 i3·:·setRandomSeed(currentTime());
19 i4·:·L·=·randomAbstractSimplicialComplex(6,3)17 i4·:·L·=·randomAbstractSimplicialComplex(6,3)
  
20 o4·=·AbstractSimplicialComplex{-1·=>·{{}}·····}18 o4·=·AbstractSimplicialComplex{-1·=>·{{}}···························}
21 ·······························0·=>·{{1},·{4}}19 ·······························0·=>·{{1},·{4},·{5},·{6}}
 20 ·······························1·=>·{{1,·4},·{1,·5},·{4,·5},·{5,·6}}
22 ·······························1·=>·{{1,·4}}21 ·······························2·=>·{{1,·4,·5}}
  
23 o4·:·AbstractSimplicialComplex22 o4·:·AbstractSimplicialComplex
24 Create·the·random·complex·Y_d(n,m)·which·has·vertex·set·[n]·and·complete·(d·âˆ’23 Create·the·random·complex·Y_d(n,m)·which·has·vertex·set·[n]·and·complete·(d·âˆ’
25 1)-skeleton,·and·has·exactly·m·d-dimensional·faces,·chosen·at·random·from·all24 1)-skeleton,·and·has·exactly·m·d-dimensional·faces,·chosen·at·random·from·all
26 binomial(binomial(n,d+1),m)·possibilities.25 binomial(binomial(n,d+1),m)·possibilities.
27 i5·:·setRandomSeed(currentTime());26 i5·:·setRandomSeed(currentTime());
28 i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)27 i6·:·M·=·randomAbstractSimplicialComplex(6,3,2)
  
29 o6·=·AbstractSimplicialComplex{-1·=>·{{}}28 o6·=·AbstractSimplicialComplex{-1·=>·{{}}
30 }29 }
31 ·······························0·=>·{{1},·{2},·{3},·{4},·{5},·{6}}30 ·······························0·=>·{{1},·{2},·{3},·{4},·{6}}
32 ·······························1·=>·{{1,·5},·{1,·6},·{2,·3},·{2,·4},·{2,·6},31 ·······························1·=>·{{1,·2},·{1,·3},·{1,·6},·{2,·3},·{2,·4},
33 {3,·4},·{4,·6},·{5,·6}}32 {2,·6},·{3,·6},·{4,·6}}
34 ·······························2·=>·{{1,·5,·6},·{2,·3,·4},·{2,·4,·6}}33 ·······························2·=>·{{1,·2,·3},·{1,·3,·6},·{2,·4,·6}}
  
35 o6·:·AbstractSimplicialComplex34 o6·:·AbstractSimplicialComplex
36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*35 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
37 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m·--·get·a·random·object36 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m·--·get·a·random·object
38 ····*·randomSquareFreeMonomialIdeal·(missing·documentation)37 ····*·randomSquareFreeMonomialIdeal·(missing·documentation)
39 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mA\x8Ab\x8bs\x8st\x8tr\x8ra\x8ac\x8ct\x8tS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*38 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mA\x8Ab\x8bs\x8st\x8tr\x8ra\x8ac\x8ct\x8tS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*
40 ····*·randomAbstractSimplicialComplex(ZZ)39 ····*·randomAbstractSimplicialComplex(ZZ)
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50 ········<table·class="examples">50 ········<table·class="examples">
51 ··········<tr>51 ··········<tr>
52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime());</code></pre>
53 </td>··········</tr>53 </td>··········</tr>
54 ··········<tr>54 ··········<tr>
55 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)55 <td>··············<pre><code·class="language-macaulay2">i2·:·K·=·randomAbstractSimplicialComplex(4)
  
56 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}56 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
57 ·······························0·=>·{{2}}57 ·······························0·=>·{{2},·{3}}
 58 ·······························1·=>·{{2,·3}}
  
58 o2·:·AbstractSimplicialComplex</code></pre>59 o2·:·AbstractSimplicialComplex</code></pre>
59 </td>··········</tr>60 </td>··········</tr>
60 ··········<tr>61 ··········<tr>
61 <td>··············<pre><code·class="language-macaulay2">i3·:·J·=·randomSubSimplicialComplex(K)62 <td>··············<pre><code·class="language-macaulay2">i3·:·J·=·randomSubSimplicialComplex(K)
  
62 o3·=·AbstractSimplicialComplex{-1·=>·{{}}}63 o3·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
63 ·······························0·=>·{{2}}64 ·······························0·=>·{{2},·{3}}
 65 ·······························1·=>·{{2,·3}}
  
64 o3·:·AbstractSimplicialComplex</code></pre>66 o3·:·AbstractSimplicialComplex</code></pre>
65 </td>··········</tr>67 </td>··········</tr>
66 ········</table>68 ········</table>
67 ······</div>69 ······</div>
68 ······<div·class="waystouse">70 ······<div·class="waystouse">
69 ········<h2>Ways·to·use·<span·class="tt">randomSubSimplicialComplex</span>:</h2>71 ········<h2>Ways·to·use·<span·class="tt">randomSubSimplicialComplex</span>:</h2>
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6 *\x8**\x8**\x8**\x8**\x8**\x8*·r\x8ra\x8an\x8nd\x8do\x8om\x8mS\x8Su\x8ub\x8bS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x·-\x8--\x8-·C\x8Cr\x8re\x8ea\x8at\x8te\x8e·a\x8a·r\x8ra\x8an\x8nd\x8do\x8om\x8m·s\x8su\x8ub\x8b-\x8-s\x8si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8l·c\x8co\x8om\x8mp\x8pl\x8le\x8ex\x8x6 *\x8**\x8**\x8**\x8**\x8**\x8*·r\x8ra\x8an\x8nd\x8do\x8om\x8mS\x8Su\x8ub\x8bS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x·-\x8--\x8-·C\x8Cr\x8re\x8ea\x8at\x8te\x8e·a\x8a·r\x8ra\x8an\x8nd\x8do\x8om\x8m·s\x8su\x8ub\x8b-\x8-s\x8si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8l·c\x8co\x8om\x8mp\x8pl\x8le\x8ex\x8x
7 *\x8**\x8**\x8**\x8**\x8**\x8*7 *\x8**\x8**\x8**\x8**\x8**\x8*
8 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*8 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
9 Creates·a·random·sub-simplicial·complex·of·a·given·simplicial·complex.9 Creates·a·random·sub-simplicial·complex·of·a·given·simplicial·complex.
10 i1·:·setRandomSeed(currentTime());10 i1·:·setRandomSeed(currentTime());
11 i2·:·K·=·randomAbstractSimplicialComplex(4)11 i2·:·K·=·randomAbstractSimplicialComplex(4)
  
12 o2·=·AbstractSimplicialComplex{-1·=>·{{}}}12 o2·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
13 ·······························0·=>·{{2}}13 ·······························0·=>·{{2},·{3}}
 14 ·······························1·=>·{{2,·3}}
  
14 o2·:·AbstractSimplicialComplex15 o2·:·AbstractSimplicialComplex
15 i3·:·J·=·randomSubSimplicialComplex(K)16 i3·:·J·=·randomSubSimplicialComplex(K)
  
16 o3·=·AbstractSimplicialComplex{-1·=>·{{}}}17 o3·=·AbstractSimplicialComplex{-1·=>·{{}}·····}
17 ·······························0·=>·{{2}}18 ·······························0·=>·{{2},·{3}}
 19 ·······························1·=>·{{2,·3}}
  
18 o3·:·AbstractSimplicialComplex20 o3·:·AbstractSimplicialComplex
19 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mS\x8Su\x8ub\x8bS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*21 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mS\x8Su\x8ub\x8bS\x8Si\x8im\x8mp\x8pl\x8li\x8ic\x8ci\x8ia\x8al\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*
20 ····*·randomSubSimplicialComplex(AbstractSimplicialComplex)22 ····*·randomSubSimplicialComplex(AbstractSimplicialComplex)
21 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
22 The·object·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8S_\x8u_\x8b_\x8S_\x8i_\x8m_\x8p_\x8l_\x8i_\x8c_\x8i_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.24 The·object·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8S_\x8u_\x8b_\x8S_\x8i_\x8m_\x8p_\x8l_\x8i_\x8c_\x8i_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out
    
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49 o8·:·BettiTally49 o8·:·BettiTally
  
50 i9·:·c=codim·I50 i9·:·c=codim·I
  
51 o9·=·451 o9·=·4
  
52 i10·:·elapsedTime·fI=res·I52 i10·:·elapsedTime·fI=res·I
53 ·--·.0170399s·elapsed53 ·--·.077924s·elapsed
  
54 ········1·······14·······33·······28·······854 ········1·······14·······33·······28·······8
55 o10·=·Pn··<--·Pn···<--·Pn···<--·Pn···<--·Pn··<--·055 o10·=·Pn··<--·Pn···<--·Pn···<--·Pn···<--·Pn··<--·0
56 ··················································56 ··················································
57 ······0·······1········2········3········4·······557 ······0·······1········2········3········4·······5
  
58 o10·:·ChainComplex58 o10·:·ChainComplex
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87 o13·:·BettiTally87 o13·:·BettiTally
  
88 i14·:·phi=map(P2,Pn,H);88 i14·:·phi=map(P2,Pn,H);
  
89 o14·:·RingMap·P2·<--·Pn89 o14·:·RingMap·P2·<--·Pn
  
90 i15·:·elapsedTime·betti(I'=trim·ker·phi)90 i15·:·elapsedTime·betti(I'=trim·ker·phi)
91 ·--·.559311s·elapsed91 ·--·1.19535s·elapsed
  
92 ·············0··192 ·············0··1
93 o15·=·total:·1·1193 o15·=·total:·1·11
94 ··········0:·1··.94 ··········0:·1··.
95 ··········1:·.··395 ··········1:·.··3
96 ··········2:·.··896 ··········2:·.··8
  
97 o15·:·BettiTally97 o15·:·BettiTally
  
98 i16·:·I'==·I98 i16·:·I'==·I
  
99 o16·=·true99 o16·=·true
  
100 i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;100 i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
101 ·--·4.25696s·elapsed101 ·--·6.12472s·elapsed
  
102 i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))102 i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
103 ··························0·1103 ··························0·1
104 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}104 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}
105 ·······················0:·1·2105 ·······················0:·1·2
106 ··························0·1106 ··························0·1
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79 ··········1:·.·.79 ··········1:·.·.
80 ··········2:·.·.80 ··········2:·.·.
81 ··········3:·.·881 ··········3:·.·8
  
82 o13·:·BettiTally82 o13·:·BettiTally
  
83 i14·:·elapsedTime·sub(I,H)83 i14·:·elapsedTime·sub(I,H)
84 ·--·.0302717s·elapsed84 ·--·.0486106s·elapsed
  
85 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)85 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)
  
86 o14·:·Ideal·of·P286 o14·:·Ideal·of·P2
  
87 i15·:·phi=map(P2,Pn,H);87 i15·:·phi=map(P2,Pn,H);
  
88 o15·:·RingMap·P2·<--·Pn88 o15·:·RingMap·P2·<--·Pn
  
89 i16·:·elapsedTime·betti(I'=trim·ker·phi)89 i16·:·elapsedTime·betti(I'=trim·ker·phi)
90 ·--·.0989118s·elapsed90 ·--·.249328s·elapsed
  
91 ·············0··191 ·············0··1
92 o16·=·total:·1·1292 o16·=·total:·1·12
93 ··········0:·1··.93 ··········0:·1··.
94 ··········1:·.·1294 ··········1:·.·12
  
95 o16·:·BettiTally95 o16·:·BettiTally
  
96 i17·:·I'==·I96 i17·:·I'==·I
  
97 o17·=·true97 o17·=·true
  
98 i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;98 i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
99 ·--·1.64072s·elapsed99 ·--·3.27592s·elapsed
  
100 i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))100 i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
101 ···························0··1101 ···························0··1
102 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}102 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}
103 ························0:·1··.103 ························0:·1··.
104 ························1:·.··.104 ························1:·.··.
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130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i9·:·c=codim·I131 <td>··············<pre><code·class="language-macaulay2">i9·:·c=codim·I
  
132 o9·=·4</code></pre>132 o9·=·4</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime·fI=res·I135 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime·fI=res·I
136 ·--·.0170399s·elapsed136 ·--·.077924s·elapsed
  
137 ········1·······14·······33·······28·······8137 ········1·······14·······33·······28·······8
138 o10·=·Pn··&lt;--·Pn···&lt;--·Pn···&lt;--·Pn···&lt;--·Pn··&lt;--·0138 o10·=·Pn··&lt;--·Pn···&lt;--·Pn···&lt;--·Pn···&lt;--·Pn··&lt;--·0
139 ··················································139 ··················································
140 ······0·······1········2········3········4·······5140 ······0·······1········2········3········4·······5
  
141 o10·:·ChainComplex</code></pre>141 o10·:·ChainComplex</code></pre>
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Offset 55, 15 lines modifiedOffset 55, 15 lines modified
55 ·········2:·.·1255 ·········2:·.·12
  
56 o8·:·BettiTally56 o8·:·BettiTally
57 i9·:·c=codim·I57 i9·:·c=codim·I
  
58 o9·=·458 o9·=·4
59 i10·:·elapsedTime·fI=res·I59 i10·:·elapsedTime·fI=res·I
60 ·--·.0170399s·elapsed60 ·--·.077924s·elapsed
  
61 ········1·······14·······33·······28·······861 ········1·······14·······33·······28·······8
62 o10·=·Pn··<--·Pn···<--·Pn···<--·Pn···<--·Pn··<--·062 o10·=·Pn··<--·Pn···<--·Pn···<--·Pn···<--·Pn··<--·0
  
63 ······0·······1········2········3········4·······563 ······0·······1········2········3········4·······5
  
64 o10·:·ChainComplex64 o10·:·ChainComplex
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./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html
    
Offset 192, 15 lines modifiedOffset 192, 15 lines modified
192 ··········<tr>192 ··········<tr>
193 <td>··············<pre><code·class="language-macaulay2">i14·:·phi=map(P2,Pn,H);193 <td>··············<pre><code·class="language-macaulay2">i14·:·phi=map(P2,Pn,H);
  
194 o14·:·RingMap·P2·&lt;--·Pn</code></pre>194 o14·:·RingMap·P2·&lt;--·Pn</code></pre>
195 </td>··········</tr>195 </td>··········</tr>
196 ··········<tr>196 ··········<tr>
197 <td>··············<pre><code·class="language-macaulay2">i15·:·elapsedTime·betti(I'=trim·ker·phi)197 <td>··············<pre><code·class="language-macaulay2">i15·:·elapsedTime·betti(I'=trim·ker·phi)
198 ·--·.559311s·elapsed198 ·--·1.19535s·elapsed
  
199 ·············0··1199 ·············0··1
200 o15·=·total:·1·11200 o15·=·total:·1·11
201 ··········0:·1··.201 ··········0:·1··.
202 ··········1:·.··3202 ··········1:·.··3
203 ··········2:·.··8203 ··········2:·.··8
  
Offset 209, 15 lines modifiedOffset 209, 15 lines modified
209 ··········<tr>209 ··········<tr>
210 <td>··············<pre><code·class="language-macaulay2">i16·:·I'==·I210 <td>··············<pre><code·class="language-macaulay2">i16·:·I'==·I
  
211 o16·=·true</code></pre>211 o16·=·true</code></pre>
212 </td>··········</tr>212 </td>··········</tr>
213 ··········<tr>213 ··········<tr>
214 <td>··············<pre><code·class="language-macaulay2">i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;214 <td>··············<pre><code·class="language-macaulay2">i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
215 ·--·4.25696s·elapsed</code></pre>215 ·--·6.12472s·elapsed</code></pre>
216 </td>··········</tr>216 </td>··········</tr>
217 ··········<tr>217 ··········<tr>
218 <td>··············<pre><code·class="language-macaulay2">i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))218 <td>··············<pre><code·class="language-macaulay2">i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
219 ··························0·1219 ··························0·1
220 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}220 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}
221 ·······················0:·1·2221 ·······················0:·1·2
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Offset 111, 28 lines modifiedOffset 111, 28 lines modified
111 ··········6:·.·7111 ··········6:·.·7
  
112 o13·:·BettiTally112 o13·:·BettiTally
113 i14·:·phi=map(P2,Pn,H);113 i14·:·phi=map(P2,Pn,H);
  
114 o14·:·RingMap·P2·<--·Pn114 o14·:·RingMap·P2·<--·Pn
115 i15·:·elapsedTime·betti(I'=trim·ker·phi)115 i15·:·elapsedTime·betti(I'=trim·ker·phi)
116 ·--·.559311s·elapsed116 ·--·1.19535s·elapsed
  
117 ·············0··1117 ·············0··1
118 o15·=·total:·1·11118 o15·=·total:·1·11
119 ··········0:·1··.119 ··········0:·1··.
120 ··········1:·.··3120 ··········1:·.··3
121 ··········2:·.··8121 ··········2:·.··8
  
122 o15·:·BettiTally122 o15·:·BettiTally
123 i16·:·I'==·I123 i16·:·I'==·I
  
124 o16·=·true124 o16·=·true
125 i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;125 i17·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
126 ·--·4.25696s·elapsed126 ·--·6.12472s·elapsed
127 i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))127 i18·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
128 ··························0·1128 ··························0·1
129 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}129 o18·=·Tally{(1,·1,·total:·1·2)·=>·5}
130 ·······················0:·1·2130 ·······················0:·1·2
131 ··························0·1131 ··························0·1
132 ············(1,·3,·total:·1·3)·=>·8132 ············(1,·3,·total:·1·3)·=>·8
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./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
    
Offset 167, 28 lines modifiedOffset 167, 28 lines modified
167 ··········2:·.·.167 ··········2:·.·.
168 ··········3:·.·8168 ··········3:·.·8
  
169 o13·:·BettiTally</code></pre>169 o13·:·BettiTally</code></pre>
170 </td>··········</tr>170 </td>··········</tr>
171 ··········<tr>171 ··········<tr>
172 <td>··············<pre><code·class="language-macaulay2">i14·:·elapsedTime·sub(I,H)172 <td>··············<pre><code·class="language-macaulay2">i14·:·elapsedTime·sub(I,H)
173 ·--·.0302717s·elapsed173 ·--·.0486106s·elapsed
  
174 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)174 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)
  
175 o14·:·Ideal·of·P2</code></pre>175 o14·:·Ideal·of·P2</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ··········<tr>177 ··········<tr>
178 <td>··············<pre><code·class="language-macaulay2">i15·:·phi=map(P2,Pn,H);178 <td>··············<pre><code·class="language-macaulay2">i15·:·phi=map(P2,Pn,H);
  
179 o15·:·RingMap·P2·&lt;--·Pn</code></pre>179 o15·:·RingMap·P2·&lt;--·Pn</code></pre>
180 </td>··········</tr>180 </td>··········</tr>
181 ··········<tr>181 ··········<tr>
182 <td>··············<pre><code·class="language-macaulay2">i16·:·elapsedTime·betti(I'=trim·ker·phi)182 <td>··············<pre><code·class="language-macaulay2">i16·:·elapsedTime·betti(I'=trim·ker·phi)
183 ·--·.0989118s·elapsed183 ·--·.249328s·elapsed
  
184 ·············0··1184 ·············0··1
185 o16·=·total:·1·12185 o16·=·total:·1·12
186 ··········0:·1··.186 ··········0:·1··.
187 ··········1:·.·12187 ··········1:·.·12
  
188 o16·:·BettiTally</code></pre>188 o16·:·BettiTally</code></pre>
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196 ··········<tr>196 ··········<tr>
197 <td>··············<pre><code·class="language-macaulay2">i17·:·I'==·I197 <td>··············<pre><code·class="language-macaulay2">i17·:·I'==·I
  
198 o17·=·true</code></pre>198 o17·=·true</code></pre>
199 </td>··········</tr>199 </td>··········</tr>
200 ··········<tr>200 ··········<tr>
201 <td>··············<pre><code·class="language-macaulay2">i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;201 <td>··············<pre><code·class="language-macaulay2">i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
202 ·--·1.64072s·elapsed</code></pre>202 ·--·3.27592s·elapsed</code></pre>
203 </td>··········</tr>203 </td>··········</tr>
204 ··········<tr>204 ··········<tr>
205 <td>··············<pre><code·class="language-macaulay2">i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))205 <td>··············<pre><code·class="language-macaulay2">i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
206 ···························0··1206 ···························0··1
207 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}207 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}
208 ························0:·1··.208 ························0:·1··.
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83 ··········0:·1·.83 ··········0:·1·.
84 ··········1:·.·.84 ··········1:·.·.
85 ··········2:·.·.85 ··········2:·.·.
86 ··········3:·.·886 ··········3:·.·8
  
87 o13·:·BettiTally87 o13·:·BettiTally
88 i14·:·elapsedTime·sub(I,H)88 i14·:·elapsedTime·sub(I,H)
89 ·--·.0302717s·elapsed89 ·--·.0486106s·elapsed
  
90 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)90 o14·=·ideal·(0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0,·0)
  
91 o14·:·Ideal·of·P291 o14·:·Ideal·of·P2
92 i15·:·phi=map(P2,Pn,H);92 i15·:·phi=map(P2,Pn,H);
  
93 o15·:·RingMap·P2·<--·Pn93 o15·:·RingMap·P2·<--·Pn
94 i16·:·elapsedTime·betti(I'=trim·ker·phi)94 i16·:·elapsedTime·betti(I'=trim·ker·phi)
95 ·--·.0989118s·elapsed95 ·--·.249328s·elapsed
  
96 ·············0··196 ·············0··1
97 o16·=·total:·1·1297 o16·=·total:·1·12
98 ··········0:·1··.98 ··········0:·1··.
99 ··········1:·.·1299 ··········1:·.·12
  
100 o16·:·BettiTally100 o16·:·BettiTally
101 i17·:·I'==·I101 i17·:·I'==·I
  
102 o17·=·true102 o17·=·true
103 i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;103 i18·:·elapsedTime·basePts=primaryDecomposition·ideal·H;
104 ·--·1.64072s·elapsed104 ·--·3.27592s·elapsed
105 i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))105 i19·:·tally·apply(basePts,c->(dim·c,·degree·c,·betti·c))
  
106 ···························0··1106 ···························0··1
107 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}107 o19·=·Tally{(0,·34,·total:·1·15)·=>·1}
108 ························0:·1··.108 ························0:·1··.
109 ························1:·.··.109 ························1:·.··.
110 ························2:·.··.110 ························2:·.··.
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./usr/share/doc/Macaulay2/AlgebraicSplines/dump/rawdocumentation.dump
    
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11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh
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./usr/share/doc/Macaulay2/AnalyzeSheafOnP1/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy
11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl
11 IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzY5LCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi11 IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzY5LCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi
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./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out
    
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114 ······|·19a+19b··-38a-16b·-18a-13b·16a+22b··|114 ······|·19a+19b··-38a-16b·-18a-13b·16a+22b··|
115 ······|·-10a-29b·39a+21b··-43a-15b·45a-34b··|115 ······|·-10a-29b·39a+21b··-43a-15b·45a-34b··|
  
116 ··············4······4116 ··············4······4
117 o13·:·Matrix·A··<--·A117 o13·:·Matrix·A··<--·A
  
118 i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))118 i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))
119 ·--·used·0.457033s·(cpu);·0.305077s·(thread);·0s·(gc)119 ·--·used·0.478185s·(cpu);·0.337257s·(thread);·0s·(gc)
  
120 ·············0·1·2120 ·············0·1·2
121 o14·=·total:·3·6·3121 o14·=·total:·3·6·3
122 ··········0:·3·.·.122 ··········0:·3·.·.
123 ··········1:·.·6·.123 ··········1:·.·6·.
124 ··········2:·.·.·3124 ··········2:·.·.·3
  
125 o14·:·BettiTally125 o14·:·BettiTally
  
126 i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))126 i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))
127 ·--·used·0.490589s·(cpu);·0.343483s·(thread);·0s·(gc)127 ·--·used·0.55658s·(cpu);·0.356307s·(thread);·0s·(gc)
  
128 ·············0·1·2128 ·············0·1·2
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130 ··········0:·3·.·.130 ··········0:·3·.·.
131 ··········1:·.·6·.131 ··········1:·.·6·.
132 ··········2:·.·.·3132 ··········2:·.·.·3
  
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229 ··············4······4229 ··············4······4
230 o13·:·Matrix·A··&lt;--·A</code></pre>230 o13·:·Matrix·A··&lt;--·A</code></pre>
231 </td>··········</tr>231 </td>··········</tr>
232 ··········<tr>232 ··········<tr>
233 <td>··············<pre><code·class="language-macaulay2">i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))233 <td>··············<pre><code·class="language-macaulay2">i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))
234 ·--·used·0.457033s·(cpu);·0.305077s·(thread);·0s·(gc)234 ·--·used·0.478185s·(cpu);·0.337257s·(thread);·0s·(gc)
  
235 ·············0·1·2235 ·············0·1·2
236 o14·=·total:·3·6·3236 o14·=·total:·3·6·3
237 ··········0:·3·.·.237 ··········0:·3·.·.
238 ··········1:·.·6·.238 ··········1:·.·6·.
239 ··········2:·.·.·3239 ··········2:·.·.·3
  
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245 ········</table>245 ········</table>
246 ········<div>246 ········<div>
247 ··········<p>With·the·form·<span·class="tt">pureResolution(p,q,D)</span>·we·can·directly·create·the·situation·of·<span·class="tt">pureResolution(M,D)</span>·where·M·is·generic·product(m_i+1)·x·#D-1+sum(m_i)·matrix·of·linear·forms·defined·over·a·ring·with·product(m_i+1)·*·#D-1+sum(m_i)·variables·of·characteristic·p,·created·by·the·script.·For·a·given·number·of·variables·in·A·this·runs·much·faster·than·taking·a·random·matrix·M.</p>247 ··········<p>With·the·form·<span·class="tt">pureResolution(p,q,D)</span>·we·can·directly·create·the·situation·of·<span·class="tt">pureResolution(M,D)</span>·where·M·is·generic·product(m_i+1)·x·#D-1+sum(m_i)·matrix·of·linear·forms·defined·over·a·ring·with·product(m_i+1)·*·#D-1+sum(m_i)·variables·of·characteristic·p,·created·by·the·script.·For·a·given·number·of·variables·in·A·this·runs·much·faster·than·taking·a·random·matrix·M.</p>
248 ········</div>248 ········</div>
249 ········<table·class="examples">249 ········<table·class="examples">
250 ··········<tr>250 ··········<tr>
251 <td>··············<pre><code·class="language-macaulay2">i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))251 <td>··············<pre><code·class="language-macaulay2">i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))
252 ·--·used·0.490589s·(cpu);·0.343483s·(thread);·0s·(gc)252 ·--·used·0.55658s·(cpu);·0.356307s·(thread);·0s·(gc)
  
253 ·············0·1·2253 ·············0·1·2
254 o15·=·total:·3·6·3254 o15·=·total:·3·6·3
255 ··········0:·3·.·.255 ··········0:·3·.·.
256 ··········1:·.·6·.256 ··········1:·.·6·.
257 ··········2:·.·.·3257 ··········2:·.·.·3
  
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162 ······|·-30a-29b·-29a-24b·-47a-39b·38a+2b···|162 ······|·-30a-29b·-29a-24b·-47a-39b·38a+2b···|
163 ······|·19a+19b··-38a-16b·-18a-13b·16a+22b··|163 ······|·19a+19b··-38a-16b·-18a-13b·16a+22b··|
164 ······|·-10a-29b·39a+21b··-43a-15b·45a-34b··|164 ······|·-10a-29b·39a+21b··-43a-15b·45a-34b··|
  
165 ··············4······4165 ··············4······4
166 o13·:·Matrix·A··<--·A166 o13·:·Matrix·A··<--·A
167 i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))167 i14·:·time·betti·(F·=·pureResolution(M,{0,2,4}))
168 ·--·used·0.457033s·(cpu);·0.305077s·(thread);·0s·(gc)168 ·--·used·0.478185s·(cpu);·0.337257s·(thread);·0s·(gc)
  
169 ·············0·1·2169 ·············0·1·2
170 o14·=·total:·3·6·3170 o14·=·total:·3·6·3
171 ··········0:·3·.·.171 ··········0:·3·.·.
172 ··········1:·.·6·.172 ··········1:·.·6·.
173 ··········2:·.·.·3173 ··········2:·.·.·3
  
174 o14·:·BettiTally174 o14·:·BettiTally
175 With·the·form·pureResolution(p,q,D)·we·can·directly·create·the·situation·of175 With·the·form·pureResolution(p,q,D)·we·can·directly·create·the·situation·of
176 pureResolution(M,D)·where·M·is·generic·product(m_i+1)·x·#D-1+sum(m_i)·matrix·of176 pureResolution(M,D)·where·M·is·generic·product(m_i+1)·x·#D-1+sum(m_i)·matrix·of
177 linear·forms·defined·over·a·ring·with·product(m_i+1)·*·#D-1+sum(m_i)·variables177 linear·forms·defined·over·a·ring·with·product(m_i+1)·*·#D-1+sum(m_i)·variables
178 of·characteristic·p,·created·by·the·script.·For·a·given·number·of·variables·in178 of·characteristic·p,·created·by·the·script.·For·a·given·number·of·variables·in
179 A·this·runs·much·faster·than·taking·a·random·matrix·M.179 A·this·runs·much·faster·than·taking·a·random·matrix·M.
180 i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))180 i15·:·time·betti·(F·=·pureResolution(11,4,{0,2,4}))
181 ·--·used·0.490589s·(cpu);·0.343483s·(thread);·0s·(gc)181 ·--·used·0.55658s·(cpu);·0.356307s·(thread);·0s·(gc)
  
182 ·············0·1·2182 ·············0·1·2
183 o15·=·total:·3·6·3183 o15·=·total:·3·6·3
184 ··········0:·3·.·.184 ··········0:·3·.·.
185 ··········1:·.·6·.185 ··········1:·.·6·.
186 ··········2:·.·.·3186 ··········2:·.·.·3
  
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./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
    
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1 --·-*-·M2-comint·-*-·hash:·13305455765671 --·-*-·M2-comint·-*-·hash:·1330545576567
  
2 i1·:·runBenchmarks·"res39"2 i1·:·runBenchmarks·"res39"
3 --·beginning·computation·Tue·Jun··3·02:45:38·-12·20253 --·beginning·computation·Tue·Jul··7·13:19:02·+14·2026
4 --·Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.1.140-1·(2025-05-22)·x86_64·GNU/Linux4 --·Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
5 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·2196.688··5 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·1996.250··
6 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.06 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.0
7 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.0989455·seconds7 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.177871·seconds
  
8 i2·:·8 i2·:·
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./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
    
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73 ········<h2>Description</h2>73 ········<h2>Description</h2>
74 ········<div>74 ········<div>
75 ··········<p>The·tests·available·are:·<br>&quot;deg2generic&quot;·--·gb·of·a·generic·ideal·of·codimension·2·and·degree·2<br>&quot;gb4by4comm&quot;·--·gb·of·the·ideal·of·generic·commuting·4·by·4·matrices·over·ZZ/101<br>&quot;gb3445&quot;·--·gb·of·an·ideal·with·elements·of·degree·3,4,4,5·in·8·variables<br>&quot;gbB148&quot;·--·gb·of·Bayesian·graph·ideal·#148<br>&quot;res39&quot;·--·res·of·a·generic·3·by·9·matrix·over·ZZ/101<br>&quot;resG25&quot;·--·res·of·the·coordinate·ring·of·Grassmannian(2,5)<br>&quot;yang-gb1&quot;·--·an·example·of·Yang-Hui·He·arising·in·string·theory<br>&quot;yang-subring&quot;·--·an·example·of·Yang-Hui·He<br></p>75 ··········<p>The·tests·available·are:·<br>&quot;deg2generic&quot;·--·gb·of·a·generic·ideal·of·codimension·2·and·degree·2<br>&quot;gb4by4comm&quot;·--·gb·of·the·ideal·of·generic·commuting·4·by·4·matrices·over·ZZ/101<br>&quot;gb3445&quot;·--·gb·of·an·ideal·with·elements·of·degree·3,4,4,5·in·8·variables<br>&quot;gbB148&quot;·--·gb·of·Bayesian·graph·ideal·#148<br>&quot;res39&quot;·--·res·of·a·generic·3·by·9·matrix·over·ZZ/101<br>&quot;resG25&quot;·--·res·of·the·coordinate·ring·of·Grassmannian(2,5)<br>&quot;yang-gb1&quot;·--·an·example·of·Yang-Hui·He·arising·in·string·theory<br>&quot;yang-subring&quot;·--·an·example·of·Yang-Hui·He<br></p>
76 ········</div>76 ········</div>
77 ········<table·class="examples">77 ········<table·class="examples">
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i1·:·runBenchmarks·&quot;res39&quot;79 <td>··············<pre><code·class="language-macaulay2">i1·:·runBenchmarks·&quot;res39&quot;
80 --·beginning·computation·Tue·Jun··3·02:45:38·-12·202580 --·beginning·computation·Tue·Jul··7·13:19:02·+14·2026
81 --·Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.1.140-1·(2025-05-22)·x86_64·GNU/Linux81 --·Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
82 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·2196.688··82 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·1996.250··
83 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.083 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.0
84 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.0989455·seconds</code></pre>84 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.177871·seconds</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ········</table>86 ········</table>
87 ······</div>87 ······</div>
88 ······<div·class="waystouse">88 ······<div·class="waystouse">
89 ········<h2>For·the·programmer</h2>89 ········<h2>For·the·programmer</h2>
90 ········<p>The·object·<a·href="_run__Benchmarks.html">runBenchmarks</a>·is·<span>a·<a·title="the·class·of·all·commands"·href="../../Macaulay2Doc/html/___Command.html">command</a></span>.</p>90 ········<p>The·object·<a·href="_run__Benchmarks.html">runBenchmarks</a>·is·<span>a·<a·title="the·class·of·all·commands"·href="../../Macaulay2Doc/html/___Command.html">command</a></span>.</p>
91 ······</div>91 ······</div>
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24 "gb3445"·--·gb·of·an·ideal·with·elements·of·degree·3,4,4,5·in·8·variables24 "gb3445"·--·gb·of·an·ideal·with·elements·of·degree·3,4,4,5·in·8·variables
25 "gbB148"·--·gb·of·Bayesian·graph·ideal·#14825 "gbB148"·--·gb·of·Bayesian·graph·ideal·#148
26 "res39"·--·res·of·a·generic·3·by·9·matrix·over·ZZ/10126 "res39"·--·res·of·a·generic·3·by·9·matrix·over·ZZ/101
27 "resG25"·--·res·of·the·coordinate·ring·of·Grassmannian(2,5)27 "resG25"·--·res·of·the·coordinate·ring·of·Grassmannian(2,5)
28 "yang-gb1"·--·an·example·of·Yang-Hui·He·arising·in·string·theory28 "yang-gb1"·--·an·example·of·Yang-Hui·He·arising·in·string·theory
29 "yang-subring"·--·an·example·of·Yang-Hui·He29 "yang-subring"·--·an·example·of·Yang-Hui·He
30 i1·:·runBenchmarks·"res39"30 i1·:·runBenchmarks·"res39"
31 --·beginning·computation·Tue·Jun··3·02:45:38·-12·202531 --·beginning·computation·Tue·Jul··7·13:19:02·+14·2026
32 --·Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian32 --·Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian
33 6.1.140-1·(2025-05-22)·x86_64·GNU/Linux33 6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
34 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·2196.68834 --·AMD·EPYC-Rome·Processor··AuthenticAMD··cpu·MHz·1996.250
35 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.035 --·Macaulay2·1.24.11,·compiled·with·gcc·14.2.0
36 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.0989455·seconds36 --·res39:·res·of·a·generic·3·by·9·matrix·over·ZZ/101:·.177871·seconds
37 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
38 The·object·_\x8r_\x8u_\x8n_\x8B_\x8e_\x8n_\x8c_\x8h_\x8m_\x8a_\x8r_\x8k_\x8s·is·a·_\x8c_\x8o_\x8m_\x8m_\x8a_\x8n_\x8d.38 The·object·_\x8r_\x8u_\x8n_\x8B_\x8e_\x8n_\x8c_\x8h_\x8m_\x8a_\x8r_\x8k_\x8s·is·a·_\x8c_\x8o_\x8m_\x8m_\x8a_\x8n_\x8d.
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./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
    
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82 <span><span·class="tt">OutputStyle</span>·(missing·documentation)<!--tag:·[bertiniParameterHomotopy,·OutputStyle]-->82 <span><span·class="tt">OutputStyle</span>·(missing·documentation)<!--tag:·[bertiniParameterHomotopy,·OutputStyle]-->
83 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;OutPoints&quot;</span>,·<span></span></span>··············</li>83 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;OutPoints&quot;</span>,·<span></span></span>··············</li>
84 ··············<li>84 ··············<li>
85 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomComplex</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>85 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomComplex</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>
86 ··············<li>86 ··············<li>
87 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomReal</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>87 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomReal</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>
88 ··············<li>88 ··············<li>
89 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-3816942-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>89 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-2789037-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>
90 ··············<li>90 ··············<li>
91 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>91 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>
92 ············</ul>92 ············</ul>
93 ··········</li>93 ··········</li>
94 ··········<li>94 ··········<li>
95 Outputs:············<ul>95 Outputs:············<ul>
96 ··············<li>96 ··············<li>
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html2text {}
    
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27 ············"OutPoints",27 ············"OutPoints",
28 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·=>·...,·default·value·{},·an·option·which·designates28 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·=>·...,·default·value·{},·an·option·which·designates
29 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real29 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real
30 ············number·or·random·complex·number30 ············number·or·random·complex·number
31 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8R_\x8e_\x8a_\x8l·=>·...,·default·value·{},·an·option·which·designates31 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8R_\x8e_\x8a_\x8l·=>·...,·default·value·{},·an·option·which·designates
32 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real32 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real
33 ············number·or·random·complex·number33 ············number·or·random·complex·number
34 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-3816942-0/0",·Option·to34 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-2789037-0/0",·Option·to
35 ············change·directory·for·file·storage.35 ············change·directory·for·file·storage.
36 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional36 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional
37 ············output37 ············output
38 ····*·Outputs:38 ····*·Outputs:
39 ··········o·S,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·whose·entries·are·lists·of·solutions·for·each39 ··········o·S,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·whose·entries·are·lists·of·solutions·for·each
40 ············target·system40 ············target·system
41 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
    
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88 ··············<li>88 ··············<li>
89 <span><span·class="tt">RandomComplex</span>·(missing·documentation)<!--tag:·[bertiniUserHomotopy,·RandomComplex]-->89 <span><span·class="tt">RandomComplex</span>·(missing·documentation)<!--tag:·[bertiniUserHomotopy,·RandomComplex]-->
90 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span></span></span>··············</li>90 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span></span></span>··············</li>
91 ··············<li>91 ··············<li>
92 <span><span·class="tt">RandomReal</span>·(missing·documentation)<!--tag:·[bertiniUserHomotopy,·RandomReal]-->92 <span><span·class="tt">RandomReal</span>·(missing·documentation)<!--tag:·[bertiniUserHomotopy,·RandomReal]-->
93 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span></span></span>··············</li>93 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span></span></span>··············</li>
94 ··············<li>94 ··············<li>
95 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-3816942-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>95 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-2789037-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>
96 ··············<li>96 ··············<li>
97 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>97 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>
98 ············</ul>98 ············</ul>
99 ··········</li>99 ··········</li>
100 ··········<li>100 ··········<li>
101 Outputs:············<ul>101 Outputs:············<ul>
102 ··············<li>102 ··············<li>
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22 ············value·{},22 ············value·{},
23 ··········o·HomVariableGroup·(missing·documentation)·=>·...,·default·value·{},23 ··········o·HomVariableGroup·(missing·documentation)·=>·...,·default·value·{},
24 ··········o·M2Precision·(missing·documentation)·=>·...,·default·value·53,24 ··········o·M2Precision·(missing·documentation)·=>·...,·default·value·53,
25 ··········o·OutputStyle·(missing·documentation)·=>·...,·default·value25 ··········o·OutputStyle·(missing·documentation)·=>·...,·default·value
26 ············"OutPoints",26 ············"OutPoints",
27 ··········o·RandomComplex·(missing·documentation)·=>·...,·default·value·{},27 ··········o·RandomComplex·(missing·documentation)·=>·...,·default·value·{},
28 ··········o·RandomReal·(missing·documentation)·=>·...,·default·value·{},28 ··········o·RandomReal·(missing·documentation)·=>·...,·default·value·{},
29 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-3816942-0/0",·Option·to29 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-2789037-0/0",·Option·to
30 ············change·directory·for·file·storage.30 ············change·directory·for·file·storage.
31 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional31 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional
32 ············output32 ············output
33 ····*·Outputs:33 ····*·Outputs:
34 ··········o·S0,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·of·solutions·to·the·target·system34 ··········o·S0,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·of·solutions·to·the·target·system
35 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*35 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
36 This·method·calls·Bertini·to·track·a·user-defined·homotopy.·The·user·needs·to36 This·method·calls·Bertini·to·track·a·user-defined·homotopy.·The·user·needs·to
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3.73 KB
./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
    
Offset 91, 15 lines modifiedOffset 91, 15 lines modified
91 <span><span·class="tt">OutputStyle</span>·(missing·documentation)<!--tag:·[bertiniZeroDimSolve,·OutputStyle]-->91 <span><span·class="tt">OutputStyle</span>·(missing·documentation)<!--tag:·[bertiniZeroDimSolve,·OutputStyle]-->
92 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;OutPoints&quot;</span>,·<span></span></span>··············</li>92 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;OutPoints&quot;</span>,·<span></span></span>··············</li>
93 ··············<li>93 ··············<li>
94 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomComplex</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>94 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomComplex</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>
95 ··············<li>95 ··············<li>
96 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomReal</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>96 <span><a·title="an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number"·href="___Bertini_spinput_spfile_spdeclarations_co_sprandom_spnumbers.html">RandomReal</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·{}</span>,·<span>an·option·which·designates·symbols/strings/variables·that·will·be·set·to·be·a·random·real·number·or·random·complex·number</span></span>··············</li>
97 ··············<li>97 ··············<li>
98 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-3816942-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>98 <span><a·title="Option·to·change·directory·for·file·storage."·href="___Top__Directory.html">TopDirectory</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·&quot;/tmp/M2-2789037-0/0&quot;</span>,·<span>Option·to·change·directory·for·file·storage.</span></span>··············</li>
99 ··············<li>99 ··············<li>
100 <span><span·class="tt">UseRegeneration</span>·(missing·documentation)<!--tag:·[bertiniZeroDimSolve,·UseRegeneration]-->100 <span><span·class="tt">UseRegeneration</span>·(missing·documentation)<!--tag:·[bertiniZeroDimSolve,·UseRegeneration]-->
101 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·-1</span>,·<span></span></span>··············</li>101 <span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·-1</span>,·<span></span></span>··············</li>
102 ··············<li>102 ··············<li>
103 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>103 <span><a·title="Option·to·silence·additional·output"·href="_bertini__Track__Homotopy_lp..._cm__Verbose_eq_gt..._rp.html">Verbose</a><span·class="tt">·=>·</span><span·class="tt">...</span>,·<span>default·value·false</span>,·<span>Option·to·silence·additional·output</span></span>··············</li>
104 ············</ul>104 ············</ul>
105 ··········</li>105 ··········</li>
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33 ············"OutPoints",33 ············"OutPoints",
34 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·=>·...,·default·value·{},·an·option·which·designates34 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·=>·...,·default·value·{},·an·option·which·designates
35 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real35 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real
36 ············number·or·random·complex·number36 ············number·or·random·complex·number
37 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8R_\x8e_\x8a_\x8l·=>·...,·default·value·{},·an·option·which·designates37 ··········o·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8R_\x8e_\x8a_\x8l·=>·...,·default·value·{},·an·option·which·designates
38 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real38 ············symbols/strings/variables·that·will·be·set·to·be·a·random·real
39 ············number·or·random·complex·number39 ············number·or·random·complex·number
40 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-3816942-0/0",·Option·to40 ··········o·_\x8T_\x8o_\x8p_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·=>·...,·default·value·"/tmp/M2-2789037-0/0",·Option·to
41 ············change·directory·for·file·storage.41 ············change·directory·for·file·storage.
42 ··········o·UseRegeneration·(missing·documentation)·=>·...,·default·value·-1,42 ··········o·UseRegeneration·(missing·documentation)·=>·...,·default·value·-1,
43 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional43 ··········o·_\x8V_\x8e_\x8r_\x8b_\x8o_\x8s_\x8e·=>·...,·default·value·false,·Option·to·silence·additional
44 ············output44 ············output
45 ····*·Outputs:45 ····*·Outputs:
46 ··········o·S,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·of·points·that·are·contained·in·the·variety·of·F46 ··········o·S,·a·_\x8l_\x8i_\x8s_\x8t,·a·list·of·points·that·are·contained·in·the·variety·of·F
47 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*47 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
⊟
759 B
./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=317 #:len=31
8 aW52ZXJzZVJpbmdBY3RvcnMoLi4uLFN1Yj0+Li4uKQ==8 aW52ZXJzZVJpbmdBY3RvcnMoLi4uLFN1Yj0+Li4uKQ==
9 #:len=2649 #:len=264
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjg0OCwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjg0OCwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW52ZXJzZVJpbmdBY3RvcnMsU3ViXSwiaW52ZXJz11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW52ZXJzZVJpbmdBY3RvcnMsU3ViXSwiaW52ZXJz
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639 B
./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out
    
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76 i8·:·A·=·action(RI,S7)76 i8·:·A·=·action(RI,S7)
  
77 o8·=·ChainComplex·with·15·actors77 o8·=·ChainComplex·with·15·actors
  
78 o8·:·ActionOnComplex78 o8·:·ActionOnComplex
  
79 i9·:·elapsedTime·c·=·character·A79 i9·:·elapsedTime·c·=·character·A
80 ·--·.345859s·elapsed80 ·--·.806165s·elapsed
  
81 o9·=·Character·over·R81 o9·=·Character·over·R
82 ······82 ······
83 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|83 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|
84 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|84 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|
85 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|85 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|
86 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|86 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|
⊟
587 B
./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
    
Offset 100, 15 lines modifiedOffset 100, 15 lines modified
100 i6·:·A=action(RI,S6)100 i6·:·A=action(RI,S6)
  
101 o6·=·ChainComplex·with·11·actors101 o6·=·ChainComplex·with·11·actors
  
102 o6·:·ActionOnComplex102 o6·:·ActionOnComplex
  
103 i7·:·elapsedTime·c=character·A103 i7·:·elapsedTime·c=character·A
104 ·--·.422058s·elapsed104 ·--·.56307s·elapsed
  
105 o7·=·Character·over·R105 o7·=·Character·over·R
106 ······106 ······
107 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|107 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|
108 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|108 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|
109 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|109 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|
110 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|110 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|
⊟
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./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
    
Offset 187, 27 lines modifiedOffset 187, 27 lines modified
187 i19·:·A2·=·action(RI2,G,Sub=>false)187 i19·:·A2·=·action(RI2,G,Sub=>false)
  
188 o19·=·ChainComplex·with·6·actors188 o19·=·ChainComplex·with·6·actors
  
189 o19·:·ActionOnComplex189 o19·:·ActionOnComplex
  
190 i20·:·elapsedTime·a1·=·character·A1190 i20·:·elapsedTime·a1·=·character·A1
191 ·--·.714376s·elapsed191 ·--·1.20761s·elapsed
  
192 o20·=·Character·over·R192 o20·=·Character·over·R
193 ·······193 ·······
194 ······(0,·{0})·=>·|·1·1·1·1·1·1·|194 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
195 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|195 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
196 ······(2,·{11})·=>·|·1·1·1·1·1·1·|196 ······(2,·{11})·=>·|·1·1·1·1·1·1·|
197 ······(2,·{13})·=>·|·1·1·1·1·1·1·|197 ······(2,·{13})·=>·|·1·1·1·1·1·1·|
  
198 o20·:·Character198 o20·:·Character
  
199 i21·:·elapsedTime·a2·=·character·A2199 i21·:·elapsedTime·a2·=·character·A2
200 ·--·28.7475s·elapsed200 ·--·36.9565s·elapsed
  
201 o21·=·Character·over·R201 o21·=·Character·over·R
202 ·······202 ·······
203 ······(0,·{0})·=>·|·1·1·1·1·1·1·|203 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
204 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|204 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|
205 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|205 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
206 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|206 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
Offset 297, 15 lines modifiedOffset 297, 15 lines modified
297 i31·:·B·=·action(M,G,Sub=>false)297 i31·:·B·=·action(M,G,Sub=>false)
  
298 o31·=·Module·with·6·actors298 o31·=·Module·with·6·actors
  
299 o31·:·ActionOnGradedModule299 o31·:·ActionOnGradedModule
  
300 i32·:·elapsedTime·b·=·character(B,21)300 i32·:·elapsedTime·b·=·character(B,21)
301 ·--·13.0288s·elapsed301 ·--·14.3239s·elapsed
  
302 o32·=·Character·over·R302 o32·=·Character·over·R
303 ·······303 ·······
304 ······(0,·{21})·=>·|·1·1·1·1·1·1·|304 ······(0,·{21})·=>·|·1·1·1·1·1·1·|
  
305 o32·:·Character305 o32·:·Character
  
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1.15 KB
./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
    
Offset 138, 15 lines modifiedOffset 138, 15 lines modified
  
138 o8·=·ChainComplex·with·15·actors138 o8·=·ChainComplex·with·15·actors
  
139 o8·:·ActionOnComplex</code></pre>139 o8·:·ActionOnComplex</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·c·=·character·A142 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·c·=·character·A
143 ·--·.345859s·elapsed143 ·--·.806165s·elapsed
  
144 o9·=·Character·over·R144 o9·=·Character·over·R
145 ······145 ······
146 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|146 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|
147 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|147 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|
148 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|148 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|
149 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|149 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|
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html2text {}
    
Offset 91, 15 lines modifiedOffset 91, 15 lines modified
91 o7·:·List91 o7·:·List
92 i8·:·A·=·action(RI,S7)92 i8·:·A·=·action(RI,S7)
  
93 o8·=·ChainComplex·with·15·actors93 o8·=·ChainComplex·with·15·actors
  
94 o8·:·ActionOnComplex94 o8·:·ActionOnComplex
95 i9·:·elapsedTime·c·=·character·A95 i9·:·elapsedTime·c·=·character·A
96 ·--·.345859s·elapsed96 ·--·.806165s·elapsed
  
97 o9·=·Character·over·R97 o9·=·Character·over·R
  
98 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|98 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·|
99 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|99 ·····(1,·{2})·=>·|·0·-1·1·-1·0·0·0·-1·2·0·2·2·2·6·14·|
100 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|100 ·····(2,·{3})·=>·|·0·1·0·0·-1·1·-1·-1·-1·-1·-1·1·-1·5·35·|
101 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|101 ·····(3,·{4})·=>·|·0·-1·0·0·1·1·1·-1·-1·1·-1·-1·-1·-5·35·|
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1.05 KB
./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
    
Offset 160, 15 lines modifiedOffset 160, 15 lines modified
  
160 o6·=·ChainComplex·with·11·actors160 o6·=·ChainComplex·with·11·actors
  
161 o6·:·ActionOnComplex</code></pre>161 o6·:·ActionOnComplex</code></pre>
162 </td>··········</tr>162 </td>··········</tr>
163 ··········<tr>163 ··········<tr>
164 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·c=character·A164 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·c=character·A
165 ·--·.422058s·elapsed165 ·--·.56307s·elapsed
  
166 o7·=·Character·over·R166 o7·=·Character·over·R
167 ······167 ······
168 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|168 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|
169 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|169 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|
170 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|170 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|
171 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|171 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|
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420 B
html2text {}
    
Offset 113, 15 lines modifiedOffset 113, 15 lines modified
113 o5·:·List113 o5·:·List
114 i6·:·A=action(RI,S6)114 i6·:·A=action(RI,S6)
  
115 o6·=·ChainComplex·with·11·actors115 o6·=·ChainComplex·with·11·actors
  
116 o6·:·ActionOnComplex116 o6·:·ActionOnComplex
117 i7·:·elapsedTime·c=character·A117 i7·:·elapsedTime·c=character·A
118 ·--·.422058s·elapsed118 ·--·.56307s·elapsed
  
119 o7·=·Character·over·R119 o7·=·Character·over·R
  
120 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|120 ·····(0,·{0})·=>·|·1·1·1·1·1·1·1·1·1·1·1·|
121 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|121 ·····(1,·{5})·=>·|·0·1·0·2·0·1·3·0·2·4·6·|
122 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|122 ·····(1,·{7})·=>·|·0·0·0·0·0·1·3·0·4·16·60·|
123 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|123 ·····(1,·{9})·=>·|·0·0·0·0·2·2·2·0·4·8·20·|
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2.45 KB
./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
    
Offset 264, 28 lines modifiedOffset 264, 28 lines modified
  
264 o19·=·ChainComplex·with·6·actors264 o19·=·ChainComplex·with·6·actors
  
265 o19·:·ActionOnComplex</code></pre>265 o19·:·ActionOnComplex</code></pre>
266 </td>··········</tr>266 </td>··········</tr>
267 ··········<tr>267 ··········<tr>
268 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·a1·=·character·A1268 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·a1·=·character·A1
269 ·--·.714376s·elapsed269 ·--·1.20761s·elapsed
  
270 o20·=·Character·over·R270 o20·=·Character·over·R
271 ·······271 ·······
272 ······(0,·{0})·=>·|·1·1·1·1·1·1·|272 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
273 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|273 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
274 ······(2,·{11})·=>·|·1·1·1·1·1·1·|274 ······(2,·{11})·=>·|·1·1·1·1·1·1·|
275 ······(2,·{13})·=>·|·1·1·1·1·1·1·|275 ······(2,·{13})·=>·|·1·1·1·1·1·1·|
  
276 o20·:·Character</code></pre>276 o20·:·Character</code></pre>
277 </td>··········</tr>277 </td>··········</tr>
278 ··········<tr>278 ··········<tr>
279 <td>··············<pre><code·class="language-macaulay2">i21·:·elapsedTime·a2·=·character·A2279 <td>··············<pre><code·class="language-macaulay2">i21·:·elapsedTime·a2·=·character·A2
280 ·--·28.7475s·elapsed280 ·--·36.9565s·elapsed
  
281 o21·=·Character·over·R281 o21·=·Character·over·R
282 ·······282 ·······
283 ······(0,·{0})·=>·|·1·1·1·1·1·1·|283 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
284 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|284 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|
285 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|285 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
286 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|286 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
Offset 397, 15 lines modifiedOffset 397, 15 lines modified
  
397 o31·=·Module·with·6·actors397 o31·=·Module·with·6·actors
  
398 o31·:·ActionOnGradedModule</code></pre>398 o31·:·ActionOnGradedModule</code></pre>
399 </td>··········</tr>399 </td>··········</tr>
400 ··········<tr>400 ··········<tr>
401 <td>··············<pre><code·class="language-macaulay2">i32·:·elapsedTime·b·=·character(B,21)401 <td>··············<pre><code·class="language-macaulay2">i32·:·elapsedTime·b·=·character(B,21)
402 ·--·13.0288s·elapsed402 ·--·14.3239s·elapsed
  
403 o32·=·Character·over·R403 o32·=·Character·over·R
404 ·······404 ·······
405 ······(0,·{21})·=>·|·1·1·1·1·1·1·|405 ······(0,·{21})·=>·|·1·1·1·1·1·1·|
  
406 o32·:·Character</code></pre>406 o32·:·Character</code></pre>
407 </td>··········</tr>407 </td>··········</tr>
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Offset 192, 26 lines modifiedOffset 192, 26 lines modified
192 o18·:·ActionOnComplex192 o18·:·ActionOnComplex
193 i19·:·A2·=·action(RI2,G,Sub=>false)193 i19·:·A2·=·action(RI2,G,Sub=>false)
  
194 o19·=·ChainComplex·with·6·actors194 o19·=·ChainComplex·with·6·actors
  
195 o19·:·ActionOnComplex195 o19·:·ActionOnComplex
196 i20·:·elapsedTime·a1·=·character·A1196 i20·:·elapsedTime·a1·=·character·A1
197 ·--·.714376s·elapsed197 ·--·1.20761s·elapsed
  
198 o20·=·Character·over·R198 o20·=·Character·over·R
  
199 ······(0,·{0})·=>·|·1·1·1·1·1·1·|199 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
200 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|200 ······(1,·{8})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
201 ······(2,·{11})·=>·|·1·1·1·1·1·1·|201 ······(2,·{11})·=>·|·1·1·1·1·1·1·|
202 ······(2,·{13})·=>·|·1·1·1·1·1·1·|202 ······(2,·{13})·=>·|·1·1·1·1·1·1·|
  
203 o20·:·Character203 o20·:·Character
204 i21·:·elapsedTime·a2·=·character·A2204 i21·:·elapsedTime·a2·=·character·A2
205 ·--·28.7475s·elapsed205 ·--·36.9565s·elapsed
  
206 o21·=·Character·over·R206 o21·=·Character·over·R
  
207 ······(0,·{0})·=>·|·1·1·1·1·1·1·|207 ······(0,·{0})·=>·|·1·1·1·1·1·1·|
208 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|208 ······(1,·{16})·=>·|·6·2·0·0·-1·-1·|
209 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|209 ······(2,·{19})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
210 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|210 ······(2,·{21})·=>·|·3·-1·0·1·a4+a2+a·-a4-a2-a-1·|
Offset 308, 15 lines modifiedOffset 308, 15 lines modified
308 i30·:·M·=·Is2·/·I2;308 i30·:·M·=·Is2·/·I2;
309 i31·:·B·=·action(M,G,Sub=>false)309 i31·:·B·=·action(M,G,Sub=>false)
  
310 o31·=·Module·with·6·actors310 o31·=·Module·with·6·actors
  
311 o31·:·ActionOnGradedModule311 o31·:·ActionOnGradedModule
312 i32·:·elapsedTime·b·=·character(B,21)312 i32·:·elapsedTime·b·=·character(B,21)
313 ·--·13.0288s·elapsed313 ·--·14.3239s·elapsed
  
314 o32·=·Character·over·R314 o32·=·Character·over·R
  
315 ······(0,·{21})·=>·|·1·1·1·1·1·1·|315 ······(0,·{21})·=>·|·1·1·1·1·1·1·|
  
316 o32·:·Character316 o32·:·Character
317 i33·:·b/T317 i33·:·b/T
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8 YmVp8 YmVp
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11 c2NyaXB0aW9uID0+ICgiYmVpIGlzIGEgc3lub255bSBmb3IgIixUT3tuZXcgRG9jdW1lbnRUYWcg11 c2NyaXB0aW9uID0+ICgiYmVpIGlzIGEgc3lub255bSBmb3IgIixUT3tuZXcgRG9jdW1lbnRUYWcg
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724 B
./usr/share/doc/Macaulay2/Binomials/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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7 #:len=157 #:len=15
8 Ymlub21pYWxJc1ByaW1l8 Ymlub21pYWxJc1ByaW1l
9 #:len=13129 #:len=1312
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGVzdCBmb3IgcHJpbWVuZXNzIG9mIGEg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGVzdCBmb3IgcHJpbWVuZXNzIG9mIGEg
11 Ymlub21pYWwgaWRlYWwiLCAibGluZW51bSIgPT4gMTU4MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJ11 Ymlub21pYWwgaWRlYWwiLCAibGluZW51bSIgPT4gMTU4MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJ
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741 B
./usr/share/doc/Macaulay2/BoijSoederberg/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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230 ··········0:·1·.·.·.·.230 ··········0:·1·.·.·.·.
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233 o22·:·BettiTally233 o22·:·BettiTally
  
234 i23·:·time·j=bruns·F.dd_3;234 i23·:·time·j=bruns·F.dd_3;
235 ·--·used·0.182853s·(cpu);·0.149762s·(thread);·0s·(gc)235 ·--·used·0.222898s·(cpu);·0.178233s·(thread);·0s·(gc)
  
236 o23·:·Ideal·of·S236 o23·:·Ideal·of·S
  
237 i24·:·betti·res·j237 i24·:·betti·res·j
  
238 ·············0·1·2·3·4238 ·············0·1·2·3·4
239 o24·=·total:·1·3·6·5·1239 o24·=·total:·1·3·6·5·1
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336 ··········1:·.·4·2·.·.336 ··········1:·.·4·2·.·.
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338 o22·:·BettiTally</code></pre>338 o22·:·BettiTally</code></pre>
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341 <td>··············<pre><code·class="language-macaulay2">i23·:·time·j=bruns·F.dd_3;341 <td>··············<pre><code·class="language-macaulay2">i23·:·time·j=bruns·F.dd_3;
342 ·--·used·0.182853s·(cpu);·0.149762s·(thread);·0s·(gc)342 ·--·used·0.222898s·(cpu);·0.178233s·(thread);·0s·(gc)
  
343 o23·:·Ideal·of·S</code></pre>343 o23·:·Ideal·of·S</code></pre>
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345 ··········<tr>345 ··········<tr>
346 <td>··············<pre><code·class="language-macaulay2">i24·:·betti·res·j346 <td>··············<pre><code·class="language-macaulay2">i24·:·betti·res·j
  
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235 o22·:·BettiTally235 o22·:·BettiTally
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238 o23·:·Ideal·of·S238 o23·:·Ideal·of·S
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240 ·············0·1·2·3·4240 ·············0·1·2·3·4
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./usr/share/doc/Macaulay2/CellularResolutions/example-output/_boundary.out
    
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34 i12·:·f·=·(cells(2,C))#0;34 i12·:·f·=·(cells(2,C))#0;
  
35 i13·:·boundary(f)35 i13·:·boundary(f)
  
36 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label36 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label
37 ······-----------------------------------------------------------------------37 ······-----------------------------------------------------------------------
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41 o13·:·List41 o13·:·List
  
42 i14·:·42 i14·:·
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./usr/share/doc/Macaulay2/CellularResolutions/example-output/_relabel__Cell__Complex.out
    
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22 i11·:·T·=·new·HashTable·from·{v0·=>·a^2*b,·v1·=>·b*c^2,·v2·=>·b^2,·v3·=>·a*c};22 i11·:·T·=·new·HashTable·from·{v0·=>·a^2*b,·v1·=>·b*c^2,·v2·=>·b^2,·v3·=>·a*c};
  
23 i12·:·relabeledC·=·relabelCellComplex(C,T);23 i12·:·relabeledC·=·relabelCellComplex(C,T);
  
24 i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)24 i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)
  
25 ········2········2······225 ·············2······2···2
26 o13·=·{b·,·a*c,·a·b,·b*c·}26 o13·=·{a*c,·a·b,·b*c·,·b·}
  
27 o13·:·List27 o13·:·List
  
28 i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)28 i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)
  
29 ········2·2···2··········2···2·2···2···229 ········2···2···2······2·2·······2·····2
30 o14·=·{a·b·,·a·b*c,·a*b*c·,·b·c·,·a·b*c·}30 o14·=·{a·b*c·,·a·b*c,·a·b·,·a*b*c·,·a*b·c}
  
31 o14·:·List31 o14·:·List
  
32 i15·:·32 i15·:·
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./usr/share/doc/Macaulay2/CellularResolutions/example-output/_scarf__Complex.out
    
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19 ····················19 ····················
20 ·····-1·····0······120 ·····-1·····0······1
  
21 o4·:·Complex21 o4·:·Complex
  
22 i5·:·cells(1,C)/cellLabel22 i5·:·cells(1,C)/cellLabel
  
23 ·········2····4·5···2······5·423 ·······2······4·5···5·4·····2
24 o5·=·{x*y·z,·x·y·,·x·y*z,·x·y·}24 o5·=·{x·y*z,·x·y·,·x·y·,·x*y·z}
  
25 o5·:·List25 o5·:·List
  
26 i6·:·cells(2,C)/cellLabel26 i6·:·cells(2,C)/cellLabel
  
27 o6·=·{}27 o6·=·{}
  
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./usr/share/doc/Macaulay2/CellularResolutions/html/_boundary.html
    
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120 <td>··············<pre><code·class="language-macaulay2">i12·:·f·=·(cells(2,C))#0;</code></pre>120 <td>··············<pre><code·class="language-macaulay2">i12·:·f·=·(cells(2,C))#0;</code></pre>
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122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i13·:·boundary(f)123 <td>··············<pre><code·class="language-macaulay2">i13·:·boundary(f)
  
124 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label124 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label
125 ······-----------------------------------------------------------------------125 ······-----------------------------------------------------------------------
126 ······1,·1),·(Cell·of·dimension·1·with·label·1,·-1),·(Cell·of·dimension·1126 ······1,·-1),·(Cell·of·dimension·1·with·label·1,·-1),·(Cell·of·dimension·1
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129 o13·:·List</code></pre>129 o13·:·List</code></pre>
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131 ········</table>131 ········</table>
132 ······</div>132 ······</div>
133 ······<div>133 ······<div>
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43 i10·:·P·=·convexHull·matrix·{{1,1,-1,-1},{1,-1,1,-1}};43 i10·:·P·=·convexHull·matrix·{{1,1,-1,-1},{1,-1,1,-1}};
44 i11·:·C·=·cellComplex(R,P);44 i11·:·C·=·cellComplex(R,P);
45 i12·:·f·=·(cells(2,C))#0;45 i12·:·f·=·(cells(2,C))#0;
46 i13·:·boundary(f)46 i13·:·boundary(f)
  
47 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label47 o13·=·{(Cell·of·dimension·1·with·label·1,·1),·(Cell·of·dimension·1·with·label
48 ······-----------------------------------------------------------------------48 ······-----------------------------------------------------------------------
49 ······1,·1),·(Cell·of·dimension·1·with·label·1,·-1),·(Cell·of·dimension·149 ······1,·-1),·(Cell·of·dimension·1·with·label·1,·-1),·(Cell·of·dimension·1
50 ······-----------------------------------------------------------------------50 ······-----------------------------------------------------------------------
51 ······with·label·1,·-1)}51 ······with·label·1,·1)}
  
52 o13·:·List52 o13·:·List
53 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*53 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
54 ····*·_\x8b_\x8o_\x8u_\x8n_\x8d_\x8a_\x8r_\x8y_\x8C_\x8e_\x8l_\x8l_\x8s_\x8(_\x8C_\x8e_\x8l_\x8l_\x8)·--·returns·the·boundary·cells·of·the·given·cell54 ····*·_\x8b_\x8o_\x8u_\x8n_\x8d_\x8a_\x8r_\x8y_\x8C_\x8e_\x8l_\x8l_\x8s_\x8(_\x8C_\x8e_\x8l_\x8l_\x8)·--·returns·the·boundary·cells·of·the·given·cell
55 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·b\x8bo\x8ou\x8un\x8nd\x8da\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*55 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·b\x8bo\x8ou\x8un\x8nd\x8da\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*
56 ····*·boundary(Cell)56 ····*·boundary(Cell)
57 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*57 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/CellularResolutions/html/_relabel__Cell__Complex.html
    
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114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i12·:·relabeledC·=·relabelCellComplex(C,T);</code></pre>116 <td>··············<pre><code·class="language-macaulay2">i12·:·relabeledC·=·relabelCellComplex(C,T);</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)119 <td>··············<pre><code·class="language-macaulay2">i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)
  
120 ········2········2······2120 ·············2······2···2
121 o13·=·{b·,·a*c,·a·b,·b*c·}121 o13·=·{a*c,·a·b,·b*c·,·b·}
  
122 o13·:·List</code></pre>122 o13·:·List</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)125 <td>··············<pre><code·class="language-macaulay2">i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)
  
126 ········2·2···2··········2···2·2···2···2126 ········2···2···2······2·2·······2·····2
127 o14·=·{a·b·,·a·b*c,·a*b*c·,·b·c·,·a·b*c·}127 o14·=·{a·b*c·,·a·b*c,·a·b·,·a*b*c·,·a*b·c}
  
128 o14·:·List</code></pre>128 o14·:·List</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ········</table>130 ········</table>
131 ······</div>131 ······</div>
132 ······<div>132 ······<div>
133 ········<h2>See·also</h2>133 ········<h2>See·also</h2>
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32 i8·:·v1·=·verts#1;32 i8·:·v1·=·verts#1;
33 i9·:·v2·=·verts#2;33 i9·:·v2·=·verts#2;
34 i10·:·v3·=·verts#3;34 i10·:·v3·=·verts#3;
35 i11·:·T·=·new·HashTable·from·{v0·=>·a^2*b,·v1·=>·b*c^2,·v2·=>·b^2,·v3·=>·a*c};35 i11·:·T·=·new·HashTable·from·{v0·=>·a^2*b,·v1·=>·b*c^2,·v2·=>·b^2,·v3·=>·a*c};
36 i12·:·relabeledC·=·relabelCellComplex(C,T);36 i12·:·relabeledC·=·relabelCellComplex(C,T);
37 i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)37 i13·:·for·c·in·cells(0,relabeledC)·list·cellLabel(c)
  
38 ········2········2······238 ·············2······2···2
39 o13·=·{b·,·a*c,·a·b,·b*c·}39 o13·=·{a*c,·a·b,·b*c·,·b·}
  
40 o13·:·List40 o13·:·List
41 i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)41 i14·:·for·c·in·cells(1,relabeledC)·list·cellLabel(c)
  
42 ········2·2···2··········2···2·2···2···242 ········2···2···2······2·2·······2·····2
43 o14·=·{a·b·,·a·b*c,·a*b*c·,·b·c·,·a·b*c·}43 o14·=·{a·b*c·,·a·b*c,·a·b·,·a*b*c·,·a*b·c}
  
44 o14·:·List44 o14·:·List
45 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*45 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
46 ····*·_\x8c_\x8e_\x8l_\x8l_\x8L_\x8a_\x8b_\x8e_\x8l·--·return·the·label·of·a·cell46 ····*·_\x8c_\x8e_\x8l_\x8l_\x8L_\x8a_\x8b_\x8e_\x8l·--·return·the·label·of·a·cell
47 ····*·_\x8R_\x8i_\x8n_\x8g_\x8M_\x8a_\x8p_\x8·_\x8*_\x8*_\x8·_\x8C_\x8e_\x8l_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·--·tensors·labels·via·a·ring·map47 ····*·_\x8R_\x8i_\x8n_\x8g_\x8M_\x8a_\x8p_\x8·_\x8*_\x8*_\x8·_\x8C_\x8e_\x8l_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·--·tensors·labels·via·a·ring·map
48 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8re\x8el\x8la\x8ab\x8be\x8el\x8lC\x8Ce\x8el\x8ll\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*48 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8re\x8el\x8la\x8ab\x8be\x8el\x8lC\x8Ce\x8el\x8ll\x8lC\x8Co\x8om\x8mp\x8pl\x8le\x8ex\x8x:\x8:·*\x8**\x8**\x8**\x8**\x8*
49 ····*·relabelCellComplex(CellComplex,HashTable)49 ····*·relabelCellComplex(CellComplex,HashTable)
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./usr/share/doc/Macaulay2/CellularResolutions/html/_scarf__Complex.html
    
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95 ·····-1·····0······195 ·····-1·····0······1
  
96 o4·:·Complex</code></pre>96 o4·:·Complex</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·cells(1,C)/cellLabel99 <td>··············<pre><code·class="language-macaulay2">i5·:·cells(1,C)/cellLabel
  
100 ·········2····4·5···2······5·4100 ·······2······4·5···5·4·····2
101 o5·=·{x*y·z,·x·y·,·x·y*z,·x·y·}101 o5·=·{x·y*z,·x·y·,·x·y·,·x*y·z}
  
102 o5·:·List</code></pre>102 o5·:·List</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i6·:·cells(2,C)/cellLabel105 <td>··············<pre><code·class="language-macaulay2">i6·:·cells(2,C)/cellLabel
  
106 o6·=·{}106 o6·=·{}
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30 o4·=·S··<--·S··<--·S30 o4·=·S··<--·S··<--·S
  
31 ·····-1·····0······131 ·····-1·····0······1
  
32 o4·:·Complex32 o4·:·Complex
33 i5·:·cells(1,C)/cellLabel33 i5·:·cells(1,C)/cellLabel
  
34 ·········2····4·5···2······5·434 ·······2······4·5···5·4·····2
35 o5·=·{x*y·z,·x·y·,·x·y*z,·x·y·}35 o5·=·{x·y*z,·x·y·,·x·y·,·x*y·z}
  
36 o5·:·List36 o5·:·List
37 i6·:·cells(2,C)/cellLabel37 i6·:·cells(2,C)/cellLabel
  
38 o6·=·{}38 o6·=·{}
  
39 o6·:·List39 o6·:·List
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=307 #:len=30
8 SG9tKENoYWluQ29tcGxleCxDaGFpbkNvbXBsZXgp8 SG9tKENoYWluQ29tcGxleCxDaGFpbkNvbXBsZXgp
9 #:len=12569 #:len=1256
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlIHRoZSBob21vbW9ycGhpc20g10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlIHRoZSBob21vbW9ycGhpc20g
11 Y29tcGxleCBvZiBhIHBhaXIgb2YgY2hhaW4gY29tcGxleGVzLiIsICJsaW5lbnVtIiA9PiA4MDcs11 Y29tcGxleCBvZiBhIHBhaXIgb2YgY2hhaW4gY29tcGxleGVzLiIsICJsaW5lbnVtIiA9PiA4MDcs
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./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_minimize.out
    
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63 o11·:·ChainComplex63 o11·:·ChainComplex
  
64 i12·:·isMinimalChainComplex·E64 i12·:·isMinimalChainComplex·E
  
65 o12·=·false65 o12·=·false
  
66 i13·:·time·m·=·minimize·(E[1]);66 i13·:·time·m·=·minimize·(E[1]);
67 ·--·used·0.218835s·(cpu);·0.177739s·(thread);·0s·(gc)67 ·--·used·0.28725s·(cpu);·0.226629s·(thread);·0s·(gc)
  
68 i14·:·isQuasiIsomorphism·m68 i14·:·isQuasiIsomorphism·m
  
69 o14·=·true69 o14·=·true
  
70 i15·:·E[1]·==·source·m70 i15·:·E[1]·==·source·m
  
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./usr/share/doc/Macaulay2/ChainComplexExtras/example-output/_resolution__Of__Chain__Complex.out
    
Offset 27, 18 lines modifiedOffset 27, 18 lines modified
27 i5·:·C·=·res(R^1/(ideal·vars·R))**(R^1/(ideal·vars·R)^5);27 i5·:·C·=·res(R^1/(ideal·vars·R))**(R^1/(ideal·vars·R)^5);
  
28 i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);28 i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);
  
29 i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-1),mods_i,substitute(matrix·C.dd_i,S));29 i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-1),mods_i,substitute(matrix·C.dd_i,S));
  
30 i8·:·time·m·=·resolutionOfChainComplex·C;30 i8·:·time·m·=·resolutionOfChainComplex·C;
31 ·--·used·0.131551s·(cpu);·0.0980714s·(thread);·0s·(gc)31 ·--·used·0.159853s·(cpu);·0.108221s·(thread);·0s·(gc)
  
32 i9·:·time·n·=·cartanEilenbergResolution·C;32 i9·:·time·n·=·cartanEilenbergResolution·C;
33 ·--·used·0.102411s·(cpu);·0.103561s·(thread);·0s·(gc)33 ·--·used·0.122496s·(cpu);·0.125744s·(thread);·0s·(gc)
  
34 i10·:·betti·source·m34 i10·:·betti·source·m
  
35 ·············0··1··2···3···4···5···6···735 ·············0··1··2···3···4···5···6···7
36 o10·=·total:·1·19·80·181·312·484·447·15636 o10·=·total:·1·19·80·181·312·484·447·156
37 ··········0:·1··3··3···1···.···.···.···.37 ··········0:·1··3··3···1···.···.···.···.
38 ··········1:·.··.··1···3···3···.···.···.38 ··········1:·.··.··1···3···3···.···.···.
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155 ········</table>155 ········</table>
156 ········<div>156 ········<div>
157 ··········<p>Now·we·minimize·the·result.·The·free·summand·we·added·to·the·end·maps·to·zero,·and·thus·is·part·of·the·minimization.</p>157 ··········<p>Now·we·minimize·the·result.·The·free·summand·we·added·to·the·end·maps·to·zero,·and·thus·is·part·of·the·minimization.</p>
158 ········</div>158 ········</div>
159 ········<table·class="examples">159 ········<table·class="examples">
160 ··········<tr>160 ··········<tr>
161 <td>··············<pre><code·class="language-macaulay2">i13·:·time·m·=·minimize·(E[1]);161 <td>··············<pre><code·class="language-macaulay2">i13·:·time·m·=·minimize·(E[1]);
162 ·--·used·0.218835s·(cpu);·0.177739s·(thread);·0s·(gc)</code></pre>162 ·--·used·0.28725s·(cpu);·0.226629s·(thread);·0s·(gc)</code></pre>
163 </td>··········</tr>163 </td>··········</tr>
164 ··········<tr>164 ··········<tr>
165 <td>··············<pre><code·class="language-macaulay2">i14·:·isQuasiIsomorphism·m165 <td>··············<pre><code·class="language-macaulay2">i14·:·isQuasiIsomorphism·m
  
166 o14·=·true</code></pre>166 o14·=·true</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
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80 o11·:·ChainComplex80 o11·:·ChainComplex
81 i12·:·isMinimalChainComplex·E81 i12·:·isMinimalChainComplex·E
  
82 o12·=·false82 o12·=·false
83 Now·we·minimize·the·result.·The·free·summand·we·added·to·the·end·maps·to·zero,83 Now·we·minimize·the·result.·The·free·summand·we·added·to·the·end·maps·to·zero,
84 and·thus·is·part·of·the·minimization.84 and·thus·is·part·of·the·minimization.
85 i13·:·time·m·=·minimize·(E[1]);85 i13·:·time·m·=·minimize·(E[1]);
86 ·--·used·0.218835s·(cpu);·0.177739s·(thread);·0s·(gc)86 ·--·used·0.28725s·(cpu);·0.226629s·(thread);·0s·(gc)
87 i14·:·isQuasiIsomorphism·m87 i14·:·isQuasiIsomorphism·m
  
88 o14·=·true88 o14·=·true
89 i15·:·E[1]·==·source·m89 i15·:·E[1]·==·source·m
  
90 o15·=·true90 o15·=·true
91 i16·:·E'·=·target·m91 i16·:·E'·=·target·m
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./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html
    
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116 <td>··············<pre><code·class="language-macaulay2">i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);</code></pre>116 <td>··············<pre><code·class="language-macaulay2">i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-1),mods_i,substitute(matrix·C.dd_i,S));</code></pre>119 <td>··············<pre><code·class="language-macaulay2">i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-1),mods_i,substitute(matrix·C.dd_i,S));</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i8·:·time·m·=·resolutionOfChainComplex·C;122 <td>··············<pre><code·class="language-macaulay2">i8·:·time·m·=·resolutionOfChainComplex·C;
123 ·--·used·0.131551s·(cpu);·0.0980714s·(thread);·0s·(gc)</code></pre>123 ·--·used·0.159853s·(cpu);·0.108221s·(thread);·0s·(gc)</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i9·:·time·n·=·cartanEilenbergResolution·C;126 <td>··············<pre><code·class="language-macaulay2">i9·:·time·n·=·cartanEilenbergResolution·C;
127 ·--·used·0.102411s·(cpu);·0.103561s·(thread);·0s·(gc)</code></pre>127 ·--·used·0.122496s·(cpu);·0.125744s·(thread);·0s·(gc)</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ··········<tr>129 ··········<tr>
130 <td>··············<pre><code·class="language-macaulay2">i10·:·betti·source·m130 <td>··············<pre><code·class="language-macaulay2">i10·:·betti·source·m
  
131 ·············0··1··2···3···4···5···6···7131 ·············0··1··2···3···4···5···6···7
132 o10·=·total:·1·19·80·181·312·484·447·156132 o10·=·total:·1·19·80·181·312·484·447·156
133 ··········0:·1··3··3···1···.···.···.···.133 ··········0:·1··3··3···1···.···.···.···.
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50 o4·:·RingMap·R·<--·S50 o4·:·RingMap·R·<--·S
51 i5·:·C·=·res(R^1/(ideal·vars·R))**(R^1/(ideal·vars·R)^5);51 i5·:·C·=·res(R^1/(ideal·vars·R))**(R^1/(ideal·vars·R)^5);
52 i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);52 i6·:·mods·=·for·i·from·0·to·max·C·list·pushForward(f,·C_i);
53 i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-53 i7·:·C·=·chainComplex·for·i·from·min·C+1·to·max·C·list·map(mods_(i-
54 1),mods_i,substitute(matrix·C.dd_i,S));54 1),mods_i,substitute(matrix·C.dd_i,S));
55 i8·:·time·m·=·resolutionOfChainComplex·C;55 i8·:·time·m·=·resolutionOfChainComplex·C;
56 ·--·used·0.131551s·(cpu);·0.0980714s·(thread);·0s·(gc)56 ·--·used·0.159853s·(cpu);·0.108221s·(thread);·0s·(gc)
57 i9·:·time·n·=·cartanEilenbergResolution·C;57 i9·:·time·n·=·cartanEilenbergResolution·C;
58 ·--·used·0.102411s·(cpu);·0.103561s·(thread);·0s·(gc)58 ·--·used·0.122496s·(cpu);·0.125744s·(thread);·0s·(gc)
59 i10·:·betti·source·m59 i10·:·betti·source·m
  
60 ·············0··1··2···3···4···5···6···760 ·············0··1··2···3···4···5···6···7
61 o10·=·total:·1·19·80·181·312·484·447·15661 o10·=·total:·1·19·80·181·312·484·447·156
62 ··········0:·1··3··3···1···.···.···.···.62 ··········0:·1··3··3···1···.···.···.···.
63 ··········1:·.··.··1···3···3···.···.···.63 ··········1:·.··.··1···3···3···.···.···.
64 ··········2:·.··1··3···3···2···.···.···.64 ··········2:·.··1··3···3···2···.···.···.
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785 B
./usr/share/doc/Macaulay2/ChainComplexOperations/dump/rawdocumentation.dump
    
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9 #:len=3359 #:len=335
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyZXZlcnNlRmFjdG9ycyxDaGFpbkNvbXBsZXgsQ2hh11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyZXZlcnNlRmFjdG9ycyxDaGFpbkNvbXBsZXgsQ2hh
⊟
752 B
./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=187 #:len=18
8 TXVsdGlQcm9qQ29vcmRSaW5n8 TXVsdGlQcm9qQ29vcmRSaW5n
9 #:len=22069 #:len=2206
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl
11 IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu11 IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu
⊟
1.05 KB
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out
    
Offset 83, 15 lines modifiedOffset 83, 15 lines modified
83 ··············2··············283 ··············2··············2
84 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)84 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)
85 ··············0·3····1·2·4···2·585 ··············0·3····1·2·4···2·5
  
86 o14·:·Ideal·of·R86 o14·:·Ideal·of·R
  
87 i15·:·time·csmK=CSM(A,K)87 i15·:·time·csmK=CSM(A,K)
88 ·--·used·1.76458s·(cpu);·0.485991s·(thread);·0s·(gc)88 ·--·used·1.89037s·(cpu);·0.39348s·(thread);·0s·(gc)
  
89 ········2·2·····2·········2····2············289 ········2·2·····2·········2····2············2
90 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h90 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
91 ········1·2·····1·2·····1·2····1·····1·2····291 ········1·2·····1·2·····1·2····1·····1·2····2
  
92 o15·:·A92 o15·:·A
  
Offset 124, 15 lines modifiedOffset 124, 15 lines modified
124 ········2·2·····2·········2·····2·············2124 ········2·2·····2·········2·····2·············2
125 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h125 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h
126 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2126 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2
  
127 o21·:·A127 o21·:·A
  
128 i22·:·time·CSM(A,K,m)128 i22·:·time·CSM(A,K,m)
129 ·--·used·0.174524s·(cpu);·0.0877111s·(thread);·0s·(gc)129 ·--·used·0.19246s·(cpu);·0.0854175s·(thread);·0s·(gc)
  
130 ········2·2·····2·········2····2············2130 ········2·2·····2·········2····2············2
131 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h131 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
132 ········1·2·····1·2·····1·2····1·····1·2····2132 ········1·2·····1·2·····1·2····1·····1·2····2
  
133 o22·:·A133 o22·:·A
  
⊟
1.35 KB
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out
    
Offset 9, 28 lines modifiedOffset 9, 28 lines modified
9 i2·:·U·=·toricProjectiveSpace·79 i2·:·U·=·toricProjectiveSpace·7
  
10 o2·=·U10 o2·=·U
  
11 o2·:·NormalToricVariety11 o2·:·NormalToricVariety
  
12 i3·:·time·CSM·U12 i3·:·time·CSM·U
13 ·--·used·0.177299s·(cpu);·0.110785s·(thread);·0s·(gc)13 ·--·used·0.187214s·(cpu);·0.108953s·(thread);·0s·(gc)
  
14 ·······7······6······5······4······3······214 ·······7······6······5······4······3······2
15 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·115 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
16 ·······7······7······7······7······7······7·····716 ·······7······7······7······7······7······7·····7
  
17 ················································ZZ[x·..x·]17 ················································ZZ[x·..x·]
18 ····················································0···718 ····················································0···7
19 o3·:·-----------------------------------------------------------------------------------------------19 o3·:·-----------------------------------------------------------------------------------------------
20 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·)20 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·)
21 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····5·····0····6·····0····721 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····5·····0····6·····0····7
  
22 i4·:·time·CSM(U,CheckSmooth=>false)22 i4·:·time·CSM(U,CheckSmooth=>false)
23 ·--·used·0.382107s·(cpu);·0.232814s·(thread);·0s·(gc)23 ·--·used·0.450526s·(cpu);·0.24553s·(thread);·0s·(gc)
  
24 ·······7······6······5······4······3······224 ·······7······6······5······4······3······2
25 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·125 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
26 ·······7······7······7······7······7······7·····726 ·······7······7······7······7······7······7·····7
  
27 ················································ZZ[x·..x·]27 ················································ZZ[x·..x·]
28 ····················································0···728 ····················································0···7
⊟
1.47 KB
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out
    
Offset 18, 29 lines modifiedOffset 18, 29 lines modified
18 i3·:·R=ZZ/32749[v_0..v_5];18 i3·:·R=ZZ/32749[v_0..v_5];
  
19 i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));19 i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
  
20 o4·:·Ideal·of·R20 o4·:·Ideal·of·R
  
21 i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)21 i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)
22 ·--·used·0.567234s·(cpu);·0.314033s·(thread);·0s·(gc)22 ·--·used·1.0027s·(cpu);·0.347547s·(thread);·0s·(gc)
  
23 ·······5······4······3······223 ·······5······4······3······2
24 o5·=·6h··+·14h··+·14h··+·10h24 o5·=·6h··+·14h··+·14h··+·10h
25 ·······1······1······1······125 ·······1······1······1······1
  
26 ·····ZZ[h·]26 ·····ZZ[h·]
27 ·········127 ·········1
28 o5·:·------28 o5·:·------
29 ········629 ········6
30 ·······h30 ·······h
31 ········131 ········1
  
32 i6·:·time·CSM(I,CompMethod=>PnResidual)32 i6·:·time·CSM(I,CompMethod=>PnResidual)
33 ·--·used·1.73912s·(cpu);·1.54989s·(thread);·0s·(gc)33 ·--·used·2.07723s·(cpu);·1.81109s·(thread);·0s·(gc)
  
34 ·······5······4······3······234 ·······5······4······3······2
35 o6·=·6H··+·14H··+·14H··+·10H35 o6·=·6H··+·14H··+·14H··+·10H
  
36 ·····ZZ[H]36 ·····ZZ[H]
37 o6·:·-----37 o6·:·-----
38 ········638 ········6
Offset 53, 29 lines modifiedOffset 53, 29 lines modified
53 i8·:·S=QQ[s_0..s_3];53 i8·:·S=QQ[s_0..s_3];
  
54 i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));54 i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
  
55 o9·:·Ideal·of·S55 o9·:·Ideal·of·S
  
56 i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)56 i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)
57 ·--·used·0.362517s·(cpu);·0.213606s·(thread);·0s·(gc)57 ·--·used·0.527489s·(cpu);·0.229569s·(thread);·0s·(gc)
  
58 ········3·····258 ········3·····2
59 o10·=·3h··+·5h59 o10·=·3h··+·5h
60 ········1·····160 ········1·····1
  
61 ······ZZ[h·]61 ······ZZ[h·]
62 ··········162 ··········1
63 o10·:·------63 o10·:·------
64 ·········464 ·········4
65 ········h65 ········h
66 ·········166 ·········1
  
67 i11·:·time·CSM(K,CompMethod=>PnResidual)67 i11·:·time·CSM(K,CompMethod=>PnResidual)
68 ·--·used·0.178296s·(cpu);·0.114979s·(thread);·0s·(gc)68 ·--·used·0.211303s·(cpu);·0.110952s·(thread);·0s·(gc)
  
69 ········3·····269 ········3·····2
70 o11·=·3H··+·5H70 o11·=·3H··+·5H
  
71 ······ZZ[H]71 ······ZZ[H]
72 o11·:·-----72 o11·:·-----
73 ·········473 ·········4
⊟
1.32 KB
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out
    
Offset 21, 20 lines modifiedOffset 21, 20 lines modified
21 ·············2························································221 ·············2························································2
22 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)22 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)
23 ·············3·········0·4········1·4·········2·4·········3·4·········423 ·············3·········0·4········1·4·········2·4·········3·4·········4
  
24 o3·:·Ideal·of·R24 o3·:·Ideal·of·R
  
25 i4·:·time·Euler(I,InputIsSmooth=>true)25 i4·:·time·Euler(I,InputIsSmooth=>true)
26 ·--·used·0.0860699s·(cpu);·0.0410961s·(thread);·0s·(gc)26 ·--·used·0.144149s·(cpu);·0.05963s·(thread);·0s·(gc)
  
27 o4·=·427 o4·=·4
  
28 i5·:·time·Euler·I28 i5·:·time·Euler·I
29 ·--·used·0.212654s·(cpu);·0.126593s·(thread);·0s·(gc)29 ·--·used·0.33655s·(cpu);·0.16484s·(thread);·0s·(gc)
  
30 o5·=·430 o5·=·4
  
31 i6·:·EulerIHash=Euler(I,Output=>HashForm);31 i6·:·EulerIHash=Euler(I,Output=>HashForm);
  
32 i7·:·A=ring·EulerIHash#"CSM"32 i7·:·A=ring·EulerIHash#"CSM"
  
Offset 62, 20 lines modifiedOffset 62, 20 lines modified
62 ·····------------------------------------------------------------------------62 ·····------------------------------------------------------------------------
63 ·····-·x·x·)63 ·····-·x·x·)
64 ········0·364 ········0·3
  
65 o9·:·Ideal·of·R65 o9·:·Ideal·of·R
  
66 i10·:·time·Euler(J,Method=>DirectCompleteInt)66 i10·:·time·Euler(J,Method=>DirectCompleteInt)
67 ·--·used·0.182533s·(cpu);·0.0976794s·(thread);·0s·(gc)67 ·--·used·0.726871s·(cpu);·0.144982s·(thread);·0s·(gc)
  
68 o10·=·268 o10·=·2
  
69 i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})69 i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
70 ·--·used·0.118114s·(cpu);·0.0652421s·(thread);·0s·(gc)70 ·--·used·0.316931s·(cpu);·0.0871421s·(thread);·0s·(gc)
  
71 o11·=·271 o11·=·2
  
72 i12·:·R=MultiProjCoordRing({2,2})72 i12·:·R=MultiProjCoordRing({2,2})
  
73 o12·=·R73 o12·=·R
  
⊟
926 B
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
    
Offset 7, 29 lines modifiedOffset 7, 29 lines modified
7 o1·:·PolynomialRing7 o1·:·PolynomialRing
  
8 i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));8 i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
  
9 o2·:·Ideal·of·R9 o2·:·Ideal·of·R
  
10 i3·:·time·CSM(I,Method=>DirectCompletInt)10 i3·:·time·CSM(I,Method=>DirectCompletInt)
11 ·--·used·3.01069s·(cpu);·1.10628s·(thread);·0s·(gc)11 ·--·used·5.17639s·(cpu);·1.3022s·(thread);·0s·(gc)
  
12 ·······2·2·····2·········212 ·······2·2·····2·········2
13 o3·=·2h·h··+·2h·h··+·5h·h13 o3·=·2h·h··+·2h·h··+·5h·h
14 ·······1·2·····1·2·····1·214 ·······1·2·····1·2·····1·2
  
15 ·····ZZ[h·..h·]15 ·····ZZ[h·..h·]
16 ·········1···216 ·········1···2
17 o3·:·----------17 o3·:·----------
18 ········3···318 ········3···3
19 ······(h·,·h·)19 ······(h·,·h·)
20 ········1···220 ········1···2
  
21 i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})21 i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
22 ·--·used·3.25754s·(cpu);·1.15785s·(thread);·0s·(gc)22 ·--·used·5.27327s·(cpu);·1.25833s·(thread);·0s·(gc)
  
23 ·······2·2·····2·········223 ·······2·2·····2·········2
24 o4·=·2h·h··+·2h·h··+·5h·h24 o4·=·2h·h··+·2h·h··+·5h·h
25 ·······1·2·····1·2·····1·225 ·······1·2·····1·2·····1·2
  
26 ·····ZZ[h·..h·]26 ·····ZZ[h·..h·]
27 ·········1···227 ·········1···2
⊟
974 B
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out
    
Offset 3, 43 lines modifiedOffset 3, 43 lines modified
3 i1·:·R·=·ZZ/32749[x_0..x_4];3 i1·:·R·=·ZZ/32749[x_0..x_4];
  
4 i2·:·I=ideal(random(2,R),random(2,R),random(1,R));4 i2·:·I=ideal(random(2,R),random(2,R),random(1,R));
  
5 o2·:·Ideal·of·R5 o2·:·Ideal·of·R
  
6 i3·:·time·CSM·I6 i3·:·time·CSM·I
7 ·--·used·0.795258s·(cpu);·0.433427s·(thread);·0s·(gc)7 ·--·used·1.28971s·(cpu);·0.515449s·(thread);·0s·(gc)
  
8 ·······38 ·······3
9 o3·=·4h9 o3·=·4h
10 ·······110 ·······1
  
11 ·····ZZ[h·]11 ·····ZZ[h·]
12 ·········112 ·········1
13 o3·:·------13 o3·:·------
14 ········514 ········5
15 ·······h15 ·······h
16 ········116 ········1
  
17 i4·:·time·CSM(I,InputIsSmooth=>true)17 i4·:·time·CSM(I,InputIsSmooth=>true)
18 ·--·used·0.0832932s·(cpu);·0.0453238s·(thread);·0s·(gc)18 ·--·used·0.120881s·(cpu);·0.0615456s·(thread);·0s·(gc)
  
19 ·······319 ·······3
20 o4·=·4h20 o4·=·4h
21 ·······121 ·······1
  
22 ·····ZZ[h·]22 ·····ZZ[h·]
23 ·········123 ·········1
24 o4·:·------24 o4·:·------
25 ········525 ········5
26 ·······h26 ·······h
27 ········127 ········1
  
28 i5·:·time·Chern·I28 i5·:·time·Chern·I
29 ·--·used·0.0359046s·(cpu);·0.0300469s·(thread);·0s·(gc)29 ·--·used·0.0359137s·(cpu);·0.037355s·(thread);·0s·(gc)
  
30 ·······330 ·······3
31 o5·=·4h31 o5·=·4h
32 ·······132 ·······1
  
33 ·····ZZ[h·]33 ·····ZZ[h·]
34 ·········134 ·········1
⊟
795 B
./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out
    
Offset 7, 29 lines modifiedOffset 7, 29 lines modified
7 o1·:·PolynomialRing7 o1·:·PolynomialRing
  
8 i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);8 i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
  
9 o2·:·Ideal·of·R9 o2·:·Ideal·of·R
  
10 i3·:·time·CSM·I10 i3·:·time·CSM·I
11 ·--·used·1.485s·(cpu);·0.82601s·(thread);·0s·(gc)11 ·--·used·2.82384s·(cpu);·0.928525s·(thread);·0s·(gc)
  
12 ········5······4·····312 ········5······4·····3
13 o3·=·12h··+·10h··+·6h13 o3·=·12h··+·10h··+·6h
14 ········1······1·····114 ········1······1·····1
  
15 ·····ZZ[h·]15 ·····ZZ[h·]
16 ·········116 ·········1
17 o3·:·------17 o3·:·------
18 ········718 ········7
19 ·······h19 ·······h
20 ········120 ········1
  
21 i4·:·time·CSM(I,Method=>DirectCompleteInt)21 i4·:·time·CSM(I,Method=>DirectCompleteInt)
22 ·--·used·0.456841s·(cpu);·0.193811s·(thread);·0s·(gc)22 ·--·used·0.847444s·(cpu);·0.231496s·(thread);·0s·(gc)
  
23 ········5······4·····323 ········5······4·····3
24 o4·=·12h··+·10h··+·6h24 o4·=·12h··+·10h··+·6h
25 ········1······1·····125 ········1······1·····1
  
26 ·····ZZ[h·]26 ·····ZZ[h·]
27 ·········127 ·········1
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Offset 221, 15 lines modifiedOffset 221, 15 lines modified
221 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)221 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)
222 ··············0·3····1·2·4···2·5222 ··············0·3····1·2·4···2·5
  
223 o14·:·Ideal·of·R</code></pre>223 o14·:·Ideal·of·R</code></pre>
224 </td>··········</tr>224 </td>··········</tr>
225 ··········<tr>225 ··········<tr>
226 <td>··············<pre><code·class="language-macaulay2">i15·:·time·csmK=CSM(A,K)226 <td>··············<pre><code·class="language-macaulay2">i15·:·time·csmK=CSM(A,K)
227 ·--·used·1.76458s·(cpu);·0.485991s·(thread);·0s·(gc)227 ·--·used·1.89037s·(cpu);·0.39348s·(thread);·0s·(gc)
  
228 ········2·2·····2·········2····2············2228 ········2·2·····2·········2····2············2
229 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h229 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
230 ········1·2·····1·2·····1·2····1·····1·2····2230 ········1·2·····1·2·····1·2····1·····1·2····2
  
231 o15·:·A</code></pre>231 o15·:·A</code></pre>
232 </td>··········</tr>232 </td>··········</tr>
Offset 274, 15 lines modifiedOffset 274, 15 lines modified
274 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h274 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h
275 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2275 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2
  
276 o21·:·A</code></pre>276 o21·:·A</code></pre>
277 </td>··········</tr>277 </td>··········</tr>
278 ··········<tr>278 ··········<tr>
279 <td>··············<pre><code·class="language-macaulay2">i22·:·time·CSM(A,K,m)279 <td>··············<pre><code·class="language-macaulay2">i22·:·time·CSM(A,K,m)
280 ·--·used·0.174524s·(cpu);·0.0877111s·(thread);·0s·(gc)280 ·--·used·0.19246s·(cpu);·0.0854175s·(thread);·0s·(gc)
  
281 ········2·2·····2·········2····2············2281 ········2·2·····2·········2····2············2
282 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h282 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
283 ········1·2·····1·2·····1·2····1·····1·2····2283 ········1·2·····1·2·····1·2····1·····1·2····2
  
284 o22·:·A</code></pre>284 o22·:·A</code></pre>
285 </td>··········</tr>285 </td>··········</tr>
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161 ··············2··············2161 ··············2··············2
162 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)162 o14·=·ideal·(x·x··-·x·x·x·,·x·x·)
163 ··············0·3····1·2·4···2·5163 ··············0·3····1·2·4···2·5
  
164 o14·:·Ideal·of·R164 o14·:·Ideal·of·R
165 i15·:·time·csmK=CSM(A,K)165 i15·:·time·csmK=CSM(A,K)
166 ·--·used·1.76458s·(cpu);·0.485991s·(thread);·0s·(gc)166 ·--·used·1.89037s·(cpu);·0.39348s·(thread);·0s·(gc)
  
167 ········2·2·····2·········2····2············2167 ········2·2·····2·········2····2············2
168 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h168 o15·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
169 ········1·2·····1·2·····1·2····1·····1·2····2169 ········1·2·····1·2·····1·2····1·····1·2····2
  
170 o15·:·A170 o15·:·A
171 i16·:·csmKHash=·CSM(A,K,Output=>HashForm)171 i16·:·csmKHash=·CSM(A,K,Output=>HashForm)
Offset 200, 15 lines modifiedOffset 200, 15 lines modified
  
200 ········2·2·····2·········2·····2·············2200 ········2·2·····2·········2·····2·············2
201 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h201 o21·=·9h·h··+·9h·h··+·9h·h··+·3h··+·7h·h··+·3h··+·3h··+·2h
202 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2202 ········1·2·····1·2·····1·2·····1·····1·2·····2·····1·····2
  
203 o21·:·A203 o21·:·A
204 i22·:·time·CSM(A,K,m)204 i22·:·time·CSM(A,K,m)
205 ·--·used·0.174524s·(cpu);·0.0877111s·(thread);·0s·(gc)205 ·--·used·0.19246s·(cpu);·0.0854175s·(thread);·0s·(gc)
  
206 ········2·2·····2·········2····2············2206 ········2·2·····2·········2····2············2
207 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h207 o22·=·7h·h··+·5h·h··+·4h·h··+·h··+·3h·h··+·h
208 ········1·2·····1·2·····1·2····1·····1·2····2208 ········1·2·····1·2·····1·2····1·····1·2····2
  
209 o22·:·A209 o22·:·A
210 In·the·case·where·the·ambient·space·is·a·toric·variety·which·is·not·a·product210 In·the·case·where·the·ambient·space·is·a·toric·variety·which·is·not·a·product
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Offset 60, 29 lines modifiedOffset 60, 29 lines modified
  
60 o2·=·U60 o2·=·U
  
61 o2·:·NormalToricVariety</code></pre>61 o2·:·NormalToricVariety</code></pre>
62 </td>··········</tr>62 </td>··········</tr>
63 ··········<tr>63 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·U64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·U
65 ·--·used·0.177299s·(cpu);·0.110785s·(thread);·0s·(gc)65 ·--·used·0.187214s·(cpu);·0.108953s·(thread);·0s·(gc)
  
66 ·······7······6······5······4······3······266 ·······7······6······5······4······3······2
67 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·167 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
68 ·······7······7······7······7······7······7·····768 ·······7······7······7······7······7······7·····7
  
69 ················································ZZ[x·..x·]69 ················································ZZ[x·..x·]
70 ····················································0···770 ····················································0···7
71 o3·:·-----------------------------------------------------------------------------------------------71 o3·:·-----------------------------------------------------------------------------------------------
72 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·)72 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·)
73 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····5·····0····6·····0····7</code></pre>73 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····5·····0····6·····0····7</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(U,CheckSmooth=>false)76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(U,CheckSmooth=>false)
77 ·--·used·0.382107s·(cpu);·0.232814s·(thread);·0s·(gc)77 ·--·used·0.450526s·(cpu);·0.24553s·(thread);·0s·(gc)
  
78 ·······7······6······5······4······3······278 ·······7······6······5······4······3······2
79 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·179 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
80 ·······7······7······7······7······7······7·····780 ·······7······7······7······7······7······7·····7
  
81 ················································ZZ[x·..x·]81 ················································ZZ[x·..x·]
82 ····················································0···782 ····················································0···7
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Offset 16, 30 lines modifiedOffset 16, 30 lines modified
16 o1·:·Package16 o1·:·Package
17 i2·:·U·=·toricProjectiveSpace·717 i2·:·U·=·toricProjectiveSpace·7
  
18 o2·=·U18 o2·=·U
  
19 o2·:·NormalToricVariety19 o2·:·NormalToricVariety
20 i3·:·time·CSM·U20 i3·:·time·CSM·U
21 ·--·used·0.177299s·(cpu);·0.110785s·(thread);·0s·(gc)21 ·--·used·0.187214s·(cpu);·0.108953s·(thread);·0s·(gc)
  
22 ·······7······6······5······4······3······222 ·······7······6······5······4······3······2
23 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·123 o3·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
24 ·······7······7······7······7······7······7·····724 ·······7······7······7······7······7······7·····7
  
25 ················································ZZ[x·..x·]25 ················································ZZ[x·..x·]
26 ····················································0···726 ····················································0···7
27 o3·:·--------------------------------------------------------------------------27 o3·:·--------------------------------------------------------------------------
28 ---------------------28 ---------------------
29 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,29 ·····(x·x·x·x·x·x·x·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,·-·x··+·x·,
30 -·x··+·x·,·-·x··+·x·)30 -·x··+·x·,·-·x··+·x·)
31 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····531 ·······0·1·2·3·4·5·6·7·····0····1·····0····2·····0····3·····0····4·····0····5
32 0····6·····0····732 0····6·····0····7
33 i4·:·time·CSM(U,CheckSmooth=>false)33 i4·:·time·CSM(U,CheckSmooth=>false)
34 ·--·used·0.382107s·(cpu);·0.232814s·(thread);·0s·(gc)34 ·--·used·0.450526s·(cpu);·0.24553s·(thread);·0s·(gc)
  
35 ·······7······6······5······4······3······235 ·······7······6······5······4······3······2
36 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·136 o4·=·8x··+·28x··+·56x··+·70x··+·56x··+·28x··+·8x··+·1
37 ·······7······7······7······7······7······7·····737 ·······7······7······7······7······7······7·····7
  
38 ················································ZZ[x·..x·]38 ················································ZZ[x·..x·]
39 ····················································0···739 ····················································0···7
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));77 <td>··············<pre><code·class="language-macaulay2">i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
  
78 o4·:·Ideal·of·R</code></pre>78 o4·:·Ideal·of·R</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)81 <td>··············<pre><code·class="language-macaulay2">i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)
82 ·--·used·0.567234s·(cpu);·0.314033s·(thread);·0s·(gc)82 ·--·used·1.0027s·(cpu);·0.347547s·(thread);·0s·(gc)
  
83 ·······5······4······3······283 ·······5······4······3······2
84 o5·=·6h··+·14h··+·14h··+·10h84 o5·=·6h··+·14h··+·14h··+·10h
85 ·······1······1······1······185 ·······1······1······1······1
  
86 ·····ZZ[h·]86 ·····ZZ[h·]
87 ·········187 ·········1
88 o5·:·------88 o5·:·------
89 ········689 ········6
90 ·······h90 ·······h
91 ········1</code></pre>91 ········1</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i6·:·time·CSM(I,CompMethod=>PnResidual)94 <td>··············<pre><code·class="language-macaulay2">i6·:·time·CSM(I,CompMethod=>PnResidual)
95 ·--·used·1.73912s·(cpu);·1.54989s·(thread);·0s·(gc)95 ·--·used·2.07723s·(cpu);·1.81109s·(thread);·0s·(gc)
  
96 ·······5······4······3······296 ·······5······4······3······2
97 o6·=·6H··+·14H··+·14H··+·10H97 o6·=·6H··+·14H··+·14H··+·10H
  
98 ·····ZZ[H]98 ·····ZZ[H]
99 o6·:·-----99 o6·:·-----
100 ········6100 ········6
Offset 116, 30 lines modifiedOffset 116, 30 lines modified
116 ··········<tr>116 ··········<tr>
117 <td>··············<pre><code·class="language-macaulay2">i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));117 <td>··············<pre><code·class="language-macaulay2">i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
  
118 o9·:·Ideal·of·S</code></pre>118 o9·:·Ideal·of·S</code></pre>
119 </td>··········</tr>119 </td>··········</tr>
120 ··········<tr>120 ··········<tr>
121 <td>··············<pre><code·class="language-macaulay2">i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)121 <td>··············<pre><code·class="language-macaulay2">i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)
122 ·--·used·0.362517s·(cpu);·0.213606s·(thread);·0s·(gc)122 ·--·used·0.527489s·(cpu);·0.229569s·(thread);·0s·(gc)
  
123 ········3·····2123 ········3·····2
124 o10·=·3h··+·5h124 o10·=·3h··+·5h
125 ········1·····1125 ········1·····1
  
126 ······ZZ[h·]126 ······ZZ[h·]
127 ··········1127 ··········1
128 o10·:·------128 o10·:·------
129 ·········4129 ·········4
130 ········h130 ········h
131 ·········1</code></pre>131 ·········1</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i11·:·time·CSM(K,CompMethod=>PnResidual)134 <td>··············<pre><code·class="language-macaulay2">i11·:·time·CSM(K,CompMethod=>PnResidual)
135 ·--·used·0.178296s·(cpu);·0.114979s·(thread);·0s·(gc)135 ·--·used·0.211303s·(cpu);·0.110952s·(thread);·0s·(gc)
  
136 ········3·····2136 ········3·····2
137 o11·=·3H··+·5H137 o11·=·3H··+·5H
  
138 ······ZZ[H]138 ······ZZ[H]
139 o11·:·-----139 o11·:·-----
140 ·········4140 ·········4
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32 using·the·regenerative·cascade·implemented·in·Bertini.·This·is·done·by·choosing32 using·the·regenerative·cascade·implemented·in·Bertini.·This·is·done·by·choosing
33 the·option·bertini,·provided·Bertini·is·_\x8i_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8e_\x8d_\x8·_\x8a_\x8n_\x8d_\x8·_\x8c_\x8o_\x8n_\x8f_\x8i_\x8g_\x8u_\x8r_\x8e_\x8d.33 the·option·bertini,·provided·Bertini·is·_\x8i_\x8n_\x8s_\x8t_\x8a_\x8l_\x8l_\x8e_\x8d_\x8·_\x8a_\x8n_\x8d_\x8·_\x8c_\x8o_\x8n_\x8f_\x8i_\x8g_\x8u_\x8r_\x8e_\x8d.
34 i3·:·R=ZZ/32749[v_0..v_5];34 i3·:·R=ZZ/32749[v_0..v_5];
35 i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));35 i4·:·I=ideal(4*v_3*v_1*v_2-8*v_1*v_3^2,v_5*(v_0*v_1*v_4-v_2^3));
  
36 o4·:·Ideal·of·R36 o4·:·Ideal·of·R
37 i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)37 i5·:·time·CSM(I,CompMethod=>ProjectiveDegree)
38 ·--·used·0.567234s·(cpu);·0.314033s·(thread);·0s·(gc)38 ·--·used·1.0027s·(cpu);·0.347547s·(thread);·0s·(gc)
  
39 ·······5······4······3······239 ·······5······4······3······2
40 o5·=·6h··+·14h··+·14h··+·10h40 o5·=·6h··+·14h··+·14h··+·10h
41 ·······1······1······1······141 ·······1······1······1······1
  
42 ·····ZZ[h·]42 ·····ZZ[h·]
43 ·········143 ·········1
44 o5·:·------44 o5·:·------
45 ········645 ········6
46 ·······h46 ·······h
47 ········147 ········1
48 i6·:·time·CSM(I,CompMethod=>PnResidual)48 i6·:·time·CSM(I,CompMethod=>PnResidual)
49 ·--·used·1.73912s·(cpu);·1.54989s·(thread);·0s·(gc)49 ·--·used·2.07723s·(cpu);·1.81109s·(thread);·0s·(gc)
  
50 ·······5······4······3······250 ·······5······4······3······2
51 o6·=·6H··+·14H··+·14H··+·10H51 o6·=·6H··+·14H··+·14H··+·10H
  
52 ·····ZZ[H]52 ·····ZZ[H]
53 o6·:·-----53 o6·:·-----
54 ········654 ········6
Offset 62, 28 lines modifiedOffset 62, 28 lines modified
  
62 o7·=·262 o7·=·2
63 i8·:·S=QQ[s_0..s_3];63 i8·:·S=QQ[s_0..s_3];
64 i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));64 i9·:·K=ideal(4*s_3*s_2-s_2^2,(s_0*s_1*s_3-s_2^3));
  
65 o9·:·Ideal·of·S65 o9·:·Ideal·of·S
66 i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)66 i10·:·time·CSM(K,CompMethod=>ProjectiveDegree)
67 ·--·used·0.362517s·(cpu);·0.213606s·(thread);·0s·(gc)67 ·--·used·0.527489s·(cpu);·0.229569s·(thread);·0s·(gc)
  
68 ········3·····268 ········3·····2
69 o10·=·3h··+·5h69 o10·=·3h··+·5h
70 ········1·····170 ········1·····1
  
71 ······ZZ[h·]71 ······ZZ[h·]
72 ··········172 ··········1
73 o10·:·------73 o10·:·------
74 ·········474 ·········4
75 ········h75 ········h
76 ·········176 ·········1
77 i11·:·time·CSM(K,CompMethod=>PnResidual)77 i11·:·time·CSM(K,CompMethod=>PnResidual)
78 ·--·used·0.178296s·(cpu);·0.114979s·(thread);·0s·(gc)78 ·--·used·0.211303s·(cpu);·0.110952s·(thread);·0s·(gc)
  
79 ········3·····279 ········3·····2
80 o11·=·3H··+·5H80 o11·=·3H··+·5H
  
81 ······ZZ[H]81 ······ZZ[H]
82 o11·:·-----82 o11·:·-----
83 ·········483 ·········4
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130 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)130 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)
131 ·············3·········0·4········1·4·········2·4·········3·4·········4131 ·············3·········0·4········1·4·········2·4·········3·4·········4
  
132 o3·:·Ideal·of·R</code></pre>132 o3·:·Ideal·of·R</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Euler(I,InputIsSmooth=>true)135 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Euler(I,InputIsSmooth=>true)
136 ·--·used·0.0860699s·(cpu);·0.0410961s·(thread);·0s·(gc)136 ·--·used·0.144149s·(cpu);·0.05963s·(thread);·0s·(gc)
  
137 o4·=·4</code></pre>137 o4·=·4</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Euler·I140 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Euler·I
141 ·--·used·0.212654s·(cpu);·0.126593s·(thread);·0s·(gc)141 ·--·used·0.33655s·(cpu);·0.16484s·(thread);·0s·(gc)
  
142 o5·=·4</code></pre>142 o5·=·4</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i6·:·EulerIHash=Euler(I,Output=>HashForm);</code></pre>145 <td>··············<pre><code·class="language-macaulay2">i6·:·EulerIHash=Euler(I,Output=>HashForm);</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
Offset 182, 21 lines modifiedOffset 182, 21 lines modified
182 ········</table>182 ········</table>
183 ········<div>183 ········<div>
184 ··········<p>Note·that·the·ideal·J·above·is·a·complete·intersection,·thus·we·may·change·the·method·option·which·may·speed·computation·in·some·cases.·We·may·also·note·that·the·ideal·generated·by·the·first·2·generators·of·I·defines·a·smooth·scheme·and·input·this·information·into·the·method.·This·may·also·improve·computation·speed.</p>184 ··········<p>Note·that·the·ideal·J·above·is·a·complete·intersection,·thus·we·may·change·the·method·option·which·may·speed·computation·in·some·cases.·We·may·also·note·that·the·ideal·generated·by·the·first·2·generators·of·I·defines·a·smooth·scheme·and·input·this·information·into·the·method.·This·may·also·improve·computation·speed.</p>
185 ········</div>185 ········</div>
186 ········<table·class="examples">186 ········<table·class="examples">
187 ··········<tr>187 ··········<tr>
188 <td>··············<pre><code·class="language-macaulay2">i10·:·time·Euler(J,Method=>DirectCompleteInt)188 <td>··············<pre><code·class="language-macaulay2">i10·:·time·Euler(J,Method=>DirectCompleteInt)
189 ·--·used·0.182533s·(cpu);·0.0976794s·(thread);·0s·(gc)189 ·--·used·0.726871s·(cpu);·0.144982s·(thread);·0s·(gc)
  
190 o10·=·2</code></pre>190 o10·=·2</code></pre>
191 </td>··········</tr>191 </td>··········</tr>
192 ··········<tr>192 ··········<tr>
193 <td>··············<pre><code·class="language-macaulay2">i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})193 <td>··············<pre><code·class="language-macaulay2">i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
194 ·--·used·0.118114s·(cpu);·0.0652421s·(thread);·0s·(gc)194 ·--·used·0.316931s·(cpu);·0.0871421s·(thread);·0s·(gc)
  
195 o11·=·2</code></pre>195 o11·=·2</code></pre>
196 </td>··········</tr>196 </td>··········</tr>
197 ········</table>197 ········</table>
198 ········<div>198 ········<div>
199 ··········<p>Now·consider·an·example·in·\PP^2·\times·\PP^2.</p>199 ··········<p>Now·consider·an·example·in·\PP^2·\times·\PP^2.</p>
200 ········</div>200 ········</div>
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Offset 75, 19 lines modifiedOffset 75, 19 lines modified
75 ·····------------------------------------------------------------------------75 ·····------------------------------------------------------------------------
76 ·············2························································276 ·············2························································2
77 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)77 ·····-·14254x··-·11226x·x··+·2653x·x··+·12365x·x··-·10226x·x··-·12696x·)
78 ·············3·········0·4········1·4·········2·4·········3·4·········478 ·············3·········0·4········1·4·········2·4·········3·4·········4
  
79 o3·:·Ideal·of·R79 o3·:·Ideal·of·R
80 i4·:·time·Euler(I,InputIsSmooth=>true)80 i4·:·time·Euler(I,InputIsSmooth=>true)
81 ·--·used·0.0860699s·(cpu);·0.0410961s·(thread);·0s·(gc)81 ·--·used·0.144149s·(cpu);·0.05963s·(thread);·0s·(gc)
  
82 o4·=·482 o4·=·4
83 i5·:·time·Euler·I83 i5·:·time·Euler·I
84 ·--·used·0.212654s·(cpu);·0.126593s·(thread);·0s·(gc)84 ·--·used·0.33655s·(cpu);·0.16484s·(thread);·0s·(gc)
  
85 o5·=·485 o5·=·4
86 i6·:·EulerIHash=Euler(I,Output=>HashForm);86 i6·:·EulerIHash=Euler(I,Output=>HashForm);
87 i7·:·A=ring·EulerIHash#"CSM"87 i7·:·A=ring·EulerIHash#"CSM"
  
88 o7·=·A88 o7·=·A
  
Offset 115, 19 lines modifiedOffset 115, 19 lines modified
115 o9·:·Ideal·of·R115 o9·:·Ideal·of·R
116 Note·that·the·ideal·J·above·is·a·complete·intersection,·thus·we·may·change·the116 Note·that·the·ideal·J·above·is·a·complete·intersection,·thus·we·may·change·the
117 method·option·which·may·speed·computation·in·some·cases.·We·may·also·note·that117 method·option·which·may·speed·computation·in·some·cases.·We·may·also·note·that
118 the·ideal·generated·by·the·first·2·generators·of·I·defines·a·smooth·scheme·and118 the·ideal·generated·by·the·first·2·generators·of·I·defines·a·smooth·scheme·and
119 input·this·information·into·the·method.·This·may·also·improve·computation119 input·this·information·into·the·method.·This·may·also·improve·computation
120 speed.120 speed.
121 i10·:·time·Euler(J,Method=>DirectCompleteInt)121 i10·:·time·Euler(J,Method=>DirectCompleteInt)
122 ·--·used·0.182533s·(cpu);·0.0976794s·(thread);·0s·(gc)122 ·--·used·0.726871s·(cpu);·0.144982s·(thread);·0s·(gc)
  
123 o10·=·2123 o10·=·2
124 i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})124 i11·:·time·Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
125 ·--·used·0.118114s·(cpu);·0.0652421s·(thread);·0s·(gc)125 ·--·used·0.316931s·(cpu);·0.0871421s·(thread);·0s·(gc)
  
126 o11·=·2126 o11·=·2
127 Now·consider·an·example·in·\PP^2·\times·\PP^2.127 Now·consider·an·example·in·\PP^2·\times·\PP^2.
128 i12·:·R=MultiProjCoordRing({2,2})128 i12·:·R=MultiProjCoordRing({2,2})
  
129 o12·=·R129 o12·=·R
  
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Offset 58, 30 lines modifiedOffset 58, 30 lines modified
58 ··········<tr>58 ··········<tr>
59 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));59 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
  
60 o2·:·Ideal·of·R</code></pre>60 o2·:·Ideal·of·R</code></pre>
61 </td>··········</tr>61 </td>··········</tr>
62 ··········<tr>62 ··········<tr>
63 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM(I,Method=>DirectCompletInt)63 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM(I,Method=>DirectCompletInt)
64 ·--·used·3.01069s·(cpu);·1.10628s·(thread);·0s·(gc)64 ·--·used·5.17639s·(cpu);·1.3022s·(thread);·0s·(gc)
  
65 ·······2·2·····2·········265 ·······2·2·····2·········2
66 o3·=·2h·h··+·2h·h··+·5h·h66 o3·=·2h·h··+·2h·h··+·5h·h
67 ·······1·2·····1·2·····1·267 ·······1·2·····1·2·····1·2
  
68 ·····ZZ[h·..h·]68 ·····ZZ[h·..h·]
69 ·········1···269 ·········1···2
70 o3·:·----------70 o3·:·----------
71 ········3···371 ········3···3
72 ······(h·,·h·)72 ······(h·,·h·)
73 ········1···2</code></pre>73 ········1···2</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
77 ·--·used·3.25754s·(cpu);·1.15785s·(thread);·0s·(gc)77 ·--·used·5.27327s·(cpu);·1.25833s·(thread);·0s·(gc)
  
78 ·······2·2·····2·········278 ·······2·2·····2·········2
79 o4·=·2h·h··+·2h·h··+·5h·h79 o4·=·2h·h··+·2h·h··+·5h·h
80 ·······1·2·····1·2·····1·280 ·······1·2·····1·2·····1·2
  
81 ·····ZZ[h·..h·]81 ·····ZZ[h·..h·]
82 ·········1···282 ·········1···2
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Offset 16, 28 lines modifiedOffset 16, 28 lines modified
16 o1·=·R16 o1·=·R
  
17 o1·:·PolynomialRing17 o1·:·PolynomialRing
18 i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));18 i2·:·I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
  
19 o2·:·Ideal·of·R19 o2·:·Ideal·of·R
20 i3·:·time·CSM(I,Method=>DirectCompletInt)20 i3·:·time·CSM(I,Method=>DirectCompletInt)
21 ·--·used·3.01069s·(cpu);·1.10628s·(thread);·0s·(gc)21 ·--·used·5.17639s·(cpu);·1.3022s·(thread);·0s·(gc)
  
22 ·······2·2·····2·········222 ·······2·2·····2·········2
23 o3·=·2h·h··+·2h·h··+·5h·h23 o3·=·2h·h··+·2h·h··+·5h·h
24 ·······1·2·····1·2·····1·224 ·······1·2·····1·2·····1·2
  
25 ·····ZZ[h·..h·]25 ·····ZZ[h·..h·]
26 ·········1···226 ·········1···2
27 o3·:·----------27 o3·:·----------
28 ········3···328 ········3···3
29 ······(h·,·h·)29 ······(h·,·h·)
30 ········1···230 ········1···2
31 i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})31 i4·:·time·CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
32 ·--·used·3.25754s·(cpu);·1.15785s·(thread);·0s·(gc)32 ·--·used·5.27327s·(cpu);·1.25833s·(thread);·0s·(gc)
  
33 ·······2·2·····2·········233 ·······2·2·····2·········2
34 o4·=·2h·h··+·2h·h··+·5h·h34 o4·=·2h·h··+·2h·h··+·5h·h
35 ·······1·2·····1·2·····1·235 ·······1·2·····1·2·····1·2
  
36 ·····ZZ[h·..h·]36 ·····ZZ[h·..h·]
37 ·········1···237 ·········1···2
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Offset 54, 30 lines modifiedOffset 54, 30 lines modified
54 ··········<tr>54 ··········<tr>
55 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(random(2,R),random(2,R),random(1,R));55 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(random(2,R),random(2,R),random(1,R));
  
56 o2·:·Ideal·of·R</code></pre>56 o2·:·Ideal·of·R</code></pre>
57 </td>··········</tr>57 </td>··········</tr>
58 ··········<tr>58 ··········<tr>
59 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·I59 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·I
60 ·--·used·0.795258s·(cpu);·0.433427s·(thread);·0s·(gc)60 ·--·used·1.28971s·(cpu);·0.515449s·(thread);·0s·(gc)
  
61 ·······361 ·······3
62 o3·=·4h62 o3·=·4h
63 ·······163 ·······1
  
64 ·····ZZ[h·]64 ·····ZZ[h·]
65 ·········165 ·········1
66 o3·:·------66 o3·:·------
67 ········567 ········5
68 ·······h68 ·······h
69 ········1</code></pre>69 ········1</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,InputIsSmooth=>true)72 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,InputIsSmooth=>true)
73 ·--·used·0.0832932s·(cpu);·0.0453238s·(thread);·0s·(gc)73 ·--·used·0.120881s·(cpu);·0.0615456s·(thread);·0s·(gc)
  
74 ·······374 ·······3
75 o4·=·4h75 o4·=·4h
76 ·······176 ·······1
  
77 ·····ZZ[h·]77 ·····ZZ[h·]
78 ·········178 ·········1
Offset 89, 15 lines modifiedOffset 89, 15 lines modified
89 ········</table>89 ········</table>
90 ········<div>90 ········<div>
91 ··········<p>Note·that·one·could,·equivalently,·use·the·command·<a·title="The·Chern·class"·href="___Chern.html">Chern</a>·instead·in·this·case.</p>91 ··········<p>Note·that·one·could,·equivalently,·use·the·command·<a·title="The·Chern·class"·href="___Chern.html">Chern</a>·instead·in·this·case.</p>
92 ········</div>92 ········</div>
93 ········<table·class="examples">93 ········<table·class="examples">
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Chern·I95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Chern·I
96 ·--·used·0.0359046s·(cpu);·0.0300469s·(thread);·0s·(gc)96 ·--·used·0.0359137s·(cpu);·0.037355s·(thread);·0s·(gc)
  
97 ·······397 ·······3
98 o5·=·4h98 o5·=·4h
99 ·······199 ·······1
  
100 ·····ZZ[h·]100 ·····ZZ[h·]
101 ·········1101 ·········1
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9 input·ideal·is·known·to·define·a·smooth·subscheme·setting·this·option·to·true9 input·ideal·is·known·to·define·a·smooth·subscheme·setting·this·option·to·true
10 will·speed·up·computations·(it·is·set·to·false·by·default).10 will·speed·up·computations·(it·is·set·to·false·by·default).
11 i1·:·R·=·ZZ/32749[x_0..x_4];11 i1·:·R·=·ZZ/32749[x_0..x_4];
12 i2·:·I=ideal(random(2,R),random(2,R),random(1,R));12 i2·:·I=ideal(random(2,R),random(2,R),random(1,R));
  
13 o2·:·Ideal·of·R13 o2·:·Ideal·of·R
14 i3·:·time·CSM·I14 i3·:·time·CSM·I
15 ·--·used·0.795258s·(cpu);·0.433427s·(thread);·0s·(gc)15 ·--·used·1.28971s·(cpu);·0.515449s·(thread);·0s·(gc)
  
16 ·······316 ·······3
17 o3·=·4h17 o3·=·4h
18 ·······118 ·······1
  
19 ·····ZZ[h·]19 ·····ZZ[h·]
20 ·········120 ·········1
21 o3·:·------21 o3·:·------
22 ········522 ········5
23 ·······h23 ·······h
24 ········124 ········1
25 i4·:·time·CSM(I,InputIsSmooth=>true)25 i4·:·time·CSM(I,InputIsSmooth=>true)
26 ·--·used·0.0832932s·(cpu);·0.0453238s·(thread);·0s·(gc)26 ·--·used·0.120881s·(cpu);·0.0615456s·(thread);·0s·(gc)
  
27 ·······327 ·······3
28 o4·=·4h28 o4·=·4h
29 ·······129 ·······1
  
30 ·····ZZ[h·]30 ·····ZZ[h·]
31 ·········131 ·········1
32 o4·:·------32 o4·:·------
33 ········533 ········5
34 ·······h34 ·······h
35 ········135 ········1
36 Note·that·one·could,·equivalently,·use·the·command·_\x8C_\x8h_\x8e_\x8r_\x8n·instead·in·this·case.36 Note·that·one·could,·equivalently,·use·the·command·_\x8C_\x8h_\x8e_\x8r_\x8n·instead·in·this·case.
37 i5·:·time·Chern·I37 i5·:·time·Chern·I
38 ·--·used·0.0359046s·(cpu);·0.0300469s·(thread);·0s·(gc)38 ·--·used·0.0359137s·(cpu);·0.037355s·(thread);·0s·(gc)
  
39 ·······339 ·······3
40 o5·=·4h40 o5·=·4h
41 ·······141 ·······1
  
42 ·····ZZ[h·]42 ·····ZZ[h·]
43 ·········143 ·········1
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./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
    
Offset 58, 30 lines modifiedOffset 58, 30 lines modified
58 ··········<tr>58 ··········<tr>
59 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);59 <td>··············<pre><code·class="language-macaulay2">i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
  
60 o2·:·Ideal·of·R</code></pre>60 o2·:·Ideal·of·R</code></pre>
61 </td>··········</tr>61 </td>··········</tr>
62 ··········<tr>62 ··········<tr>
63 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·I63 <td>··············<pre><code·class="language-macaulay2">i3·:·time·CSM·I
64 ·--·used·1.485s·(cpu);·0.82601s·(thread);·0s·(gc)64 ·--·used·2.82384s·(cpu);·0.928525s·(thread);·0s·(gc)
  
65 ········5······4·····365 ········5······4·····3
66 o3·=·12h··+·10h··+·6h66 o3·=·12h··+·10h··+·6h
67 ········1······1·····167 ········1······1·····1
  
68 ·····ZZ[h·]68 ·····ZZ[h·]
69 ·········169 ·········1
70 o3·:·------70 o3·:·------
71 ········771 ········7
72 ·······h72 ·······h
73 ········1</code></pre>73 ········1</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,Method=>DirectCompleteInt)76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·CSM(I,Method=>DirectCompleteInt)
77 ·--·used·0.456841s·(cpu);·0.193811s·(thread);·0s·(gc)77 ·--·used·0.847444s·(cpu);·0.231496s·(thread);·0s·(gc)
  
78 ········5······4·····378 ········5······4·····3
79 o4·=·12h··+·10h··+·6h79 o4·=·12h··+·10h··+·6h
80 ········1······1·····180 ········1······1·····1
  
81 ·····ZZ[h·]81 ·····ZZ[h·]
82 ·········182 ·········1
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html2text {}
    
Offset 18, 28 lines modifiedOffset 18, 28 lines modified
18 o1·=·R18 o1·=·R
  
19 o1·:·PolynomialRing19 o1·:·PolynomialRing
20 i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);20 i2·:·I=ideal(random(2,R),random(1,R),R_0*R_1*R_6-R_0^3);
  
21 o2·:·Ideal·of·R21 o2·:·Ideal·of·R
22 i3·:·time·CSM·I22 i3·:·time·CSM·I
23 ·--·used·1.485s·(cpu);·0.82601s·(thread);·0s·(gc)23 ·--·used·2.82384s·(cpu);·0.928525s·(thread);·0s·(gc)
  
24 ········5······4·····324 ········5······4·····3
25 o3·=·12h··+·10h··+·6h25 o3·=·12h··+·10h··+·6h
26 ········1······1·····126 ········1······1·····1
  
27 ·····ZZ[h·]27 ·····ZZ[h·]
28 ·········128 ·········1
29 o3·:·------29 o3·:·------
30 ········730 ········7
31 ·······h31 ·······h
32 ········132 ········1
33 i4·:·time·CSM(I,Method=>DirectCompleteInt)33 i4·:·time·CSM(I,Method=>DirectCompleteInt)
34 ·--·used·0.456841s·(cpu);·0.193811s·(thread);·0s·(gc)34 ·--·used·0.847444s·(cpu);·0.231496s·(thread);·0s·(gc)
  
35 ········5······4·····335 ········5······4·····3
36 o4·=·12h··+·10h··+·6h36 o4·=·12h··+·10h··+·6h
37 ········1······1·····137 ········1······1·····1
  
38 ·····ZZ[h·]38 ·····ZZ[h·]
39 ·········139 ·········1
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./usr/share/doc/Macaulay2/Chordal/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=187 #:len=18
8 UmluZ01hcCBDaG9yZGFsTmV08 UmluZ01hcCBDaG9yZGFsTmV0
9 #:len=14249 #:len=1424
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgcmluZyBtYXAgdG8gYSBjaG9y10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgcmluZyBtYXAgdG8gYSBjaG9y
11 ZGFsIG5ldHdvcmsiLCAibGluZW51bSIgPT4gODg5LCBJbnB1dHMgPT4ge1NQQU57VFR7ImYifSwi11 ZGFsIG5ldHdvcmsiLCAibGluZW51bSIgPT4gODg5LCBJbnB1dHMgPT4ge1NQQU57VFR7ImYifSwi
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./usr/share/doc/Macaulay2/Chordal/example-output/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.out
    
Offset 16, 32 lines modifiedOffset 16, 32 lines modified
  
16 o2·:·Digraph16 o2·:·Digraph
  
17 i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};17 i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
  
18 i4·:·tree·=·elimTree·G18 i4·:·tree·=·elimTree·G
  
19 o4·=·ElimTree{a·=>·c}19 o4·=·ElimTree{a·=>·b···}
20 ··············b·=>·c20 ··············b·=>·c
21 ··············c·=>·d21 ··············c·=>·d
22 ··············d·=>·b22 ··············d·=>·null
  
23 o4·:·ElimTree23 o4·:·ElimTree
  
24 i5·:·rnk·=·hashTable{"a0"=>a,·"a1"=>a,·"b0"=>b,·"b1"=>b,·"b2"=>b,24 i5·:·rnk·=·hashTable{"a0"=>a,·"a1"=>a,·"b0"=>b,·"b1"=>b,·"b2"=>b,
25 ·····················"c0"=>c,·"d0"=>d,·"c1"=>c,·"d1"=>d};25 ·····················"c0"=>c,·"d0"=>d,·"c1"=>c,·"d1"=>d};
  
26 i6·:·eqs·=·hashTable{"a0"·=>·({a},{}),·"a1"·=>·({},{}),26 i6·:·eqs·=·hashTable{"a0"·=>·({a},{}),·"a1"·=>·({},{}),
27 ·····················"b0"·=>·({b},{}),·"b1"·=>·({},{}),·"b2"·=>·({b},{}),27 ·····················"b0"·=>·({b},{}),·"b1"·=>·({},{}),·"b2"·=>·({b},{}),
28 ·····················"c0"·=>·({c},{}),·"c1"·=>·({},{}),28 ·····················"c0"·=>·({c},{}),·"c1"·=>·({},{}),
29 ·····················"d0"·=>·({},{}),·"d1"·=>·({d},{})·};29 ·····················"d0"·=>·({},{}),·"d1"·=>·({d},{})·};
  
30 i7·:·chordalNet(eqs,rnk,tree,DG)30 i7·:·chordalNet(eqs,rnk,tree,DG)
  
31 o7·=·ChordalNet{·a·=>·{a,··}····}31 o7·=·ChordalNet{·d·=>·{·,·d}····}
32 ·················c·=>·{·,·c} 
33 ·················d·=>·{·,·d} 
34 ·················b·=>·{b,··,·b}32 ·················b·=>·{b,··,·b}
 33 ·················a·=>·{a,··}
 34 ·················c·=>·{·,·c}
  
35 o7·:·ChordalNet35 o7·:·ChordalNet
  
36 i8·:·36 i8·:·
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./usr/share/doc/Macaulay2/Chordal/example-output/_components_lp__Chordal__Net_cm__Z__Z_rp.out
    
Offset 27, 14 lines modifiedOffset 27, 14 lines modified
27 o5·:·HashTable27 o5·:·HashTable
  
28 i6·:·components(N)28 i6·:·components(N)
  
29 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}················································································································································································}29 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}················································································································································································}
30 ·····························1···3···5···7···········2···4···6···830 ·····························1···3···5···7···········2···4···6···8
31 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·)}31 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·)}
32 ·····························1···3···4···6···7···········1···2···4···6···7···········1···2···4···5···7···········2···3···5···6···8···········1···3···5···6···8···········1···3···4···6···8···········2···4···5···7···8···········2···3···5···7···832 ·····························1···2···4···5···7···········1···3···4···6···7···········1···2···4···6···7···········2···3···5···7···8···········2···4···5···7···8···········2···3···5···6···8···········1···3···5···6···8···········1···3···4···6···8
33 ···············6·=>·{}33 ···············6·=>·{}
34 ···············7·=>·{}34 ···············7·=>·{}
  
35 o6·:·HashTable35 o6·:·HashTable
  
36 i7·:·36 i7·:·
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./usr/share/doc/Macaulay2/Chordal/html/_chordal__Net_lp__Hash__Table_cm__Hash__Table_cm__Elim__Tree_cm__Digraph_rp.html
    
Offset 99, 18 lines modifiedOffset 99, 18 lines modified
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};</code></pre>101 <td>··············<pre><code·class="language-macaulay2">i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i4·:·tree·=·elimTree·G104 <td>··············<pre><code·class="language-macaulay2">i4·:·tree·=·elimTree·G
  
105 o4·=·ElimTree{a·=>·c}105 o4·=·ElimTree{a·=>·b···}
106 ··············b·=>·c106 ··············b·=>·c
107 ··············c·=>·d107 ··············c·=>·d
108 ··············d·=>·b108 ··············d·=>·null
  
109 o4·:·ElimTree</code></pre>109 o4·:·ElimTree</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i5·:·rnk·=·hashTable{&quot;a0&quot;=>a,·&quot;a1&quot;=>a,·&quot;b0&quot;=>b,·&quot;b1&quot;=>b,·&quot;b2&quot;=>b,112 <td>··············<pre><code·class="language-macaulay2">i5·:·rnk·=·hashTable{&quot;a0&quot;=>a,·&quot;a1&quot;=>a,·&quot;b0&quot;=>b,·&quot;b1&quot;=>b,·&quot;b2&quot;=>b,
113 ·····················&quot;c0&quot;=>c,·&quot;d0&quot;=>d,·&quot;c1&quot;=>c,·&quot;d1&quot;=>d};</code></pre>113 ·····················&quot;c0&quot;=>c,·&quot;d0&quot;=>d,·&quot;c1&quot;=>c,·&quot;d1&quot;=>d};</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
Offset 119, 18 lines modifiedOffset 119, 18 lines modified
119 ·····················&quot;b0&quot;·=>·({b},{}),·&quot;b1&quot;·=>·({},{}),·&quot;b2&quot;·=>·({b},{}),119 ·····················&quot;b0&quot;·=>·({b},{}),·&quot;b1&quot;·=>·({},{}),·&quot;b2&quot;·=>·({b},{}),
120 ·····················&quot;c0&quot;·=>·({c},{}),·&quot;c1&quot;·=>·({},{}),120 ·····················&quot;c0&quot;·=>·({c},{}),·&quot;c1&quot;·=>·({},{}),
121 ·····················&quot;d0&quot;·=>·({},{}),·&quot;d1&quot;·=>·({d},{})·};</code></pre>121 ·····················&quot;d0&quot;·=>·({},{}),·&quot;d1&quot;·=>·({d},{})·};</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i7·:·chordalNet(eqs,rnk,tree,DG)124 <td>··············<pre><code·class="language-macaulay2">i7·:·chordalNet(eqs,rnk,tree,DG)
  
125 o7·=·ChordalNet{·a·=>·{a,··}····}125 o7·=·ChordalNet{·d·=>·{·,·d}····}
126 ·················c·=>·{·,·c} 
127 ·················d·=>·{·,·d} 
128 ·················b·=>·{b,··,·b}126 ·················b·=>·{b,··,·b}
 127 ·················a·=>·{a,··}
 128 ·················c·=>·{·,·c}
  
129 o7·:·ChordalNet</code></pre>129 o7·:·ChordalNet</code></pre>
130 </td>··········</tr>130 </td>··········</tr>
131 ········</table>131 ········</table>
132 ········<pre></pre>132 ········<pre></pre>
133 ······</div>133 ······</div>
134 ······<div>134 ······<div>
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Offset 33, 32 lines modifiedOffset 33, 32 lines modified
33 ·············d0·=>·{}33 ·············d0·=>·{}
34 ·············d1·=>·{}34 ·············d1·=>·{}
  
35 o2·:·Digraph35 o2·:·Digraph
36 i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};36 i3·:·G·=·chordalGraph·digraph·hashTable{a=>{b,c},b=>{c},c=>{d},d=>{}};
37 i4·:·tree·=·elimTree·G37 i4·:·tree·=·elimTree·G
  
38 o4·=·ElimTree{a·=>·c}38 o4·=·ElimTree{a·=>·b···}
39 ··············b·=>·c39 ··············b·=>·c
40 ··············c·=>·d40 ··············c·=>·d
41 ··············d·=>·b41 ··············d·=>·null
  
42 o4·:·ElimTree42 o4·:·ElimTree
43 i5·:·rnk·=·hashTable{"a0"=>a,·"a1"=>a,·"b0"=>b,·"b1"=>b,·"b2"=>b,43 i5·:·rnk·=·hashTable{"a0"=>a,·"a1"=>a,·"b0"=>b,·"b1"=>b,·"b2"=>b,
44 ·····················"c0"=>c,·"d0"=>d,·"c1"=>c,·"d1"=>d};44 ·····················"c0"=>c,·"d0"=>d,·"c1"=>c,·"d1"=>d};
45 i6·:·eqs·=·hashTable{"a0"·=>·({a},{}),·"a1"·=>·({},{}),45 i6·:·eqs·=·hashTable{"a0"·=>·({a},{}),·"a1"·=>·({},{}),
46 ·····················"b0"·=>·({b},{}),·"b1"·=>·({},{}),·"b2"·=>·({b},{}),46 ·····················"b0"·=>·({b},{}),·"b1"·=>·({},{}),·"b2"·=>·({b},{}),
47 ·····················"c0"·=>·({c},{}),·"c1"·=>·({},{}),47 ·····················"c0"·=>·({c},{}),·"c1"·=>·({},{}),
48 ·····················"d0"·=>·({},{}),·"d1"·=>·({d},{})·};48 ·····················"d0"·=>·({},{}),·"d1"·=>·({d},{})·};
49 i7·:·chordalNet(eqs,rnk,tree,DG)49 i7·:·chordalNet(eqs,rnk,tree,DG)
  
50 o7·=·ChordalNet{·a·=>·{a,··}····}50 o7·=·ChordalNet{·d·=>·{·,·d}····}
51 ·················c·=>·{·,·c} 
52 ·················d·=>·{·,·d} 
53 ·················b·=>·{b,··,·b}51 ·················b·=>·{b,··,·b}
 52 ·················a·=>·{a,··}
 53 ·················c·=>·{·,·c}
  
54 o7·:·ChordalNet54 o7·:·ChordalNet
55 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*55 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
56 ····*·_\x8d_\x8i_\x8s_\x8p_\x8l_\x8a_\x8y_\x8N_\x8e_\x8t·--·displays·a·chordal·network·using·Graphivz56 ····*·_\x8d_\x8i_\x8s_\x8p_\x8l_\x8a_\x8y_\x8N_\x8e_\x8t·--·displays·a·chordal·network·using·Graphivz
57 ····*·_\x8d_\x8i_\x8g_\x8r_\x8a_\x8p_\x8h_\x8(_\x8C_\x8h_\x8o_\x8r_\x8d_\x8a_\x8l_\x8N_\x8e_\x8t_\x8)·--·digraph·associated·to·a·chordal·network57 ····*·_\x8d_\x8i_\x8g_\x8r_\x8a_\x8p_\x8h_\x8(_\x8C_\x8h_\x8o_\x8r_\x8d_\x8a_\x8l_\x8N_\x8e_\x8t_\x8)·--·digraph·associated·to·a·chordal·network
58 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*58 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
59 ····*·_\x8c_\x8h_\x8o_\x8r_\x8d_\x8a_\x8l_\x8N_\x8e_\x8t_\x8(_\x8H_\x8a_\x8s_\x8h_\x8T_\x8a_\x8b_\x8l_\x8e_\x8,_\x8H_\x8a_\x8s_\x8h_\x8T_\x8a_\x8b_\x8l_\x8e_\x8,_\x8E_\x8l_\x8i_\x8m_\x8T_\x8r_\x8e_\x8e_\x8,_\x8D_\x8i_\x8g_\x8r_\x8a_\x8p_\x8h_\x8)·--·construct·chordal59 ····*·_\x8c_\x8h_\x8o_\x8r_\x8d_\x8a_\x8l_\x8N_\x8e_\x8t_\x8(_\x8H_\x8a_\x8s_\x8h_\x8T_\x8a_\x8b_\x8l_\x8e_\x8,_\x8H_\x8a_\x8s_\x8h_\x8T_\x8a_\x8b_\x8l_\x8e_\x8,_\x8E_\x8l_\x8i_\x8m_\x8T_\x8r_\x8e_\x8e_\x8,_\x8D_\x8i_\x8g_\x8r_\x8a_\x8p_\x8h_\x8)·--·construct·chordal
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./usr/share/doc/Macaulay2/Chordal/html/_components_lp__Chordal__Net_cm__Z__Z_rp.html
    
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109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i6·:·components(N)111 <td>··············<pre><code·class="language-macaulay2">i6·:·components(N)
  
112 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}················································································································································································}112 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}················································································································································································}
113 ·····························1···3···5···7···········2···4···6···8113 ·····························1···3···5···7···········2···4···6···8
114 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·)}114 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·)}
115 ·····························1···3···4···6···7···········1···2···4···6···7···········1···2···4···5···7···········2···3···5···6···8···········1···3···5···6···8···········1···3···4···6···8···········2···4···5···7···8···········2···3···5···7···8115 ·····························1···2···4···5···7···········1···3···4···6···7···········1···2···4···6···7···········2···3···5···7···8···········2···4···5···7···8···········2···3···5···6···8···········1···3···5···6···8···········1···3···4···6···8
116 ···············6·=>·{}116 ···············6·=>·{}
117 ···············7·=>·{}117 ···············7·=>·{}
  
118 o6·:·HashTable</code></pre>118 o6·:·HashTable</code></pre>
119 </td>··········</tr>119 </td>··········</tr>
120 ········</table>120 ········</table>
121 ········<pre></pre>121 ········<pre></pre>
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Offset 44, 17 lines modifiedOffset 44, 17 lines modified
44 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}44 o6·=·HashTable{4·=>·{ideal·(x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·)}
45 }45 }
46 ·····························1···3···5···7···········2···4···6···846 ·····························1···3···5···7···········2···4···6···8
47 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),47 ···············5·=>·{ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),
48 ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,48 ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,
49 x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,49 x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,·x·,·x·),·ideal·(x·,·x·,·x·,
50 x·,·x·)}50 x·,·x·)}
51 ·····························1···3···4···6···7···········1···2···4···6···751 ·····························1···2···4···5···7···········1···3···4···6···7
52 1···2···4···5···7···········2···3···5···6···8···········1···3···5···6···852 1···2···4···6···7···········2···3···5···7···8···········2···4···5···7···8
53 1···3···4···6···8···········2···4···5···7···8···········2···3···5···7···853 2···3···5···6···8···········1···3···5···6···8···········1···3···4···6···8
54 ···············6·=>·{}54 ···············6·=>·{}
55 ···············7·=>·{}55 ···············7·=>·{}
  
56 o6·:·HashTable56 o6·:·HashTable
57 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*57 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
58 It·is·assumed·that·the·_\x8c_\x8h_\x8a_\x8i_\x8n_\x8s·of·the·network·define·prime·ideals.58 It·is·assumed·that·the·_\x8c_\x8h_\x8a_\x8i_\x8n_\x8s·of·the·network·define·prime·ideals.
59 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*59 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/Classic/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=217 #:len=21
8 bW9ub21pYWxJZGVhbChTdHJpbmcp8 bW9ub21pYWxJZGVhbChTdHJpbmcp
9 #:len=11499 #:len=1149
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibWFrZSBhIG1vbm9taWFsIGlkZWFsIHVz10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibWFrZSBhIG1vbm9taWFsIGlkZWFsIHVz
11 aW5nIGNsYXNzaWMgTWFjYXVsYXkgc3ludGF4IiwgImxpbmVudW0iID0+IDE0NCwgSW5wdXRzID0+11 aW5nIGNsYXNzaWMgTWFjYXVsYXkgc3ludGF4IiwgImxpbmVudW0iID0+IDE0NCwgSW5wdXRzID0+
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./usr/share/doc/Macaulay2/CodingTheory/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=147 #:len=14
8 VmFuaXNoaW5nSWRlYWw=8 VmFuaXNoaW5nSWRlYWw=
9 #:len=12379 #:len=1237
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV210 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV2
11 YWx1YXRpb24gY29kZSIsICJsaW5lbnVtIiA9PiA1MTU4LCBJbnB1dHMgPT4ge1NQQU57VFR7IkMi11 YWx1YXRpb24gY29kZSIsICJsaW5lbnVtIiA9PiA1MTU4LCBJbnB1dHMgPT4ge1NQQU57VFR7IkMi
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./usr/share/doc/Macaulay2/CodingTheory/example-output/_hamming__Code.out
    
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2 i1·:·C1·=·hammingCode(2,3);2 i1·:·C1·=·hammingCode(2,3);
  
3 i2·:·C1.ParityCheckMatrix3 i2·:·C1.ParityCheckMatrix
  
4 o2·=·|·1·1·1·1·0·0·0·|4 o2·=·|·1·1·1·1·0·0·0·|
5 ·····|·0·1·0·1·1·1·0·|5 ·····|·0·1·0·1·1·1·0·|
6 ·····|·1·0·0·1·1·0·1·|6 ·····|·0·1·1·0·0·1·1·|
  
7 ··················3···········77 ··················3···········7
8 o2·:·Matrix·(GF·2)··<--·(GF·2)8 o2·:·Matrix·(GF·2)··<--·(GF·2)
  
9 i3·:·9 i3·:·
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./usr/share/doc/Macaulay2/CodingTheory/example-output/_messages.out
    
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2 i1·:·F=GF(4,Variable=>a);2 i1·:·F=GF(4,Variable=>a);
  
3 i2·:·R=linearCode(F,{{1,1,1}});3 i2·:·R=linearCode(F,{{1,1,1}});
  
4 i3·:·messages·R4 i3·:·messages·R
  
5 o3·=·{{a},·{1},·{a·+·1},·{0}}5 o3·=·{{1},·{a},·{a·+·1},·{0}}
  
6 o3·:·List6 o3·:·List
  
7 i4·:·messages·hammingCode(2,3)7 i4·:·messages·hammingCode(2,3)
  
8 o4·=·{{1,·0,·0,·0},·{1,·0,·0,·1},·{1,·0,·1,·0},·{1,·0,·1,·1},·{1,·1,·1,·0},8 o4·=·{{1,·0,·0,·0},·{1,·0,·0,·1},·{1,·0,·1,·0},·{1,·0,·1,·1},·{1,·1,·1,·0},
9 ·····------------------------------------------------------------------------9 ·····------------------------------------------------------------------------
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47 ···············|·0·|·=>·|·0·|47 ···············|·0·|·=>·|·0·|
48 ···············|·1·|····|·0·|48 ···············|·1·|····|·0·|
49 ···············|·1·|····|·0·|49 ···············|·1·|····|·0·|
50 ························|·0·|50 ························|·0·|
51 ························|·1·|51 ························|·1·|
52 ························|·0·|52 ························|·0·|
53 ························|·0·|53 ························|·0·|
54 ···············|·1·|·=>·|·1·|54 ···············|·1·|·=>·|·0·|
55 ···············|·0·|····|·0·| 
56 ···············|·0·|····|·0·|55 ···············|·0·|····|·0·|
 56 ···············|·0·|····|·1·|
57 ························|·0·|57 ························|·0·|
58 ························|·0·|58 ························|·0·|
59 ························|·0·|59 ························|·0·|
60 ························|·0·|60 ························|·0·|
61 ···············|·1·|·=>·|·0·|61 ···············|·1·|·=>·|·1·|
62 ···············|·0·|····|·1·|62 ···············|·0·|····|·0·|
63 ···············|·1·|····|·0·|63 ···············|·1·|····|·0·|
64 ························|·0·|64 ························|·0·|
65 ························|·0·|65 ························|·0·|
66 ························|·0·|66 ························|·0·|
67 ························|·0·|67 ························|·0·|
68 ···············|·1·|·=>·|·0·|68 ···············|·1·|·=>·|·0·|
69 ···············|·1·|····|·0·|69 ···············|·1·|····|·1·|
70 ···············|·0·|····|·1·|70 ···············|·0·|····|·0·|
71 ························|·0·|71 ························|·0·|
72 ························|·0·|72 ························|·0·|
73 ························|·0·|73 ························|·0·|
74 ························|·0·|74 ························|·0·|
75 ···············|·1·|·=>·|·0·|75 ···············|·1·|·=>·|·0·|
76 ···············|·1·|····|·0·|76 ···············|·1·|····|·0·|
77 ···············|·1·|····|·0·|77 ···············|·1·|····|·0·|
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77 <td>··············<pre><code·class="language-macaulay2">i1·:·C1·=·hammingCode(2,3);</code></pre>77 <td>··············<pre><code·class="language-macaulay2">i1·:·C1·=·hammingCode(2,3);</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i2·:·C1.ParityCheckMatrix80 <td>··············<pre><code·class="language-macaulay2">i2·:·C1.ParityCheckMatrix
  
81 o2·=·|·1·1·1·1·0·0·0·|81 o2·=·|·1·1·1·1·0·0·0·|
82 ·····|·0·1·0·1·1·1·0·|82 ·····|·0·1·0·1·1·1·0·|
83 ·····|·1·0·0·1·1·0·1·|83 ·····|·0·1·1·0·0·1·1·|
  
84 ··················3···········784 ··················3···········7
85 o2·:·Matrix·(GF·2)··&lt;--·(GF·2)</code></pre>85 o2·:·Matrix·(GF·2)··&lt;--·(GF·2)</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ········</table>87 ········</table>
88 ······</div>88 ······</div>
89 ······<div·class="waystouse">89 ······<div·class="waystouse">
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15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 Returns·the·Hamming·code·$C$·over·GF(q)·whose·dual·has·dimension·s.16 Returns·the·Hamming·code·$C$·over·GF(q)·whose·dual·has·dimension·s.
17 i1·:·C1·=·hammingCode(2,3);17 i1·:·C1·=·hammingCode(2,3);
18 i2·:·C1.ParityCheckMatrix18 i2·:·C1.ParityCheckMatrix
  
19 o2·=·|·1·1·1·1·0·0·0·|19 o2·=·|·1·1·1·1·0·0·0·|
20 ·····|·0·1·0·1·1·1·0·|20 ·····|·0·1·0·1·1·1·0·|
21 ·····|·1·0·0·1·1·0·1·|21 ·····|·0·1·1·0·0·1·1·|
  
22 ··················3···········722 ··················3···········7
23 o2·:·Matrix·(GF·2)··<--·(GF·2)23 o2·:·Matrix·(GF·2)··<--·(GF·2)
24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ha\x8am\x8mm\x8mi\x8in\x8ng\x8gC\x8Co\x8od\x8de\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ha\x8am\x8mm\x8mi\x8in\x8ng\x8gC\x8Co\x8od\x8de\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*
25 ····*·hammingCode(ZZ,ZZ)25 ····*·hammingCode(ZZ,ZZ)
26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
27 The·object·_\x8h_\x8a_\x8m_\x8m_\x8i_\x8n_\x8g_\x8C_\x8o_\x8d_\x8e·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.27 The·object·_\x8h_\x8a_\x8m_\x8m_\x8i_\x8n_\x8g_\x8C_\x8o_\x8d_\x8e·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·R=linearCode(F,{{1,1,1}});</code></pre>78 <td>··············<pre><code·class="language-macaulay2">i2·:·R=linearCode(F,{{1,1,1}});</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·messages·R81 <td>··············<pre><code·class="language-macaulay2">i3·:·messages·R
  
82 o3·=·{{a},·{1},·{a·+·1},·{0}}82 o3·=·{{1},·{a},·{a·+·1},·{0}}
  
83 o3·:·List</code></pre>83 o3·:·List</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ········</table>85 ········</table>
86 ········<table·class="examples">86 ········<table·class="examples">
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·messages·hammingCode(2,3)88 <td>··············<pre><code·class="language-macaulay2">i4·:·messages·hammingCode(2,3)
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Offset 15, 15 lines modifiedOffset 15, 15 lines modified
15 Given·a·code·C·of·dimension·$k$·over·a·finite·field·$F$,·this·function·returns15 Given·a·code·C·of·dimension·$k$·over·a·finite·field·$F$,·this·function·returns
16 the·list·that·contains·all·the·elements·of·$F^k$.·Every·element·of·the·list·can16 the·list·that·contains·all·the·elements·of·$F^k$.·Every·element·of·the·list·can
17 be·used·to·encode·a·message·using·the·linear·code·C.17 be·used·to·encode·a·message·using·the·linear·code·C.
18 i1·:·F=GF(4,Variable=>a);18 i1·:·F=GF(4,Variable=>a);
19 i2·:·R=linearCode(F,{{1,1,1}});19 i2·:·R=linearCode(F,{{1,1,1}});
20 i3·:·messages·R20 i3·:·messages·R
  
21 o3·=·{{a},·{1},·{a·+·1},·{0}}21 o3·=·{{1},·{a},·{a·+·1},·{0}}
  
22 o3·:·List22 o3·:·List
23 i4·:·messages·hammingCode(2,3)23 i4·:·messages·hammingCode(2,3)
  
24 o4·=·{{1,·0,·0,·0},·{1,·0,·0,·1},·{1,·0,·1,·0},·{1,·0,·1,·1},·{1,·1,·1,·0},24 o4·=·{{1,·0,·0,·0},·{1,·0,·0,·1},·{1,·0,·1,·0},·{1,·0,·1,·1},·{1,·1,·1,·0},
25 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
26 ·····{1,·1,·1,·1},·{0,·1,·1,·0},·{0,·1,·0,·1},·{0,·1,·0,·0},·{0,·1,·1,·1},26 ·····{1,·1,·1,·1},·{0,·1,·1,·0},·{0,·1,·0,·1},·{0,·1,·0,·0},·{0,·1,·1,·1},
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
    
Offset 184, 15 lines modifiedOffset 184, 15 lines modified
184 ······{0,·-1,·0,·0,·0,·-1},·{0,·0,·-1,·0,·0,·-1},·{0,·0,·0,·-1,·0,·-1},·{0,184 ······{0,·-1,·0,·0,·0,·-1},·{0,·0,·-1,·0,·0,·-1},·{0,·0,·0,·-1,·0,·-1},·{0,
185 ······-----------------------------------------------------------------------185 ······-----------------------------------------------------------------------
186 ······0,·0,·0,·-1,·-1}}186 ······0,·0,·0,·-1,·-1}}
  
187 o19·:·List187 o19·:·List
  
188 i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)188 i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)
189 ·--·3.75056s·elapsed189 ·--·8.88343s·elapsed
  
190 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},190 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},
191 ······-----------------------------------------------------------------------191 ······-----------------------------------------------------------------------
192 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,192 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,
193 ······-----------------------------------------------------------------------193 ······-----------------------------------------------------------------------
194 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,194 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,
195 ······-----------------------------------------------------------------------195 ······-----------------------------------------------------------------------
Offset 265, 45 lines modifiedOffset 265, 45 lines modified
265 i22·:·degree(X_3·+·X_7·+·X_8)265 i22·:·degree(X_3·+·X_7·+·X_8)
  
266 o22·=·{0,·0,·1,·2,·0,·-1}266 o22·=·{0,·0,·1,·2,·0,·-1}
  
267 o22·:·List267 o22·:·List
  
268 i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)268 i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)
269 ·--·.427203s·elapsed269 ·--·1.05383s·elapsed
  
270 o23·=·{1,·0,·0,·0,·0}270 o23·=·{1,·0,·0,·0,·0}
  
271 o23·:·List271 o23·:·List
  
272 i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,0,-1))272 i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,0,-1))
273 ·--·171.554s·elapsed273 ·--·133.472s·elapsed
  
274 o24·=·{1,·0,·0,·0,·0}274 o24·=·{1,·0,·0,·0,·0}
  
275 o24·:·List275 o24·:·List
  
276 i25·:·assert(cohomvec1·==·cohomvec2)276 i25·:·assert(cohomvec1·==·cohomvec2)
  
277 i26·:·degree(X_3·+·X_7·-·X_8)277 i26·:·degree(X_3·+·X_7·-·X_8)
  
278 o26·=·{0,·0,·1,·2,·-2,·-1}278 o26·=·{0,·0,·1,·2,·-2,·-1}
  
279 o26·:·List279 o26·:·List
  
280 i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)280 i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)
281 ·--·.677634s·elapsed281 ·--·.632905s·elapsed
  
282 o27·=·{0,·0,·0,·0,·0}282 o27·=·{0,·0,·0,·0,·0}
  
283 o27·:·List283 o27·:·List
  
284 i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,-2,-1))284 i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,-2,-1))
285 ·--·.553347s·elapsed285 ·--·.434069s·elapsed
286 ·--·.553395s·elapsed286 ·--·.434115s·elapsed
  
287 o28·=·{0,·0,·0,·0,·0}287 o28·=·{0,·0,·0,·0,·0}
  
288 o28·:·List288 o28·:·List
  
289 i29·:·assert(cohomvec1·==·cohomvec2)289 i29·:·assert(cohomvec1·==·cohomvec2)
  
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263 ······-----------------------------------------------------------------------263 ······-----------------------------------------------------------------------
264 ······0,·0,·0,·-1,·-1}}264 ······0,·0,·0,·-1,·-1}}
  
265 o19·:·List</code></pre>265 o19·:·List</code></pre>
266 </td>··········</tr>266 </td>··········</tr>
267 ··········<tr>267 ··········<tr>
268 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)268 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)
269 ·--·3.75056s·elapsed269 ·--·8.88343s·elapsed
  
270 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},270 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},
271 ······-----------------------------------------------------------------------271 ······-----------------------------------------------------------------------
272 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,272 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,
273 ······-----------------------------------------------------------------------273 ······-----------------------------------------------------------------------
274 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,274 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,
275 ······-----------------------------------------------------------------------275 ······-----------------------------------------------------------------------
Offset 347, 23 lines modifiedOffset 347, 23 lines modified
  
347 o22·=·{0,·0,·1,·2,·0,·-1}347 o22·=·{0,·0,·1,·2,·0,·-1}
  
348 o22·:·List</code></pre>348 o22·:·List</code></pre>
349 </td>··········</tr>349 </td>··········</tr>
350 ··········<tr>350 ··········<tr>
351 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)351 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)
352 ·--·.427203s·elapsed352 ·--·1.05383s·elapsed
  
353 o23·=·{1,·0,·0,·0,·0}353 o23·=·{1,·0,·0,·0,·0}
  
354 o23·:·List</code></pre>354 o23·:·List</code></pre>
355 </td>··········</tr>355 </td>··········</tr>
356 ··········<tr>356 ··········<tr>
357 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,0,-1))357 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,0,-1))
358 ·--·171.554s·elapsed358 ·--·133.472s·elapsed
  
359 o24·=·{1,·0,·0,·0,·0}359 o24·=·{1,·0,·0,·0,·0}
  
360 o24·:·List</code></pre>360 o24·:·List</code></pre>
361 </td>··········</tr>361 </td>··········</tr>
362 ··········<tr>362 ··········<tr>
363 <td>··············<pre><code·class="language-macaulay2">i25·:·assert(cohomvec1·==·cohomvec2)</code></pre>363 <td>··············<pre><code·class="language-macaulay2">i25·:·assert(cohomvec1·==·cohomvec2)</code></pre>
Offset 373, 24 lines modifiedOffset 373, 24 lines modified
  
373 o26·=·{0,·0,·1,·2,·-2,·-1}373 o26·=·{0,·0,·1,·2,·-2,·-1}
  
374 o26·:·List</code></pre>374 o26·:·List</code></pre>
375 </td>··········</tr>375 </td>··········</tr>
376 ··········<tr>376 ··········<tr>
377 <td>··············<pre><code·class="language-macaulay2">i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)377 <td>··············<pre><code·class="language-macaulay2">i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)
378 ·--·.677634s·elapsed378 ·--·.632905s·elapsed
  
379 o27·=·{0,·0,·0,·0,·0}379 o27·=·{0,·0,·0,·0,·0}
  
380 o27·:·List</code></pre>380 o27·:·List</code></pre>
381 </td>··········</tr>381 </td>··········</tr>
382 ··········<tr>382 ··········<tr>
383 <td>··············<pre><code·class="language-macaulay2">i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,-2,-1))383 <td>··············<pre><code·class="language-macaulay2">i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X(0,0,1,2,-2,-1))
384 ·--·.553347s·elapsed384 ·--·.434069s·elapsed
385 ·--·.553395s·elapsed385 ·--·.434115s·elapsed
  
386 o28·=·{0,·0,·0,·0,·0}386 o28·=·{0,·0,·0,·0,·0}
  
387 o28·:·List</code></pre>387 o28·:·List</code></pre>
388 </td>··········</tr>388 </td>··········</tr>
389 ··········<tr>389 ··········<tr>
390 <td>··············<pre><code·class="language-macaulay2">i29·:·assert(cohomvec1·==·cohomvec2)</code></pre>390 <td>··············<pre><code·class="language-macaulay2">i29·:·assert(cohomvec1·==·cohomvec2)</code></pre>
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2.02 KB
html2text {}
    
Offset 182, 15 lines modifiedOffset 182, 15 lines modified
182 ······-----------------------------------------------------------------------182 ······-----------------------------------------------------------------------
183 ······{0,·-1,·0,·0,·0,·-1},·{0,·0,·-1,·0,·0,·-1},·{0,·0,·0,·-1,·0,·-1},·{0,183 ······{0,·-1,·0,·0,·0,·-1},·{0,·0,·-1,·0,·0,·-1},·{0,·0,·0,·-1,·0,·-1},·{0,
184 ······-----------------------------------------------------------------------184 ······-----------------------------------------------------------------------
185 ······0,·0,·0,·-1,·-1}}185 ······0,·0,·0,·-1,·-1}}
  
186 o19·:·List186 o19·:·List
187 i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)187 i20·:·elapsedTime·hvecs·=·cohomCalg(X,·D2)
188 ·--·3.75056s·elapsed188 ·--·8.88343s·elapsed
  
189 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},189 o20·=·{{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},
190 ······-----------------------------------------------------------------------190 ······-----------------------------------------------------------------------
191 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,191 ······{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·1,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,
192 ······-----------------------------------------------------------------------192 ······-----------------------------------------------------------------------
193 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,193 ······0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,·0,·0,·0},·{0,·0,
194 ······-----------------------------------------------------------------------194 ······-----------------------------------------------------------------------
Offset 262, 42 lines modifiedOffset 262, 42 lines modified
262 ·······················{2,·2,·3,·1,·-4,·-6}·=>·{{0,·1,·0,·0,·0},·{{1,·1x1*x2}}}262 ·······················{2,·2,·3,·1,·-4,·-6}·=>·{{0,·1,·0,·0,·0},·{{1,·1x1*x2}}}
263 i22·:·degree(X_3·+·X_7·+·X_8)263 i22·:·degree(X_3·+·X_7·+·X_8)
  
264 o22·=·{0,·0,·1,·2,·0,·-1}264 o22·=·{0,·0,·1,·2,·0,·-1}
  
265 o22·:·List265 o22·:·List
266 i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)266 i23·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·+·X_8)
267 ·--·.427203s·elapsed267 ·--·1.05383s·elapsed
  
268 o23·=·{1,·0,·0,·0,·0}268 o23·=·{1,·0,·0,·0,·0}
  
269 o23·:·List269 o23·:·List
270 i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X270 i24·:·elapsedTime·cohomvec2·=·for·j·from·0·to·dim·X·list·rank·HH^j(X,·OO_X
271 (0,0,1,2,0,-1))271 (0,0,1,2,0,-1))
272 ·--·171.554s·elapsed272 ·--·133.472s·elapsed
  
273 o24·=·{1,·0,·0,·0,·0}273 o24·=·{1,·0,·0,·0,·0}
  
274 o24·:·List274 o24·:·List
275 i25·:·assert(cohomvec1·==·cohomvec2)275 i25·:·assert(cohomvec1·==·cohomvec2)
276 i26·:·degree(X_3·+·X_7·-·X_8)276 i26·:·degree(X_3·+·X_7·-·X_8)
  
277 o26·=·{0,·0,·1,·2,·-2,·-1}277 o26·=·{0,·0,·1,·2,·-2,·-1}
  
278 o26·:·List278 o26·:·List
279 i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)279 i27·:·elapsedTime·cohomvec1·=·cohomCalg(X_3·+·X_7·-·X_8)
280 ·--·.677634s·elapsed280 ·--·.632905s·elapsed
  
281 o27·=·{0,·0,·0,·0,·0}281 o27·=·{0,·0,·0,·0,·0}
  
282 o27·:·List282 o27·:·List
283 i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j283 i28·:·elapsedTime·cohomvec2·=·elapsedTime·for·j·from·0·to·dim·X·list·rank·HH^j
284 (X,·OO_X(0,0,1,2,-2,-1))284 (X,·OO_X(0,0,1,2,-2,-1))
285 ·--·.553347s·elapsed285 ·--·.434069s·elapsed
286 ·--·.553395s·elapsed286 ·--·.434115s·elapsed
  
287 o28·=·{0,·0,·0,·0,·0}287 o28·=·{0,·0,·0,·0,·0}
  
288 o28·:·List288 o28·:·List
289 i29·:·assert(cohomvec1·==·cohomvec2)289 i29·:·assert(cohomvec1·==·cohomvec2)
290 _\x8c_\x8o_\x8h_\x8o_\x8m_\x8C_\x8a_\x8l_\x8g·computes·cohomology·vectors·by·calling·CohomCalg.·It·also·stashes290 _\x8c_\x8o_\x8h_\x8o_\x8m_\x8C_\x8a_\x8l_\x8g·computes·cohomology·vectors·by·calling·CohomCalg.·It·also·stashes
291 it's·results·in·the·toric·variety's·cache·table,·so·computations·need·not·be291 it's·results·in·the·toric·variety's·cache·table,·so·computations·need·not·be
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753 B
./usr/share/doc/Macaulay2/CoincidentRootLoci/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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6 #·End·of·header6 #·End·of·header
7 #:len=247 #:len=24
8 Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp8 Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp
9 #:len=2979 #:len=297
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs
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787 B
./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=287 #:len=28
8 RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==8 RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==
9 #:len=3499 #:len=349
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2NSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2NSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRWlzZW5idWRTaGFtYXNoVG90YWwsTW9kdWxlKSwi11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRWlzZW5idWRTaGFtYXNoVG90YWwsTW9kdWxlKSwi
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2.72 KB
./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out
    
Offset 35, 15 lines modifiedOffset 35, 15 lines modified
35 o5·:·QuotientRing35 o5·:·QuotientRing
  
36 i6·:·len·=·1036 i6·:·len·=·10
  
37 o6·=·1037 o6·=·10
  
38 i7·:·time·G·=·EisenbudShamash(ff,F,len)38 i7·:·time·G·=·EisenbudShamash(ff,F,len)
39 ·--·used·7.17212s·(cpu);·4.93034s·(thread);·0s·(gc)39 ·--·used·7.98961s·(cpu);·4.96509s·(thread);·0s·(gc)
  
40 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S···\28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/····S···\68·····/····S···\7640 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S···\28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/····S···\68·····/····S···\76
41 o7·=·|--------|··<--·|--------|··<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|41 o7·=·|--------|··<--·|--------|··<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|
42 ·····|··2···3·|······|··2···3·|······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|42 ·····|··2···3·|······|··2···3·|······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|
43 ·····|(x·,·x·)|······|(x·,·x·)|······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|43 ·····|(x·,·x·)|······|(x·,·x·)|······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|
44 ·····\··0···1·/······\··0···1·/······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/44 ·····\··0···1·/······\··0···1·/······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/
45 ··············································································································································································45 ··············································································································································································
Offset 140, 37 lines modifiedOffset 140, 37 lines modified
140 i19·:·R1·=·R/ideal·ff140 i19·:·R1·=·R/ideal·ff
  
141 o19·=·R1141 o19·=·R1
  
142 o19·:·QuotientRing142 o19·:·QuotientRing
  
143 i20·:·FF·=·time·Shamash(R1,F,4)143 i20·:·FF·=·time·Shamash(R1,F,4)
144 ·--·used·0.155454s·(cpu);·0.0883096s·(thread);·0s·(gc)144 ·--·used·0.171796s·(cpu);·0.0848261s·(thread);·0s·(gc)
  
145 ········1·······6·······18·······38·······66145 ········1·······6·······18·······38·······66
146 o20·=·R1··<--·R1··<--·R1···<--·R1···<--·R1146 o20·=·R1··<--·R1··<--·R1···<--·R1···<--·R1
147 ·········································147 ·········································
148 ······0·······1·······2········3········4148 ······0·······1·······2········3········4
  
149 o20·:·ChainComplex149 o20·:·ChainComplex
  
150 i21·:·GG·=·time·EisenbudShamash(ff,F,4)150 i21·:·GG·=·time·EisenbudShamash(ff,F,4)
151 ·--·used·1.04644s·(cpu);·0.710844s·(thread);·0s·(gc)151 ·--·used·1.22823s·(cpu);·0.740067s·(thread);·0s·(gc)
  
152 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66152 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66
153 o21·=·|--|··<--·|--|··<--·|--|···<--·|--|···<--·|--|153 o21·=·|--|··<--·|--|··<--·|--|···<--·|--|···<--·|--|
154 ······|·3|······|·3|······|·3|·······|·3|·······|·3|154 ······|·3|······|·3|······|·3|·······|·3|·······|·3|
155 ······\c·/······\c·/······\c·/·······\c·/·······\c·/155 ······\c·/······\c·/······\c·/·······\c·/·······\c·/
156 ·················································156 ·················································
157 ······0·········1·········2··········3··········4157 ······0·········1·········2··········3··········4
  
158 o21·:·ChainComplex158 o21·:·ChainComplex
  
159 i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)159 i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)
160 ·--·used·1.11465s·(cpu);·0.770192s·(thread);·0s·(gc)160 ·--·used·1.31332s·(cpu);·0.763836s·(thread);·0s·(gc)
  
161 ········1·······6·······18·······38·······66161 ········1·······6·······18·······38·······66
162 o22·=·R1··<--·R1··<--·R1···<--·R1···<--·R1162 o22·=·R1··<--·R1··<--·R1···<--·R1···<--·R1
163 ·········································163 ·········································
164 ······-2······-1······0········1········2164 ······-2······-1······0········1········2
  
165 o22·:·ChainComplex165 o22·:·ChainComplex
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731 B
./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out
    
Offset 1, 17 lines modifiedOffset 1, 17 lines modified
1 --·-*-·M2-comint·-*-·hash:·17317413654323116141 --·-*-·M2-comint·-*-·hash:·1731741365432311614
  
2 i1·:·setRandomSeed·02 i1·:·setRandomSeed·0
  
3 o1·=·03 o1·=·0
  
4 i2·:·sumTwoMonomials(2,3)4 i2·:·sumTwoMonomials(2,3)
5 ·--·used·1.30279s·(cpu);·0.430089s·(thread);·0s·(gc)5 ·--·used·1.2017s·(cpu);·0.417163s·(thread);·0s·(gc)
6 ·--·used·0.452189s·(cpu);·0.147033s·(thread);·0s·(gc)6 ·--·used·0.414563s·(cpu);·0.129049s·(thread);·0s·(gc)
7 ·--·used·0.000128484s·(cpu);·1.041e-06s·(thread);·0s·(gc)7 ·--·used·0.000169227s·(cpu);·2.946e-06s·(thread);·0s·(gc)
8 28 2
9 Tally{{{2,·2},·{1,·2}}·=>·3}9 Tally{{{2,·2},·{1,·2}}·=>·3}
  
10 310 3
11 Tally{{{2,·2},·{1,·2}}·=>·1}11 Tally{{{2,·2},·{1,·2}}·=>·1}
  
12 412 4
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Offset 1, 23 lines modifiedOffset 1, 23 lines modified
1 --·-*-·M2-comint·-*-·hash:·17317413668843597531 --·-*-·M2-comint·-*-·hash:·1731741366884359753
  
2 i1·:·setRandomSeed·02 i1·:·setRandomSeed·0
  
3 o1·=·03 o1·=·0
  
4 i2·:·twoMonomials(2,3)4 i2·:·twoMonomials(2,3)
5 ·--·used·2.30661s·(cpu);·0.709914s·(thread);·0s·(gc)5 ·--·used·2.22133s·(cpu);·0.786163s·(thread);·0s·(gc)
6 26 2
7 Tally{{{1,·1}}·=>·2········}7 Tally{{{1,·1}}·=>·2········}
8 ······{{2,·2},·{1,·2}}·=>·48 ······{{2,·2},·{1,·2}}·=>·4
  
9 ·--·used·1.22706s·(cpu);·0.417905s·(thread);·0s·(gc)9 ·--·used·1.21473s·(cpu);·0.442702s·(thread);·0s·(gc)
10 310 3
11 Tally{{{2,·2},·{1,·2}}·=>·2}11 Tally{{{2,·2},·{1,·2}}·=>·2}
12 ······{{3,·3},·{2,·3}}·=>·112 ······{{3,·3},·{2,·3}}·=>·1
  
13 ·--·used·0.428494s·(cpu);·0.143748s·(thread);·0s·(gc)13 ·--·used·0.464983s·(cpu);·0.173951s·(thread);·0s·(gc)
14 414 4
15 Tally{{{2,·2},·{1,·2}}·=>·1}15 Tally{{{2,·2},·{1,·2}}·=>·1}
  
  
16 i3·:·16 i3·:·
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Offset 121, 15 lines modifiedOffset 121, 15 lines modified
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i6·:·len·=·10122 <td>··············<pre><code·class="language-macaulay2">i6·:·len·=·10
  
123 o6·=·10</code></pre>123 o6·=·10</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i7·:·time·G·=·EisenbudShamash(ff,F,len)126 <td>··············<pre><code·class="language-macaulay2">i7·:·time·G·=·EisenbudShamash(ff,F,len)
127 ·--·used·7.17212s·(cpu);·4.93034s·(thread);·0s·(gc)127 ·--·used·7.98961s·(cpu);·4.96509s·(thread);·0s·(gc)
  
128 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S···\28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/····S···\68·····/····S···\76128 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S···\28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/····S···\68·····/····S···\76
129 o7·=·|--------|··&lt;--·|--------|··&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|129 o7·=·|--------|··&lt;--·|--------|··&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|···&lt;--·|--------|
130 ·····|··2···3·|······|··2···3·|······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|130 ·····|··2···3·|······|··2···3·|······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|·······|··2···3·|
131 ·····|(x·,·x·)|······|(x·,·x·)|······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|131 ·····|(x·,·x·)|······|(x·,·x·)|······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|·······|(x·,·x·)|
132 ·····\··0···1·/······\··0···1·/······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/132 ·····\··0···1·/······\··0···1·/······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/·······\··0···1·/
133 ··············································································································································································133 ··············································································································································································
Offset 259, 26 lines modifiedOffset 259, 26 lines modified
  
259 o19·=·R1259 o19·=·R1
  
260 o19·:·QuotientRing</code></pre>260 o19·:·QuotientRing</code></pre>
261 </td>··········</tr>261 </td>··········</tr>
262 ··········<tr>262 ··········<tr>
263 <td>··············<pre><code·class="language-macaulay2">i20·:·FF·=·time·Shamash(R1,F,4)263 <td>··············<pre><code·class="language-macaulay2">i20·:·FF·=·time·Shamash(R1,F,4)
264 ·--·used·0.155454s·(cpu);·0.0883096s·(thread);·0s·(gc)264 ·--·used·0.171796s·(cpu);·0.0848261s·(thread);·0s·(gc)
  
265 ········1·······6·······18·······38·······66265 ········1·······6·······18·······38·······66
266 o20·=·R1··&lt;--·R1··&lt;--·R1···&lt;--·R1···&lt;--·R1266 o20·=·R1··&lt;--·R1··&lt;--·R1···&lt;--·R1···&lt;--·R1
267 ·········································267 ·········································
268 ······0·······1·······2········3········4268 ······0·······1·······2········3········4
  
269 o20·:·ChainComplex</code></pre>269 o20·:·ChainComplex</code></pre>
270 </td>··········</tr>270 </td>··········</tr>
271 ··········<tr>271 ··········<tr>
272 <td>··············<pre><code·class="language-macaulay2">i21·:·GG·=·time·EisenbudShamash(ff,F,4)272 <td>··············<pre><code·class="language-macaulay2">i21·:·GG·=·time·EisenbudShamash(ff,F,4)
273 ·--·used·1.04644s·(cpu);·0.710844s·(thread);·0s·(gc)273 ·--·used·1.22823s·(cpu);·0.740067s·(thread);·0s·(gc)
  
274 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66274 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66
275 o21·=·|--|··&lt;--·|--|··&lt;--·|--|···&lt;--·|--|···&lt;--·|--|275 o21·=·|--|··&lt;--·|--|··&lt;--·|--|···&lt;--·|--|···&lt;--·|--|
276 ······|·3|······|·3|······|·3|·······|·3|·······|·3|276 ······|·3|······|·3|······|·3|·······|·3|·······|·3|
277 ······\c·/······\c·/······\c·/·······\c·/·······\c·/277 ······\c·/······\c·/······\c·/·······\c·/·······\c·/
278 ·················································278 ·················································
279 ······0·········1·········2··········3··········4279 ······0·········1·········2··········3··········4
Offset 288, 15 lines modifiedOffset 288, 15 lines modified
288 ········</table>288 ········</table>
289 ········<div>289 ········<div>
290 ··········<p>The·function·also·deals·correctly·with·complexes·F·where·min·F·is·not·0:</p>290 ··········<p>The·function·also·deals·correctly·with·complexes·F·where·min·F·is·not·0:</p>
291 ········</div>291 ········</div>
292 ········<table·class="examples">292 ········<table·class="examples">
293 ··········<tr>293 ··········<tr>
294 <td>··············<pre><code·class="language-macaulay2">i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)294 <td>··············<pre><code·class="language-macaulay2">i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)
295 ·--·used·1.11465s·(cpu);·0.770192s·(thread);·0s·(gc)295 ·--·used·1.31332s·(cpu);·0.763836s·(thread);·0s·(gc)
  
296 ········1·······6·······18·······38·······66296 ········1·······6·······18·······38·······66
297 o22·=·R1··&lt;--·R1··&lt;--·R1···&lt;--·R1···&lt;--·R1297 o22·=·R1··&lt;--·R1··&lt;--·R1···&lt;--·R1···&lt;--·R1
298 ·········································298 ·········································
299 ······-2······-1······0········1········2299 ······-2······-1······0········1········2
  
300 o22·:·ChainComplex</code></pre>300 o22·:·ChainComplex</code></pre>
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Offset 51, 15 lines modifiedOffset 51, 15 lines modified
51 o5·=·R51 o5·=·R
  
52 o5·:·QuotientRing52 o5·:·QuotientRing
53 i6·:·len·=·1053 i6·:·len·=·10
  
54 o6·=·1054 o6·=·10
55 i7·:·time·G·=·EisenbudShamash(ff,F,len)55 i7·:·time·G·=·EisenbudShamash(ff,F,len)
56 ·--·used·7.17212s·(cpu);·4.93034s·(thread);·0s·(gc)56 ·--·used·7.98961s·(cpu);·4.96509s·(thread);·0s·(gc)
  
57 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S57 ·····/····S···\1·····/····S···\5·····/····S···\12·····/····S···\20·····/····S
58 \28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/58 \28·····/····S···\36·····/····S···\44·····/····S···\52·····/····S···\60·····/
59 S···\68·····/····S···\7659 S···\68·····/····S···\76
60 o7·=·|--------|··<--·|--------|··<--·|--------|···<--·|--------|···<--·|-------60 o7·=·|--------|··<--·|--------|··<--·|--------|···<--·|--------|···<--·|-------
61 -|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|-61 -|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|--------|···<--·|-
62 -------|···<--·|--------|62 -------|···<--·|--------|
Offset 167, 36 lines modifiedOffset 167, 36 lines modified
167 o18·:·Matrix·R··<--·R167 o18·:·Matrix·R··<--·R
168 i19·:·R1·=·R/ideal·ff168 i19·:·R1·=·R/ideal·ff
  
169 o19·=·R1169 o19·=·R1
  
170 o19·:·QuotientRing170 o19·:·QuotientRing
171 i20·:·FF·=·time·Shamash(R1,F,4)171 i20·:·FF·=·time·Shamash(R1,F,4)
172 ·--·used·0.155454s·(cpu);·0.0883096s·(thread);·0s·(gc)172 ·--·used·0.171796s·(cpu);·0.0848261s·(thread);·0s·(gc)
  
173 ········1·······6·······18·······38·······66173 ········1·······6·······18·······38·······66
174 o20·=·R1··<--·R1··<--·R1···<--·R1···<--·R1174 o20·=·R1··<--·R1··<--·R1···<--·R1···<--·R1
  
175 ······0·······1·······2········3········4175 ······0·······1·······2········3········4
  
176 o20·:·ChainComplex176 o20·:·ChainComplex
177 i21·:·GG·=·time·EisenbudShamash(ff,F,4)177 i21·:·GG·=·time·EisenbudShamash(ff,F,4)
178 ·--·used·1.04644s·(cpu);·0.710844s·(thread);·0s·(gc)178 ·--·used·1.22823s·(cpu);·0.740067s·(thread);·0s·(gc)
  
179 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66179 ······/·R\1·····/·R\6·····/·R\18·····/·R\38·····/·R\66
180 o21·=·|--|··<--·|--|··<--·|--|···<--·|--|···<--·|--|180 o21·=·|--|··<--·|--|··<--·|--|···<--·|--|···<--·|--|
181 ······|·3|······|·3|······|·3|·······|·3|·······|·3|181 ······|·3|······|·3|······|·3|·······|·3|·······|·3|
182 ······\c·/······\c·/······\c·/·······\c·/·······\c·/182 ······\c·/······\c·/······\c·/·······\c·/·······\c·/
  
183 ······0·········1·········2··········3··········4183 ······0·········1·········2··········3··········4
  
184 o21·:·ChainComplex184 o21·:·ChainComplex
185 The·function·also·deals·correctly·with·complexes·F·where·min·F·is·not·0:185 The·function·also·deals·correctly·with·complexes·F·where·min·F·is·not·0:
186 i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)186 i22·:·GG·=·time·EisenbudShamash(R1,F[2],4)
187 ·--·used·1.11465s·(cpu);·0.770192s·(thread);·0s·(gc)187 ·--·used·1.31332s·(cpu);·0.763836s·(thread);·0s·(gc)
  
188 ········1·······6·······18·······38·······66188 ········1·······6·······18·······38·······66
189 o22·=·R1··<--·R1··<--·R1···<--·R1···<--·R1189 o22·=·R1··<--·R1··<--·R1···<--·R1···<--·R1
  
190 ······-2······-1······0········1········2190 ······-2······-1······0········1········2
  
191 o22·:·ChainComplex191 o22·:·ChainComplex
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed·077 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed·0
  
78 o1·=·0</code></pre>78 o1·=·0</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·sumTwoMonomials(2,3)81 <td>··············<pre><code·class="language-macaulay2">i2·:·sumTwoMonomials(2,3)
82 ·--·used·1.30279s·(cpu);·0.430089s·(thread);·0s·(gc)82 ·--·used·1.2017s·(cpu);·0.417163s·(thread);·0s·(gc)
83 ·--·used·0.452189s·(cpu);·0.147033s·(thread);·0s·(gc)83 ·--·used·0.414563s·(cpu);·0.129049s·(thread);·0s·(gc)
84 ·--·used·0.000128484s·(cpu);·1.041e-06s·(thread);·0s·(gc)84 ·--·used·0.000169227s·(cpu);·2.946e-06s·(thread);·0s·(gc)
85 285 2
86 Tally{{{2,·2},·{1,·2}}·=>·3}86 Tally{{{2,·2},·{1,·2}}·=>·3}
  
87 387 3
88 Tally{{{2,·2},·{1,·2}}·=>·1}88 Tally{{{2,·2},·{1,·2}}·=>·1}
  
89 489 4
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18 =·S/(d-th·powers·of·the·variables),·with·full·complexity·(=c);·that·is,·for·an18 =·S/(d-th·powers·of·the·variables),·with·full·complexity·(=c);·that·is,·for·an
19 appropriate·syzygy·M·of·M0·=·R/(m1+m2)·where·m1·and·m2·are·monomials·of·the19 appropriate·syzygy·M·of·M0·=·R/(m1+m2)·where·m1·and·m2·are·monomials·of·the
20 same·degree.20 same·degree.
21 i1·:·setRandomSeed·021 i1·:·setRandomSeed·0
  
22 o1·=·022 o1·=·0
23 i2·:·sumTwoMonomials(2,3)23 i2·:·sumTwoMonomials(2,3)
24 ·--·used·1.30279s·(cpu);·0.430089s·(thread);·0s·(gc)24 ·--·used·1.2017s·(cpu);·0.417163s·(thread);·0s·(gc)
25 ·--·used·0.452189s·(cpu);·0.147033s·(thread);·0s·(gc)25 ·--·used·0.414563s·(cpu);·0.129049s·(thread);·0s·(gc)
26 ·--·used·0.000128484s·(cpu);·1.041e-06s·(thread);·0s·(gc)26 ·--·used·0.000169227s·(cpu);·2.946e-06s·(thread);·0s·(gc)
27 227 2
28 Tally{{{2,·2},·{1,·2}}·=>·3}28 Tally{{{2,·2},·{1,·2}}·=>·3}
  
29 329 3
30 Tally{{{2,·2},·{1,·2}}·=>·1}30 Tally{{{2,·2},·{1,·2}}·=>·1}
  
31 431 4
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Offset 82, 25 lines modifiedOffset 82, 25 lines modified
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed·083 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed·0
  
84 o1·=·0</code></pre>84 o1·=·0</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i2·:·twoMonomials(2,3)87 <td>··············<pre><code·class="language-macaulay2">i2·:·twoMonomials(2,3)
88 ·--·used·2.30661s·(cpu);·0.709914s·(thread);·0s·(gc)88 ·--·used·2.22133s·(cpu);·0.786163s·(thread);·0s·(gc)
89 289 2
90 Tally{{{1,·1}}·=>·2········}90 Tally{{{1,·1}}·=>·2········}
91 ······{{2,·2},·{1,·2}}·=>·491 ······{{2,·2},·{1,·2}}·=>·4
  
92 ·--·used·1.22706s·(cpu);·0.417905s·(thread);·0s·(gc)92 ·--·used·1.21473s·(cpu);·0.442702s·(thread);·0s·(gc)
93 393 3
94 Tally{{{2,·2},·{1,·2}}·=>·2}94 Tally{{{2,·2},·{1,·2}}·=>·2}
95 ······{{3,·3},·{2,·3}}·=>·195 ······{{3,·3},·{2,·3}}·=>·1
  
96 ·--·used·0.428494s·(cpu);·0.143748s·(thread);·0s·(gc)96 ·--·used·0.464983s·(cpu);·0.173951s·(thread);·0s·(gc)
97 497 4
98 Tally{{{2,·2},·{1,·2}}·=>·1}</code></pre>98 Tally{{{2,·2},·{1,·2}}·=>·1}</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ········</table>100 ········</table>
101 ······</div>101 ······</div>
102 ······<div>102 ······<div>
103 ········<h2>See·also</h2>103 ········<h2>See·also</h2>
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html2text {}
    
Offset 20, 25 lines modifiedOffset 20, 25 lines modified
20 monomials·in·R·=·S/(d-th·powers·of·the·variables),·with·full·complexity·(=c);20 monomials·in·R·=·S/(d-th·powers·of·the·variables),·with·full·complexity·(=c);
21 that·is,·for·an·appropriate·syzygy·M·of·M0·=·R/(m1,·m2)·where·m1·and·m2·are21 that·is,·for·an·appropriate·syzygy·M·of·M0·=·R/(m1,·m2)·where·m1·and·m2·are
22 monomials·of·the·same·degree.22 monomials·of·the·same·degree.
23 i1·:·setRandomSeed·023 i1·:·setRandomSeed·0
  
24 o1·=·024 o1·=·0
25 i2·:·twoMonomials(2,3)25 i2·:·twoMonomials(2,3)
26 ·--·used·2.30661s·(cpu);·0.709914s·(thread);·0s·(gc)26 ·--·used·2.22133s·(cpu);·0.786163s·(thread);·0s·(gc)
27 227 2
28 Tally{{{1,·1}}·=>·2········}28 Tally{{{1,·1}}·=>·2········}
29 ······{{2,·2},·{1,·2}}·=>·429 ······{{2,·2},·{1,·2}}·=>·4
  
30 ·--·used·1.22706s·(cpu);·0.417905s·(thread);·0s·(gc)30 ·--·used·1.21473s·(cpu);·0.442702s·(thread);·0s·(gc)
31 331 3
32 Tally{{{2,·2},·{1,·2}}·=>·2}32 Tally{{{2,·2},·{1,·2}}·=>·2}
33 ······{{3,·3},·{2,·3}}·=>·133 ······{{3,·3},·{2,·3}}·=>·1
  
34 ·--·used·0.428494s·(cpu);·0.143748s·(thread);·0s·(gc)34 ·--·used·0.464983s·(cpu);·0.173951s·(thread);·0s·(gc)
35 435 4
36 Tally{{{2,·2},·{1,·2}}·=>·1}36 Tally{{{2,·2},·{1,·2}}·=>·1}
37 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
38 ····*·_\x8t_\x8w_\x8o_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8s·--·tally·the·sequences·of·BRanks·for·certain·examples38 ····*·_\x8t_\x8w_\x8o_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8s·--·tally·the·sequences·of·BRanks·for·certain·examples
39 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8tw\x8wo\x8oM\x8Mo\x8on\x8no\x8om\x8mi\x8ia\x8al\x8ls\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*39 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8tw\x8wo\x8oM\x8Mo\x8on\x8no\x8om\x8mi\x8ia\x8al\x8ls\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
40 ····*·twoMonomials(ZZ,ZZ)40 ····*·twoMonomials(ZZ,ZZ)
41 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/Complexes/dump/rawdocumentation.dump
    
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748 B
./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
    
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759 B
./usr/share/doc/Macaulay2/ConvexInterface/dump/rawdocumentation.dump
    
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744 B
./usr/share/doc/Macaulay2/ConwayPolynomials/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
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799 B
./usr/share/doc/Macaulay2/CorrespondenceScrolls/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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726 B
./usr/share/doc/Macaulay2/CotangentSchubert/dump/rawdocumentation.dump
    
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8 TGFiZWxz8 TGFiZWxz
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728 B
./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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1.66 KB
./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
    
Offset 13, 26 lines modifiedOffset 13, 26 lines modified
13 o2·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)13 o2·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)
14 ···············1····0·2·····1·2····0·3·····2····1·314 ···············1····0·2·····1·2····0·3·····2····1·3
  
15 o2·:·Ideal·of·GF·78125[x·..x·]15 o2·:·Ideal·of·GF·78125[x·..x·]
16 ························0···416 ························0···4
  
17 i3·:·time·ChernSchwartzMacPherson·C17 i3·:·time·ChernSchwartzMacPherson·C
18 ·--·used·1.18342s·(cpu);·0.815881s·(thread);·0s·(gc)18 ·--·used·1.33858s·(cpu);·0.886344s·(thread);·0s·(gc)
  
19 ·······4·····3·····219 ·······4·····3·····2
20 o3·=·3H··+·5H··+·3H20 o3·=·3H··+·5H··+·3H
  
21 ·····ZZ[H]21 ·····ZZ[H]
22 o3·:·-----22 o3·:·-----
23 ········523 ········5
24 ·······H24 ·······H
  
25 i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)25 i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)
26 ·--·used·2.50688s·(cpu);·0.787772s·(thread);·0s·(gc)26 ·--·used·2.28539s·(cpu);·0.809176s·(thread);·0s·(gc)
27 Certify:·output·certified!27 Certify:·output·certified!
  
28 ·······4·····3·····228 ·······4·····3·····2
29 o4·=·3H··+·5H··+·3H29 o4·=·3H··+·5H··+·3H
  
30 ·····ZZ[H]30 ·····ZZ[H]
31 o4·:·-----31 o4·:·-----
Offset 62, 26 lines modifiedOffset 62, 26 lines modified
62 ········0,2·1,3····0,1·2,362 ········0,2·1,3····0,1·2,3
  
63 ················ZZ63 ················ZZ
64 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]64 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]
65 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,465 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4
  
66 i9·:·time·ChernClass·G66 i9·:·time·ChernClass·G
67 ·--·used·0.166395s·(cpu);·0.129677s·(thread);·0s·(gc)67 ·--·used·0.169689s·(cpu);·0.118999s·(thread);·0s·(gc)
  
68 ········9······8······7······6······5······4·····368 ········9······8······7······6······5······4·····3
69 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H69 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
70 ·····ZZ[H]70 ·····ZZ[H]
71 o9·:·-----71 o9·:·-----
72 ·······1072 ·······10
73 ······H73 ······H
  
74 i10·:·time·ChernClass(G,Certify=>true)74 i10·:·time·ChernClass(G,Certify=>true)
75 ·--·used·0.226004s·(cpu);·0.0496908s·(thread);·0s·(gc)75 ·--·used·0.1636s·(cpu);·0.027524s·(thread);·0s·(gc)
76 Certify:·output·certified!76 Certify:·output·certified!
  
77 ·········9······8······7······6······5······4·····377 ·········9······8······7······6······5······4·····3
78 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H78 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
79 ······ZZ[H]79 ······ZZ[H]
80 o10·:·-----80 o10·:·-----
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14.0 KB
./usr/share/doc/Macaulay2/Cremona/example-output/___Cremona.out
    
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1 --·-*-·M2-comint·-*-·hash:·104334092679444218251 --·-*-·M2-comint·-*-·hash:·10433409267944421825
  
2 i1·:·ZZ/300007[t_0..t_6];2 i1·:·ZZ/300007[t_0..t_6];
  
3 i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})3 i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
4 ·--·used·0.00403256s·(cpu);·0.0041249s·(thread);·0s·(gc)4 ·--·used·0.00400105s·(cpu);·0.0034921s·(thread);·0s·(gc)
  
5 ············ZZ··············ZZ················3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····25 ············ZZ··············ZZ················3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····2
6 o2·=·map·(------[t·..t·],·------[x·..x·],·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})6 o2·=·map·(------[t·..t·],·------[x·..x·],·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})
7 ··········300007··0···6···300007··0···9·······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·67 ··········300007··0···6···300007··0···9·······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·6
  
8 ···············ZZ·················ZZ8 ···············ZZ·················ZZ
9 o2·:·RingMap·------[t·..t·]·<--·------[x·..x·]9 o2·:·RingMap·------[t·..t·]·<--·------[x·..x·]
10 ·············300007··0···6······300007··0···910 ·············300007··0···6······300007··0···9
  
11 i3·:·time·J·=·kernel(phi,2)11 i3·:·time·J·=·kernel(phi,2)
12 ·--·used·0.0398989s·(cpu);·0.0391807s·(thread);·0s·(gc)12 ·--·used·0.0383873s·(cpu);·0.0398905s·(thread);·0s·(gc)
  
13 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x·13 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x·
14 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·414 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4
15 ·····------------------------------------------------------------------------15 ·····------------------------------------------------------------------------
16 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)16 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
17 ········1·6····0·8···2·4····1·5····0·717 ········1·6····0·8···2·4····1·5····0·7
  
18 ················ZZ18 ················ZZ
19 o3·:·Ideal·of·------[x·..x·]19 o3·:·Ideal·of·------[x·..x·]
20 ··············300007··0···920 ··············300007··0···9
  
21 i4·:·time·degreeMap·phi21 i4·:·time·degreeMap·phi
22 ·--·used·0.0909701s·(cpu);·0.0436168s·(thread);·0s·(gc)22 ·--·used·0.0857462s·(cpu);·0.0404878s·(thread);·0s·(gc)
  
23 o4·=·123 o4·=·1
  
24 i5·:·time·projectiveDegrees·phi24 i5·:·time·projectiveDegrees·phi
25 ·--·used·0.34797s·(cpu);·0.305061s·(thread);·0s·(gc)25 ·--·used·0.459454s·(cpu);·0.3758s·(thread);·0s·(gc)
  
26 o5·=·{1,·3,·9,·17,·21,·15,·5}26 o5·=·{1,·3,·9,·17,·21,·15,·5}
  
27 o5·:·List27 o5·:·List
  
28 i6·:·time·projectiveDegrees(phi,NumDegrees=>0)28 i6·:·time·projectiveDegrees(phi,NumDegrees=>0)
29 ·--·used·0.0521797s·(cpu);·0.0526681s·(thread);·0s·(gc)29 ·--·used·0.0645226s·(cpu);·0.0651927s·(thread);·0s·(gc)
  
30 o6·=·{5}30 o6·=·{5}
  
31 o6·:·List31 o6·:·List
  
32 i7·:·time·phi·=·toMap(phi,Dominant=>J)32 i7·:·time·phi·=·toMap(phi,Dominant=>J)
33 ·--·used·0.000225799s·(cpu);·0.00189703s·(thread);·0s·(gc)33 ·--·used·0.000414056s·(cpu);·0.00209904s·(thread);·0s·(gc)
  
34 ·······································································ZZ34 ·······································································ZZ
35 ·····································································------[x·..x·]35 ·····································································------[x·..x·]
36 ············ZZ·······················································300007··0···9··················································3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····236 ············ZZ·······················································300007··0···9··················································3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····2
37 o7·=·map·(------[t·..t·],·----------------------------------------------------------------------------------------------------,·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})37 o7·=·map·(------[t·..t·],·----------------------------------------------------------------------------------------------------,·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})
38 ··········300007··0···6···(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·638 ··········300007··0···6···(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·6
39 ····························6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·739 ····························6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
Offset 59, 15 lines modifiedOffset 59, 15 lines modified
59 ···········································································------[x·..x·]59 ···········································································------[x·..x·]
60 ···············ZZ··························································300007··0···960 ···············ZZ··························································300007··0···9
61 o7·:·RingMap·------[t·..t·]·<--·----------------------------------------------------------------------------------------------------61 o7·:·RingMap·------[t·..t·]·<--·----------------------------------------------------------------------------------------------------
62 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)62 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
63 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·763 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
  
64 i8·:·time·psi·=·inverseMap·phi64 i8·:·time·psi·=·inverseMap·phi
65 ·--·used·0.468032s·(cpu);·0.425579s·(thread);·0s·(gc)65 ·--·used·0.439703s·(cpu);·0.39935s·(thread);·0s·(gc)
  
66 ·······················································ZZ66 ·······················································ZZ
67 ·····················································------[x·..x·]67 ·····················································------[x·..x·]
68 ·····················································300007··0···9················································ZZ··············3················2···············2····2························2··························2·····2········2·······························2···································2···············2·············2·······················3·················································2·················2····2··································2····2·················2·················································3·························2······2····2······2··············································268 ·····················································300007··0···9················································ZZ··············3················2···············2····2························2··························2·····2········2·······························2···································2···············2·············2·······················3·················································2·················2····2··································2····2·················2·················································3·························2······2····2······2··············································2
69 o8·=·map·(----------------------------------------------------------------------------------------------------,·------[t·..t·],·{x··-·2x·x·x··+·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x·x··-·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··-·2x·x·,·x·x··-·x·x··-·x·x·x··+·x·x·x··+·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x·x··-·x·x·x··+·x·x··-·x·x·x··+·x·x··-·x·x·x··-·x·x·x·,·x··-·x·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x·,·x·x··-·x·x·x··+·x·x··+·x·x··-·x·x·x··-·x·x·x··-·x·x·x·,·x·x··-·x·x··-·x·x·x··+·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x··-·2x·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··-·x·x·x··-·2x·x·})69 o8·=·map·(----------------------------------------------------------------------------------------------------,·------[t·..t·],·{x··-·2x·x·x··+·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x·x··-·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··-·2x·x·,·x·x··-·x·x··-·x·x·x··+·x·x·x··+·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x·x··-·x·x·x··+·x·x··-·x·x·x··+·x·x··-·x·x·x··-·x·x·x·,·x··-·x·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x·,·x·x··-·x·x·x··+·x·x··+·x·x··-·x·x·x··-·x·x·x··-·x·x·x·,·x·x··-·x·x··-·x·x·x··+·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x··-·2x·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··-·x·x·x··-·2x·x·})
70 ··········(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)··300007··0···6·····2·····1·2·3····0·3····1·2·5····0·5····1·6····0·2·6····0·4·6····1·7·····0·2·7····0·4·7·····0·9···2·3····1·3····1·2·6····0·3·6····0·5·6····1·8·····0·2·8····0·4·8····0·1·9···2·3····1·3·6····0·6····0·3·8····1·9····0·2·9····0·4·9···3····1·3·8····0·6·8····1·2·9·····0·3·9····0·5·9···3·6····2·3·8····0·8····2·9····1·3·9····0·6·9····0·7·9···3·6····3·8····2·6·8····1·8····2·3·9····2·5·9·····1·6·9····1·7·9····0·8·9···6·····3·6·8····5·6·8····2·8····4·8····3·9····5·9····2·6·9·····4·6·9····2·7·9····4·7·9·····0·970 ··········(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)··300007··0···6·····2·····1·2·3····0·3····1·2·5····0·5····1·6····0·2·6····0·4·6····1·7·····0·2·7····0·4·7·····0·9···2·3····1·3····1·2·6····0·3·6····0·5·6····1·8·····0·2·8····0·4·8····0·1·9···2·3····1·3·6····0·6····0·3·8····1·9····0·2·9····0·4·9···3····1·3·8····0·6·8····1·2·9·····0·3·9····0·5·9···3·6····2·3·8····0·8····2·9····1·3·9····0·6·9····0·7·9···3·6····3·8····2·6·8····1·8····2·3·9····2·5·9·····1·6·9····1·7·9····0·8·9···6·····3·6·8····5·6·8····2·8····4·8····3·9····5·9····2·6·9·····4·6·9····2·7·9····4·7·9·····0·9
71 ············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·771 ············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
Offset 76, 32 lines modifiedOffset 76, 32 lines modified
76 ························································------[x·..x·]76 ························································------[x·..x·]
77 ························································300007··0···9···················································ZZ77 ························································300007··0···9···················································ZZ
78 o8·:·RingMap·----------------------------------------------------------------------------------------------------·<--·------[t·..t·]78 o8·:·RingMap·----------------------------------------------------------------------------------------------------·<--·------[t·..t·]
79 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···679 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···6
80 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·780 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
  
81 i9·:·time·isInverseMap(phi,psi)81 i9·:·time·isInverseMap(phi,psi)
82 ·--·used·0.00801057s·(cpu);·0.00771425s·(thread);·0s·(gc)82 ·--·used·0.00783973s·(cpu);·0.0072009s·(thread);·0s·(gc)
  
83 o9·=·true83 o9·=·true
  
84 i10·:·time·degreeMap·psi84 i10·:·time·degreeMap·psi
85 ·--·used·0.201606s·(cpu);·0.15919s·(thread);·0s·(gc)85 ·--·used·0.234928s·(cpu);·0.182881s·(thread);·0s·(gc)
  
86 o10·=·186 o10·=·1
  
87 i11·:·time·projectiveDegrees·psi87 i11·:·time·projectiveDegrees·psi
88 ·--·used·5.09383s·(cpu);·4.83436s·(thread);·0s·(gc)88 ·--·used·5.06976s·(cpu);·4.68993s·(thread);·0s·(gc)
  
89 o11·=·{5,·15,·21,·17,·9,·3,·1}89 o11·=·{5,·15,·21,·17,·9,·3,·1}
  
90 o11·:·List90 o11·:·List
  
91 i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})91 i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
92 ·--·used·0.000721897s·(cpu);·0.00192268s·(thread);·0s·(gc)92 ·--·used·0.00180521s·(cpu);·0.00182693s·(thread);·0s·(gc)
  
93 o12·=·--·rational·map·--93 o12·=·--·rational·map·--
94 ·····················ZZ94 ·····················ZZ
95 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])95 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
96 ···················300007··0···1···2···3···4···5···696 ···················300007··0···1···2···3···4···5···6
97 ·····················ZZ97 ·····················ZZ
98 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])98 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 147, 15 lines modifiedOffset 147, 15 lines modified
147 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t147 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t
148 ··························4·····3·4·5····2·5····3·6····2·4·6148 ··························4·····3·4·5····2·5····3·6····2·4·6
149 ······················}149 ······················}
  
150 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)150 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)
  
151 i13·:·time·phi·=·rationalMap(phi,Dominant=>2)151 i13·:·time·phi·=·rationalMap(phi,Dominant=>2)
152 ·--·used·0.108803s·(cpu);·0.0695419s·(thread);·0s·(gc)152 ·--·used·0.10811s·(cpu);·0.068337s·(thread);·0s·(gc)
  
153 o13·=·--·rational·map·--153 o13·=·--·rational·map·--
154 ·····················ZZ154 ·····················ZZ
155 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])155 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
156 ···················300007··0···1···2···3···4···5···6156 ···················300007··0···1···2···3···4···5···6
157 ···································ZZ157 ···································ZZ
158 ······target:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])·defined·by158 ······target:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])·defined·by
Offset 217, 15 lines modifiedOffset 217, 15 lines modified
217 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t217 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t
218 ··························4·····3·4·5····2·5····3·6····2·4·6218 ··························4·····3·4·5····2·5····3·6····2·4·6
219 ······················}219 ······················}
  
220 o13·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)220 o13·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)
Max diff block lines reached; 3582/14213 bytes (25.20%) of diff not shown.
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775 B
./usr/share/doc/Macaulay2/Cremona/example-output/___Euler__Characteristic.out
    
Offset 3, 18 lines modifiedOffset 3, 18 lines modified
3 i1·:·I·=·Grassmannian(1,4,CoefficientRing=>ZZ/190181);3 i1·:·I·=·Grassmannian(1,4,CoefficientRing=>ZZ/190181);
  
4 ················ZZ4 ················ZZ
5 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]5 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]
6 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,46 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4
  
7 i2·:·time·EulerCharacteristic·I7 i2·:·time·EulerCharacteristic·I
8 ·--·used·0.20939s·(cpu);·0.13637s·(thread);·0s·(gc)8 ·--·used·0.235163s·(cpu);·0.14315s·(thread);·0s·(gc)
  
9 o2·=·109 o2·=·10
  
10 i3·:·time·EulerCharacteristic(I,Certify=>true)10 i3·:·time·EulerCharacteristic(I,Certify=>true)
11 ·--·used·0.242261s·(cpu);·0.0362552s·(thread);·0s·(gc)11 ·--·used·0.123625s·(cpu);·0.020704s·(thread);·0s·(gc)
12 Certify:·output·certified!12 Certify:·output·certified!
  
13 o3·=·1013 o3·=·10
  
14 i4·:·14 i4·:·
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1.01 KB
./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp!.out
    
Offset 8, 15 lines modifiedOffset 8, 15 lines modified
  
8 o3·=·rational·map·defined·by·forms·of·degree·28 o3·=·rational·map·defined·by·forms·of·degree·2
9 ·····source·variety:·PP^59 ·····source·variety:·PP^5
10 ·····target·variety:·PP^510 ·····target·variety:·PP^5
11 ·····coefficient·ring:·QQ11 ·····coefficient·ring:·QQ
  
12 i4·:·time·phi!·;12 i4·:·time·phi!·;
13 ·--·used·0.397259s·(cpu);·0.0971215s·(thread);·0s·(gc)13 ·--·used·0.409125s·(cpu);·0.117687s·(thread);·0s·(gc)
  
14 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))14 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))
  
15 i5·:·describe·phi15 i5·:·describe·phi
  
16 o5·=·rational·map·defined·by·forms·of·degree·216 o5·=·rational·map·defined·by·forms·of·degree·2
17 ·····source·variety:·PP^517 ·····source·variety:·PP^5
Offset 37, 15 lines modifiedOffset 37, 15 lines modified
  
37 o8·=·rational·map·defined·by·forms·of·degree·237 o8·=·rational·map·defined·by·forms·of·degree·2
38 ·····source·variety:·PP^438 ·····source·variety:·PP^4
39 ·····target·variety:·PP^539 ·····target·variety:·PP^5
40 ·····coefficient·ring:·QQ40 ·····coefficient·ring:·QQ
  
41 i9·:·time·phi!·;41 i9·:·time·phi!·;
42 ·--·used·0.269141s·(cpu);·0.0534929s·(thread);·0s·(gc)42 ·--·used·0.321513s·(cpu);·0.0796105s·(thread);·0s·(gc)
  
43 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)43 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)
  
44 i10·:·describe·phi44 i10·:·describe·phi
  
45 o10·=·rational·map·defined·by·forms·of·degree·245 o10·=·rational·map·defined·by·forms·of·degree·2
46 ······source·variety:·PP^446 ······source·variety:·PP^4
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724 B
./usr/share/doc/Macaulay2/Cremona/example-output/___Rational__Map_sp^_st_st_sp__Ideal.out
    
Offset 67, 15 lines modifiedOffset 67, 15 lines modified
67 ·····-·a*c·+·e·········-·b*c·+·f67 ·····-·a*c·+·e·········-·b*c·+·f
68 ·····----------*v,·x·+·----------*v)68 ·····----------*v,·x·+·----------*v)
69 ······d*e·-·a*f·········d*e·-·a*f69 ······d*e·-·a*f·········d*e·-·a*f
  
70 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]70 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]
  
71 i6·:·time·phi^**·q71 i6·:·time·phi^**·q
72 ·--·used·0.172677s·(cpu);·0.134122s·(thread);·0s·(gc)72 ·--·used·0.240581s·(cpu);·0.186416s·(thread);·0s·(gc)
  
73 ················-e········-d········-c········-b········-a73 ················-e········-d········-c········-b········-a
74 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)74 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)
75 ·················f·········f·········f·········f·········f75 ·················f·········f·········f·········f·········f
  
76 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]76 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]
  
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5.0 KB
./usr/share/doc/Macaulay2/Cremona/example-output/___Segre__Class.out
    
Offset 47, 49 lines modifiedOffset 47, 49 lines modified
47 ··········································································P747 ··········································································P7
48 o3·:·Ideal·of·-------------------------------------------------------------------------------------------------------------------------48 o3·:·Ideal·of·-------------------------------------------------------------------------------------------------------------------------
49 ···············2·2················2·2········································2·2····················································2·249 ···············2·2················2·2········································2·2····················································2·2
50 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x··+·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x50 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x··+·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x
51 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····1·6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·751 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····1·6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·7
  
52 i4·:·time·SegreClass·X52 i4·:·time·SegreClass·X
53 ·--·used·0.423006s·(cpu);·0.346287s·(thread);·0s·(gc)53 ·--·used·0.503888s·(cpu);·0.411667s·(thread);·0s·(gc)
  
54 ··········7········6·······5·······4······354 ··········7········6·······5·······4······3
55 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H55 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
56 ·····ZZ[H]56 ·····ZZ[H]
57 o4·:·-----57 o4·:·-----
58 ········858 ········8
59 ·······H59 ·······H
  
60 i5·:·time·SegreClass·lift(X,P7)60 i5·:·time·SegreClass·lift(X,P7)
61 ·--·used·0.253089s·(cpu);·0.216709s·(thread);·0s·(gc)61 ·--·used·0.2606s·(cpu);·0.211402s·(thread);·0s·(gc)
  
62 ··········7········6·······5······4······362 ··········7········6·······5······4······3
63 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H63 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
64 ·····ZZ[H]64 ·····ZZ[H]
65 o5·:·-----65 o5·:·-----
66 ········866 ········8
67 ·······H67 ·······H
  
68 i6·:·time·SegreClass(X,Certify=>true)68 i6·:·time·SegreClass(X,Certify=>true)
69 ·--·used·0.294334s·(cpu);·0.0596635s·(thread);·0s·(gc)69 ·--·used·0.230385s·(cpu);·0.035137s·(thread);·0s·(gc)
70 Certify:·output·certified!70 Certify:·output·certified!
  
71 ··········7········6·······5·······4······371 ··········7········6·······5·······4······3
72 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H72 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
73 ·····ZZ[H]73 ·····ZZ[H]
74 o6·:·-----74 o6·:·-----
75 ········875 ········8
76 ·······H76 ·······H
  
77 i7·:·time·SegreClass(lift(X,P7),Certify=>true)77 i7·:·time·SegreClass(lift(X,P7),Certify=>true)
78 ·--·used·0.38928s·(cpu);·0.138091s·(thread);·0s·(gc)78 ·--·used·0.341144s·(cpu);·0.121427s·(thread);·0s·(gc)
79 Certify:·output·certified!79 Certify:·output·certified!
  
80 ··········7········6·······5······4······380 ··········7········6·······5······4······3
81 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H81 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
82 ·····ZZ[H]82 ·····ZZ[H]
83 o7·:·-----83 o7·:·-----
Offset 105, 15 lines modifiedOffset 105, 15 lines modified
105 ·······ZZ105 ·······ZZ
106 o9·=·------[x·..x·]106 o9·=·------[x·..x·]
107 ·····100003··0···6107 ·····100003··0···6
  
108 o9·:·PolynomialRing108 o9·:·PolynomialRing
  
109 i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)109 i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
110 ·--·used·0.0455862s·(cpu);·0.0477137s·(thread);·0s·(gc)110 ·--·used·0.0599694s·(cpu);·0.0579296s·(thread);·0s·(gc)
  
111 ························································ZZ111 ························································ZZ
112 ······················································------[y·..y·]112 ······················································------[y·..y·]
113 ······················································100003··0···9················································ZZ··············2······························2113 ······················································100003··0···9················································ZZ··············2······························2
114 o10·=·map·(----------------------------------------------------------------------------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y·,·y··-·y·y··-·y·y·,·y·y··+·y·y··-·y·y·,·y·y··-·y·y·,·y·y··-·y·y··-·y·y·,·y·y··-·y·y··-·y·y·})114 o10·=·map·(----------------------------------------------------------------------------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y·,·y··-·y·y··-·y·y·,·y·y··+·y·y··-·y·y·,·y·y··-·y·y·,·y·y··-·y·y··-·y·y·,·y·y··-·y·y··-·y·y·})
115 ···········(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)··100003··0···6·····3····0·5····1·6···3·4····1·7···4····2·7····0·9···2·5····3·5····1·8···4·5····1·9···4·8····2·9····3·9···7·8····4·9····6·9115 ···········(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)··100003··0···6·····3····0·5····1·6···3·4····1·7···4····2·7····0·9···2·5····3·5····1·8···4·5····1·9···4·8····2·9····3·9···7·8····4·9····6·9
116 ·············5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5116 ·············5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5
Offset 122, 15 lines modifiedOffset 122, 15 lines modified
122 ·························································------[y·..y·]122 ·························································------[y·..y·]
123 ·························································100003··0···9···················································ZZ123 ·························································100003··0···9···················································ZZ
124 o10·:·RingMap·----------------------------------------------------------------------------------------------------·<--·------[x·..x·]124 o10·:·RingMap·----------------------------------------------------------------------------------------------------·<--·------[x·..x·]
125 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6125 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6
126 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5126 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5
  
127 i11·:·time·SegreClass·phi127 i11·:·time·SegreClass·phi
128 ·--·used·0.269223s·(cpu);·0.19767s·(thread);·0s·(gc)128 ·--·used·0.305646s·(cpu);·0.212694s·(thread);·0s·(gc)
  
129 ·········9······8······7······6·····5129 ·········9······8······7······6·····5
130 o11·=·23H··-·42H··+·36H··-·22H··+·9H130 o11·=·23H··-·42H··+·36H··-·22H··+·9H
  
131 ······ZZ[H]131 ······ZZ[H]
132 o11·:·-----132 o11·:·-----
133 ········10133 ········10
Offset 150, 27 lines modifiedOffset 150, 27 lines modified
150 ··························································100003··0···9150 ··························································100003··0···9
151 o12·:·Ideal·of·----------------------------------------------------------------------------------------------------151 o12·:·Ideal·of·----------------------------------------------------------------------------------------------------
152 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)152 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
153 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5153 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5
  
154 i13·:·--·Segre·class·of·B·in·G(1,4)154 i13·:·--·Segre·class·of·B·in·G(1,4)
155 ······time·SegreClass·B155 ······time·SegreClass·B
156 ·--·used·0.241335s·(cpu);·0.200347s·(thread);·0s·(gc)156 ·--·used·0.260493s·(cpu);·0.213822s·(thread);·0s·(gc)
  
157 ·········9······8······7······6·····5157 ·········9······8······7······6·····5
158 o13·=·23H··-·42H··+·36H··-·22H··+·9H158 o13·=·23H··-·42H··+·36H··-·22H··+·9H
  
159 ······ZZ[H]159 ······ZZ[H]
160 o13·:·-----160 o13·:·-----
161 ········10161 ········10
162 ·······H162 ·······H
  
163 i14·:·--·Segre·class·of·B·in·P^9163 i14·:·--·Segre·class·of·B·in·P^9
164 ······time·SegreClass·lift(B,ambient·ring·B)164 ······time·SegreClass·lift(B,ambient·ring·B)
165 ·--·used·0.767014s·(cpu);·0.613042s·(thread);·0s·(gc)165 ·--·used·0.883692s·(cpu);·0.687767s·(thread);·0s·(gc)
  
166 ···········9·······8·······7······6·····5166 ···········9·······8·······7······6·····5
167 o14·=·2764H··-·984H··+·294H··-·67H··+·9H167 o14·=·2764H··-·984H··+·294H··-·67H··+·9H
  
168 ······ZZ[H]168 ······ZZ[H]
169 o14·:·-----169 o14·:·-----
170 ········10170 ········10
⊟
3.13 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_abstract__Rational__Map.out
    
Offset 17, 32 lines modifiedOffset 17, 32 lines modified
  
17 o3·=·QQ[u·..u·]17 o3·=·QQ[u·..u·]
18 ·········0···518 ·········0···5
  
19 o3·:·PolynomialRing19 o3·:·PolynomialRing
  
20 i4·:·time·psi·=·abstractRationalMap(P4,P5,f)20 i4·:·time·psi·=·abstractRationalMap(P4,P5,f)
21 ·--·used·0.0039265s·(cpu);·0.000510879s·(thread);·0s·(gc)21 ·--·used·1.6471e-05s·(cpu);·0.000453851s·(thread);·0s·(gc)
  
22 o4·=·--·rational·map·--22 o4·=·--·rational·map·--
23 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])23 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
24 ······················0···1···2···3···424 ······················0···1···2···3···4
25 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])25 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
26 ······················0···1···2···3···4···526 ······················0···1···2···3···4···5
27 ·····defining·forms:·given·by·a·function27 ·····defining·forms:·given·by·a·function
  
28 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)28 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)
  
29 i5·:·time·projectiveDegrees(psi,3)29 i5·:·time·projectiveDegrees(psi,3)
30 ·--·used·0.160223s·(cpu);·0.120764s·(thread);·0s·(gc)30 ·--·used·0.192038s·(cpu);·0.123195s·(thread);·0s·(gc)
  
31 o5·=·231 o5·=·2
  
32 i6·:·time·rationalMap·psi32 i6·:·time·rationalMap·psi
33 ·--·used·0.32006s·(cpu);·0.279947s·(thread);·0s·(gc)33 ·--·used·0.336602s·(cpu);·0.298803s·(thread);·0s·(gc)
  
34 o6·=·--·rational·map·--34 o6·=·--·rational·map·--
35 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])35 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
36 ······················0···1···2···3···436 ······················0···1···2···3···4
37 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])37 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
38 ······················0···1···2···3···4···538 ······················0···1···2···3···4···5
39 ·····defining·forms:·{39 ·····defining·forms:·{
Offset 113, 48 lines modifiedOffset 113, 48 lines modified
113 ················1····0·2·····1·2····0·3·····2····1·3113 ················1····0·2·····1·2····0·3·····2····1·3
  
114 ·················ZZ114 ·················ZZ
115 o13·:·Ideal·of·-----[x·..x·]115 o13·:·Ideal·of·-----[x·..x·]
116 ···············65521··0···3116 ···············65521··0···3
  
117 i14·:·time·T·=·abstractRationalMap(I,"OADP")117 i14·:·time·T·=·abstractRationalMap(I,"OADP")
118 ·--·used·0.0344235s·(cpu);·0.0350582s·(thread);·0s·(gc)118 ·--·used·0.0408175s·(cpu);·0.0442475s·(thread);·0s·(gc)
  
119 o14·=·--·rational·map·--119 o14·=·--·rational·map·--
120 ·····················ZZ120 ·····················ZZ
121 ······source:·Proj(-----[x·,·x·,·x·,·x·])121 ······source:·Proj(-----[x·,·x·,·x·,·x·])
122 ···················65521··0···1···2···3122 ···················65521··0···1···2···3
123 ·····················ZZ123 ·····················ZZ
124 ······target:·Proj(-----[x·,·x·,·x·,·x·])124 ······target:·Proj(-----[x·,·x·,·x·,·x·])
125 ···················65521··0···1···2···3125 ···················65521··0···1···2···3
126 ······defining·forms:·given·by·a·function126 ······defining·forms:·given·by·a·function
  
127 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)127 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)
  
128 i15·:·time·projectiveDegrees(T,2)128 i15·:·time·projectiveDegrees(T,2)
129 ·--·used·2.42403s·(cpu);·1.58854s·(thread);·0s·(gc)129 ·--·used·2.9282s·(cpu);·1.83756s·(thread);·0s·(gc)
  
130 o15·=·3130 o15·=·3
  
131 i16·:·time·T2·=·T·*·T131 i16·:·time·T2·=·T·*·T
132 ·--·used·0.000178328s·(cpu);·1.8668e-05s·(thread);·0s·(gc)132 ·--·used·0.000163336s·(cpu);·1.8605e-05s·(thread);·0s·(gc)
  
133 o16·=·--·rational·map·--133 o16·=·--·rational·map·--
134 ·····················ZZ134 ·····················ZZ
135 ······source:·Proj(-----[x·,·x·,·x·,·x·])135 ······source:·Proj(-----[x·,·x·,·x·,·x·])
136 ···················65521··0···1···2···3136 ···················65521··0···1···2···3
137 ·····················ZZ137 ·····················ZZ
138 ······target:·Proj(-----[x·,·x·,·x·,·x·])138 ······target:·Proj(-----[x·,·x·,·x·,·x·])
139 ···················65521··0···1···2···3139 ···················65521··0···1···2···3
140 ······defining·forms:·given·by·a·function140 ······defining·forms:·given·by·a·function
  
141 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)141 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)
  
142 i17·:·time·projectiveDegrees(T2,2)142 i17·:·time·projectiveDegrees(T2,2)
143 ·--·used·3.82737s·(cpu);·2.55243s·(thread);·0s·(gc)143 ·--·used·4.03348s·(cpu);·2.4503s·(thread);·0s·(gc)
  
144 o17·=·1144 o17·=·1
  
145 i18·:·p·=·apply(3,i->random(ZZ/65521))|{1}145 i18·:·p·=·apply(3,i->random(ZZ/65521))|{1}
  
146 o18·=·{28963,·31975,·-30172,·1}146 o18·=·{28963,·31975,·-30172,·1}
  
Offset 169, 15 lines modifiedOffset 169, 15 lines modified
169 i20·:·T·q169 i20·:·T·q
  
170 o20·=·{28963,·31975,·-30172,·1}170 o20·=·{28963,·31975,·-30172,·1}
  
171 o20·:·List171 o20·:·List
  
172 i21·:·time·f·=·rationalMap·T172 i21·:·time·f·=·rationalMap·T
173 ·--·used·3.17764s·(cpu);·2.16018s·(thread);·0s·(gc)173 ·--·used·3.35844s·(cpu);·2.13863s·(thread);·0s·(gc)
  
174 o21·=·--·rational·map·--174 o21·=·--·rational·map·--
175 ·····················ZZ175 ·····················ZZ
176 ······source:·Proj(-----[x·,·x·,·x·,·x·])176 ······source:·Proj(-----[x·,·x·,·x·,·x·])
177 ···················65521··0···1···2···3177 ···················65521··0···1···2···3
178 ·····················ZZ178 ·····················ZZ
179 ······target:·Proj(-----[x·,·x·,·x·,·x·])179 ······target:·Proj(-----[x·,·x·,·x·,·x·])
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./usr/share/doc/Macaulay2/Cremona/example-output/_approximate__Inverse__Map.out
    
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44 ······················x·x··-·x·x44 ······················x·x··-·x·x
45 ·······················1·2····0·445 ·······················1·2····0·4
46 ·····················}46 ·····················}
  
47 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)47 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)
  
48 i3·:·time·psi·=·approximateInverseMap·phi48 i3·:·time·psi·=·approximateInverseMap·phi
49 ·--·used·0.211611s·(cpu);·0.16808s·(thread);·0s·(gc)49 ·--·used·0.256069s·(cpu);·0.198245s·(thread);·0s·(gc)
50 --·approximateInverseMap:·step·1·of·1050 --·approximateInverseMap:·step·1·of·10
51 --·approximateInverseMap:·step·2·of·1051 --·approximateInverseMap:·step·2·of·10
52 --·approximateInverseMap:·step·3·of·1052 --·approximateInverseMap:·step·3·of·10
53 --·approximateInverseMap:·step·4·of·1053 --·approximateInverseMap:·step·4·of·10
54 --·approximateInverseMap:·step·5·of·1054 --·approximateInverseMap:·step·5·of·10
55 --·approximateInverseMap:·step·6·of·1055 --·approximateInverseMap:·step·6·of·10
56 --·approximateInverseMap:·step·7·of·1056 --·approximateInverseMap:·step·7·of·10
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106 ·····················}106 ·····················}
  
107 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)107 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)
  
108 i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)108 i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)
  
109 i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);109 i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);
110 ·--·used·0.182534s·(cpu);·0.139433s·(thread);·0s·(gc)110 ·--·used·0.218309s·(cpu);·0.148777s·(thread);·0s·(gc)
111 --·approximateInverseMap:·step·1·of·3111 --·approximateInverseMap:·step·1·of·3
112 --·approximateInverseMap:·step·2·of·3112 --·approximateInverseMap:·step·2·of·3
113 --·approximateInverseMap:·step·3·of·3113 --·approximateInverseMap:·step·3·of·3
  
114 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)114 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)
  
115 i6·:·assert(psi·==·psi')115 i6·:·assert(psi·==·psi')
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189 ·······················0······1·4······0·5······1·5·····2·5······3·5······4·5······5······0·6······1·6······2·6······4·6······5·6······6······0·7······1·7······2·7·····3·7······4·7······5·7······6·7·····0·8······1·8······3·8·····4·8······5·8······6·8·····7·8189 ·······················0······1·4······0·5······1·5·····2·5······3·5······4·5······5······0·6······1·6······2·6······4·6······5·6······6······0·7······1·7······2·7·····3·7······4·7······5·7······6·7·····0·8······1·8······3·8·····4·8······5·8······6·8·····7·8
190 ·····················}190 ·····················}
  
191 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)191 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)
  
192 i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time·192 i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time·
193 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)193 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)
194 ·--·used·1.59839s·(cpu);·1.39582s·(thread);·0s·(gc)194 ·--·used·2.30278s·(cpu);·1.94875s·(thread);·0s·(gc)
195 --·approximateInverseMap:·step·1·of·3195 --·approximateInverseMap:·step·1·of·3
196 --·approximateInverseMap:·step·2·of·3196 --·approximateInverseMap:·step·2·of·3
197 --·approximateInverseMap:·step·3·of·3197 --·approximateInverseMap:·step·3·of·3
  
198 o8·=·--·rational·map·--198 o8·=·--·rational·map·--
199 ································ZZ199 ································ZZ
200 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by200 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by
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254 i9·:·--·but...254 i9·:·--·but...
255 ·····phi·*·psi·==·1255 ·····phi·*·psi·==·1
  
256 o9·=·false256 o9·=·false
  
257 i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify257 i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify
258 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)258 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
259 ·--·used·2.43036s·(cpu);·2.12624s·(thread);·0s·(gc)259 ·--·used·2.9462s·(cpu);·2.54577s·(thread);·0s·(gc)
260 --·approximateInverseMap:·step·1·of·3260 --·approximateInverseMap:·step·1·of·3
261 --·approximateInverseMap:·step·2·of·3261 --·approximateInverseMap:·step·2·of·3
262 --·approximateInverseMap:·step·3·of·3262 --·approximateInverseMap:·step·3·of·3
263 Certify:·output·certified!263 Certify:·output·certified!
  
264 o10·=·--·rational·map·--264 o10·=·--·rational·map·--
265 ·································ZZ265 ·································ZZ
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./usr/share/doc/Macaulay2/Cremona/example-output/_degree__Map.out
    
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9 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10 o4·=·map·(ringP8,·ringP14,·{-·95x··+·181x·x··+·1028x··-·1384x·x··-·1455x·x··+·559x··-·502x·x··+·1264x·x··-·162x·x··+·1209x··-·180x·x··-·504x·x··-·1168x·x··-·676x·x··+·501x··+·73x·x··+·1263x·x··+·1035x·x··+·844x·x··+·1593x·x··+·785x··+·982x·x··-·412x·x··+·1335x·x··+·1136x·x··+·826x·x··+·1078x·x··+·1158x··+·335x·x··-·982x·x··-·1479x·x··-·15x·x··+·1363x·x··+·1397x·x··-·575x·x··-·71x··+·1255x·x··-·1138x·x··-·1590x·x··+·604x·x··+·1182x·x··-·63x·x··-·1382x·x··-·1255x·x··-·613x·,·-·1444x··+·575x·x··+·767x··-·1495x·x··+·1631x·x··-·217x··-·294x·x··-·1511x·x··-·504x·x··-·1284x··-·1459x·x··+·152x·x··+·141x·x··-·10x·x··-·95x··+·1056x·x··+·654x·x··+·1397x·x··-·930x·x··+·578x·x··-·696x··+·759x·x··+·733x·x··+·505x·x··-·609x·x··+·526x·x··-·659x·x··+·846x··+·1253x·x··-·1519x·x··+·635x·x··+·576x·x··+·54x·x··-·1261x·x··-·822x·x··-·257x··-·986x·x··+·356x·x··-·1488x·x··-·1561x·x··-·850x·x··-·85x·x··-·1350x·x··-·783x·x··-·1335x·,·-·871x··+·1006x·x··-·1399x··-·1636x·x··-·699x·x··-·769x··-·307x·x··-·1645x·x··-·502x·x··-·719x··+·1405x·x··+·870x·x··-·1133x·x··+·425x·x··-·1203x··-·1601x·x··+·117x·x··-·382x·x··+·318x·x··-·117x·x··-·560x··+·1135x·x··+·1468x·x··+·869x·x··-·943x·x··-·335x·x··-·1218x·x··+·201x··-·11x·x··+·540x·x··-·710x·x··-·489x·x··+·1605x·x··+·1663x·x··-·423x·x··+·1246x··+·97x·x··-·644x·x··+·1655x·x··+·1219x·x··+·1476x·x··+·1355x·x··+·1594x·x··+·893x·x··+·1150x·,·-·143x··+·1240x·x··-·1042x··+·1649x·x··+·1024x·x··+·794x··+·1442x·x··-·1263x·x··+·537x·x··-·82x··-·734x·x··-·1569x·x··-·798x·x··-·366x·x··+·1289x··-·569x·x··-·254x·x··+·237x·x··-·1234x·x··-·807x·x··+·264x··-·202x·x··-·616x·x··+·44x·x··+·1465x·x··+·685x·x··+·1630x·x··-·406x··-·123x·x··-·4x·x··+·1583x·x··+·1235x·x··+·162x·x··+·1034x·x··-·1035x·x··+·737x··+·660x·x··+·1459x·x··-·359x·x··-·1291x·x··+·1638x·x··-·325x·x··-·631x·x··+·73x·x··-·1471x·,·-·1340x··+·31x·x··-·994x··-·880x·x··-·89x·x··+·574x··+·760x·x··-·1054x·x··+·772x·x··-·239x··-·443x·x··+·1240x·x··+·637x·x··-·1423x·x··+·320x··-·1363x·x··-·1139x·x··-·158x·x··-·325x·x··-·1578x·x··+·32x··+·695x·x··+·305x·x··+·1012x·x··+·1492x·x··+·1290x·x··+·1579x·x··-·342x··-·83x·x··-·104x·x··+·998x·x··-·92x·x··+·1554x·x··+·201x·x··-·237x·x··+·160x··-·228x·x··-·543x·x··-·1147x·x··-·376x·x··+·1313x·x··+·603x·x··+·106x·x··-·1361x·x··+·699x·,·-·228x··-·1510x·x··+·277x··-·4x·x··-·22x·x··-·1526x··+·234x·x··+·969x·x··+·1253x·x··-·1426x··-·1474x·x··+·947x·x··+·194x·x··-·316x·x··-·988x··-·1211x·x··+·1087x·x··+·536x·x··-·491x·x··+·870x·x··-·659x··+·1490x·x··-·469x·x··+·1190x·x··+·807x·x··+·650x·x··+·448x·x··-·1353x··-·218x·x··+·759x·x··-·253x·x··+·830x·x··-·1080x·x··-·143x·x··-·1313x·x··-·374x··-·180x·x··+·741x·x··+·742x·x··-·1254x·x··+·458x·x··-·345x·x··+·597x·x··+·1567x·x··-·31x·,·1120x··+·709x·x··-·1538x··-·1048x·x··-·162x·x··-·1518x··-·73x·x··+·380x·x··+·533x·x··-·286x··+·1374x·x··-·74x·x··-·22x·x··+·1535x·x··-·1071x··-·839x·x··-·560x·x··+·928x·x··+·335x·x··-·1008x·x··+·810x··-·448x·x··-·357x·x··-·107x·x··+·40x·x··+·784x·x··-·1423x·x··+·1276x··+·147x·x··+·443x·x··-·598x·x··-·1077x·x··-·1214x·x··+·322x·x··-·1408x·x··+·72x··-·63x·x··-·1513x·x··-·791x·x··+·11x·x··+·77x·x··+·836x·x··-·1100x·x··+·1637x·x··-·788x·,·1331x··+·318x·x··-·704x··+·51x·x··+·275x·x··+·1149x··+·1526x·x··+·768x·x··+·414x·x··-·782x··-·262x·x··+·686x·x··-·380x·x··+·1377x·x··+·1077x··+·1650x·x··-·1129x·x··-·508x·x··+·846x·x··+·1513x·x··+·460x··-·1626x·x··-·1024x·x··+·862x·x··+·1352x·x··-·188x·x··-·1382x·x··-·650x··+·55x·x··-·326x·x··+·1037x·x··+·705x·x··-·667x·x··+·1483x·x··+·1661x·x··-·1652x··-·1052x·x··-·692x·x··-·542x·x··+·162x·x··+·582x·x··-·1369x·x··+·934x·x··+·1392x·x··+·1227x·,·-·346x··+·1408x·x··-·1225x··-·1536x·x··-·1028x·x··-·985x··-·210x·x··-·1312x·x··+·915x·x··+·1633x··-·202x·x··-·1636x·x··-·1653x·x··-·480x·x··-·1260x··-·813x·x··-·1623x·x··-·1429x·x··+·1094x·x··-·747x·x··+·955x··+·898x·x··-·795x·x··-·35x·x··-·566x·x··+·1631x·x··-·324x·x··+·926x··-·132x·x··-·9x·x··-·1290x·x··-·543x·x··+·902x·x··+·735x·x··-·342x·x··-·400x··+·900x·x··-·463x·x··+·694x·x··-·1262x·x··-·1449x·x··-·448x·x··-·1402x·x··-·731x·x··-·996x·,·301x··+·166x·x··-·955x··-·739x·x··-·1199x·x··-·319x··+·1047x·x··-·532x·x··+·902x·x··+·1195x··-·663x·x··+·1215x·x··-·534x·x··-·332x·x··-·973x··+·772x·x··-·308x·x··+·315x·x··-·454x·x··-·483x·x··-·239x··-·1313x·x··-·419x·x··-·1340x·x··-·1388x·x··-·1340x·x··-·1665x·x··-·333x··-·465x·x··-·1084x·x··+·676x·x··-·1612x·x··-·288x·x··+·11x·x··-·1170x·x··-·189x··+·498x·x··-·889x·x··+·693x·x··+·1460x·x··-·473x·x··-·414x·x··-·122x·x··-·1659x·x··-·1421x·,·14x··-·1049x·x··+·1506x··+·1235x·x··+·642x·x··-·1034x··+·460x·x··+·150x·x··+·760x·x··-·1246x··-·1407x·x··+·1570x·x··+·1403x·x··-·1610x·x··-·431x··+·574x·x··+·893x·x··-·657x·x··+·417x·x··+·1362x·x··+·224x··+·268x·x··+·1097x·x··+·1132x·x··+·148x·x··+·1331x·x··-·77x·x··-·756x··+·228x·x··+·136x·x··-·1484x·x··-·1478x·x··-·13x·x··+·1620x·x··-·701x·x··-·769x··-·760x·x··-·492x·x··-·1077x·x··-·1249x·x··-·834x·x··-·395x·x··-·1358x·x··-·988x·x··+·113x·,·-·1634x··-·13x·x··+·805x··-·21x·x··-·1655x·x··+·1479x··-·1510x·x··-·646x·x··+·225x·x··-·1411x··+·1227x·x··-·1108x·x··+·1291x·x··-·59x·x··-·142x··+·586x·x··-·676x·x··+·655x·x··-·1476x·x··+·453x·x··-·1076x··-·1152x·x··+·1373x·x··-·1191x·x··-·416x·x··+·699x·x··+·317x·x··+·825x··-·1560x·x··-·488x·x··-·1035x·x··-·1561x·x··-·644x·x··-·1178x·x··-·1320x·x··+·158x··+·889x·x··+·1444x·x··-·1486x·x··-·1211x·x··+·1269x·x··-·1228x·x··+·568x·x··+·1591x·x··+·1207x·,·105x··-·538x·x··-·1222x··-·277x·x··+·716x·x··-·1067x··-·428x·x··+·154x·x··-·469x·x··+·77x··+·538x·x··-·179x·x··+·921x·x··-·223x·x··+·1093x··-·262x·x··+·1299x·x··+·631x·x··+·1486x·x··-·1280x·x··-·121x··-·50x·x··-·978x·x··-·694x·x··-·531x·x··+·505x·x··+·1412x·x··-·1061x··+·1202x·x··+·448x·x··-·187x·x··+·1276x·x··-·121x·x··+·1361x·x··+·697x·x··+·682x··+·1592x·x··+·705x·x··-·227x·x··-·7x·x··-·1423x·x··-·1446x·x··-·1578x·x··+·1511x·x··+·917x·,·1270x··-·391x·x··-·1116x··-·287x·x··+·653x·x··+·1643x··+·1623x·x··+·514x·x··-·14x·x··-·90x··+·1232x·x··-·1434x·x··+·1296x·x··+·1522x·x··+·136x··-·623x·x··-·607x·x··+·18x·x··+·896x·x··-·29x·x··+·1059x··-·1053x·x··+·1643x·x··+·1652x·x··-·1190x·x··-·1073x·x··+·1470x·x··-·944x··-·93x·x··-·187x·x··-·994x·x··-·1415x·x··-·229x·x··-·796x·x··+·1642x·x··+·1600x··-·344x·x··+·905x·x··+·1032x·x··-·538x·x··-·891x·x··+·1243x·x··+·1290x·x··+·490x·x··-·1148x·,·1613x··+·175x·x··-·1346x··-·1000x·x··-·1217x·x··-·729x··-·1296x·x··+·1456x·x··+·745x·x··+·539x··+·525x·x··-·811x·x··+·753x·x··+·1362x·x··+·1629x··-·840x·x··+·513x·x··+·429x·x··+·842x·x··+·1414x·x··-·308x··+·1415x·x··-·1461x·x··-·1135x·x··+·701x·x··+·766x·x··+·785x·x··+·1503x··+·147x·x··+·929x·x··-·1220x·x··-·853x·x··+·493x·x··+·226x·x··+·1416x·x··+·280x··-·7x·x··+·1632x·x··+·520x·x··+·1259x·x··+·157x·x··+·1596x·x··+·655x·x··-·42x·x··-·586x·})10 o4·=·map·(ringP8,·ringP14,·{-·95x··+·181x·x··+·1028x··-·1384x·x··-·1455x·x··+·559x··-·502x·x··+·1264x·x··-·162x·x··+·1209x··-·180x·x··-·504x·x··-·1168x·x··-·676x·x··+·501x··+·73x·x··+·1263x·x··+·1035x·x··+·844x·x··+·1593x·x··+·785x··+·982x·x··-·412x·x··+·1335x·x··+·1136x·x··+·826x·x··+·1078x·x··+·1158x··+·335x·x··-·982x·x··-·1479x·x··-·15x·x··+·1363x·x··+·1397x·x··-·575x·x··-·71x··+·1255x·x··-·1138x·x··-·1590x·x··+·604x·x··+·1182x·x··-·63x·x··-·1382x·x··-·1255x·x··-·613x·,·-·1444x··+·575x·x··+·767x··-·1495x·x··+·1631x·x··-·217x··-·294x·x··-·1511x·x··-·504x·x··-·1284x··-·1459x·x··+·152x·x··+·141x·x··-·10x·x··-·95x··+·1056x·x··+·654x·x··+·1397x·x··-·930x·x··+·578x·x··-·696x··+·759x·x··+·733x·x··+·505x·x··-·609x·x··+·526x·x··-·659x·x··+·846x··+·1253x·x··-·1519x·x··+·635x·x··+·576x·x··+·54x·x··-·1261x·x··-·822x·x··-·257x··-·986x·x··+·356x·x··-·1488x·x··-·1561x·x··-·850x·x··-·85x·x··-·1350x·x··-·783x·x··-·1335x·,·-·871x··+·1006x·x··-·1399x··-·1636x·x··-·699x·x··-·769x··-·307x·x··-·1645x·x··-·502x·x··-·719x··+·1405x·x··+·870x·x··-·1133x·x··+·425x·x··-·1203x··-·1601x·x··+·117x·x··-·382x·x··+·318x·x··-·117x·x··-·560x··+·1135x·x··+·1468x·x··+·869x·x··-·943x·x··-·335x·x··-·1218x·x··+·201x··-·11x·x··+·540x·x··-·710x·x··-·489x·x··+·1605x·x··+·1663x·x··-·423x·x··+·1246x··+·97x·x··-·644x·x··+·1655x·x··+·1219x·x··+·1476x·x··+·1355x·x··+·1594x·x··+·893x·x··+·1150x·,·-·143x··+·1240x·x··-·1042x··+·1649x·x··+·1024x·x··+·794x··+·1442x·x··-·1263x·x··+·537x·x··-·82x··-·734x·x··-·1569x·x··-·798x·x··-·366x·x··+·1289x··-·569x·x··-·254x·x··+·237x·x··-·1234x·x··-·807x·x··+·264x··-·202x·x··-·616x·x··+·44x·x··+·1465x·x··+·685x·x··+·1630x·x··-·406x··-·123x·x··-·4x·x··+·1583x·x··+·1235x·x··+·162x·x··+·1034x·x··-·1035x·x··+·737x··+·660x·x··+·1459x·x··-·359x·x··-·1291x·x··+·1638x·x··-·325x·x··-·631x·x··+·73x·x··-·1471x·,·-·1340x··+·31x·x··-·994x··-·880x·x··-·89x·x··+·574x··+·760x·x··-·1054x·x··+·772x·x··-·239x··-·443x·x··+·1240x·x··+·637x·x··-·1423x·x··+·320x··-·1363x·x··-·1139x·x··-·158x·x··-·325x·x··-·1578x·x··+·32x··+·695x·x··+·305x·x··+·1012x·x··+·1492x·x··+·1290x·x··+·1579x·x··-·342x··-·83x·x··-·104x·x··+·998x·x··-·92x·x··+·1554x·x··+·201x·x··-·237x·x··+·160x··-·228x·x··-·543x·x··-·1147x·x··-·376x·x··+·1313x·x··+·603x·x··+·106x·x··-·1361x·x··+·699x·,·-·228x··-·1510x·x··+·277x··-·4x·x··-·22x·x··-·1526x··+·234x·x··+·969x·x··+·1253x·x··-·1426x··-·1474x·x··+·947x·x··+·194x·x··-·316x·x··-·988x··-·1211x·x··+·1087x·x··+·536x·x··-·491x·x··+·870x·x··-·659x··+·1490x·x··-·469x·x··+·1190x·x··+·807x·x··+·650x·x··+·448x·x··-·1353x··-·218x·x··+·759x·x··-·253x·x··+·830x·x··-·1080x·x··-·143x·x··-·1313x·x··-·374x··-·180x·x··+·741x·x··+·742x·x··-·1254x·x··+·458x·x··-·345x·x··+·597x·x··+·1567x·x··-·31x·,·1120x··+·709x·x··-·1538x··-·1048x·x··-·162x·x··-·1518x··-·73x·x··+·380x·x··+·533x·x··-·286x··+·1374x·x··-·74x·x··-·22x·x··+·1535x·x··-·1071x··-·839x·x··-·560x·x··+·928x·x··+·335x·x··-·1008x·x··+·810x··-·448x·x··-·357x·x··-·107x·x··+·40x·x··+·784x·x··-·1423x·x··+·1276x··+·147x·x··+·443x·x··-·598x·x··-·1077x·x··-·1214x·x··+·322x·x··-·1408x·x··+·72x··-·63x·x··-·1513x·x··-·791x·x··+·11x·x··+·77x·x··+·836x·x··-·1100x·x··+·1637x·x··-·788x·,·1331x··+·318x·x··-·704x··+·51x·x··+·275x·x··+·1149x··+·1526x·x··+·768x·x··+·414x·x··-·782x··-·262x·x··+·686x·x··-·380x·x··+·1377x·x··+·1077x··+·1650x·x··-·1129x·x··-·508x·x··+·846x·x··+·1513x·x··+·460x··-·1626x·x··-·1024x·x··+·862x·x··+·1352x·x··-·188x·x··-·1382x·x··-·650x··+·55x·x··-·326x·x··+·1037x·x··+·705x·x··-·667x·x··+·1483x·x··+·1661x·x··-·1652x··-·1052x·x··-·692x·x··-·542x·x··+·162x·x··+·582x·x··-·1369x·x··+·934x·x··+·1392x·x··+·1227x·,·-·346x··+·1408x·x··-·1225x··-·1536x·x··-·1028x·x··-·985x··-·210x·x··-·1312x·x··+·915x·x··+·1633x··-·202x·x··-·1636x·x··-·1653x·x··-·480x·x··-·1260x··-·813x·x··-·1623x·x··-·1429x·x··+·1094x·x··-·747x·x··+·955x··+·898x·x··-·795x·x··-·35x·x··-·566x·x··+·1631x·x··-·324x·x··+·926x··-·132x·x··-·9x·x··-·1290x·x··-·543x·x··+·902x·x··+·735x·x··-·342x·x··-·400x··+·900x·x··-·463x·x··+·694x·x··-·1262x·x··-·1449x·x··-·448x·x··-·1402x·x··-·731x·x··-·996x·,·301x··+·166x·x··-·955x··-·739x·x··-·1199x·x··-·319x··+·1047x·x··-·532x·x··+·902x·x··+·1195x··-·663x·x··+·1215x·x··-·534x·x··-·332x·x··-·973x··+·772x·x··-·308x·x··+·315x·x··-·454x·x··-·483x·x··-·239x··-·1313x·x··-·419x·x··-·1340x·x··-·1388x·x··-·1340x·x··-·1665x·x··-·333x··-·465x·x··-·1084x·x··+·676x·x··-·1612x·x··-·288x·x··+·11x·x··-·1170x·x··-·189x··+·498x·x··-·889x·x··+·693x·x··+·1460x·x··-·473x·x··-·414x·x··-·122x·x··-·1659x·x··-·1421x·,·14x··-·1049x·x··+·1506x··+·1235x·x··+·642x·x··-·1034x··+·460x·x··+·150x·x··+·760x·x··-·1246x··-·1407x·x··+·1570x·x··+·1403x·x··-·1610x·x··-·431x··+·574x·x··+·893x·x··-·657x·x··+·417x·x··+·1362x·x··+·224x··+·268x·x··+·1097x·x··+·1132x·x··+·148x·x··+·1331x·x··-·77x·x··-·756x··+·228x·x··+·136x·x··-·1484x·x··-·1478x·x··-·13x·x··+·1620x·x··-·701x·x··-·769x··-·760x·x··-·492x·x··-·1077x·x··-·1249x·x··-·834x·x··-·395x·x··-·1358x·x··-·988x·x··+·113x·,·-·1634x··-·13x·x··+·805x··-·21x·x··-·1655x·x··+·1479x··-·1510x·x··-·646x·x··+·225x·x··-·1411x··+·1227x·x··-·1108x·x··+·1291x·x··-·59x·x··-·142x··+·586x·x··-·676x·x··+·655x·x··-·1476x·x··+·453x·x··-·1076x··-·1152x·x··+·1373x·x··-·1191x·x··-·416x·x··+·699x·x··+·317x·x··+·825x··-·1560x·x··-·488x·x··-·1035x·x··-·1561x·x··-·644x·x··-·1178x·x··-·1320x·x··+·158x··+·889x·x··+·1444x·x··-·1486x·x··-·1211x·x··+·1269x·x··-·1228x·x··+·568x·x··+·1591x·x··+·1207x·,·105x··-·538x·x··-·1222x··-·277x·x··+·716x·x··-·1067x··-·428x·x··+·154x·x··-·469x·x··+·77x··+·538x·x··-·179x·x··+·921x·x··-·223x·x··+·1093x··-·262x·x··+·1299x·x··+·631x·x··+·1486x·x··-·1280x·x··-·121x··-·50x·x··-·978x·x··-·694x·x··-·531x·x··+·505x·x··+·1412x·x··-·1061x··+·1202x·x··+·448x·x··-·187x·x··+·1276x·x··-·121x·x··+·1361x·x··+·697x·x··+·682x··+·1592x·x··+·705x·x··-·227x·x··-·7x·x··-·1423x·x··-·1446x·x··-·1578x·x··+·1511x·x··+·917x·,·1270x··-·391x·x··-·1116x··-·287x·x··+·653x·x··+·1643x··+·1623x·x··+·514x·x··-·14x·x··-·90x··+·1232x·x··-·1434x·x··+·1296x·x··+·1522x·x··+·136x··-·623x·x··-·607x·x··+·18x·x··+·896x·x··-·29x·x··+·1059x··-·1053x·x··+·1643x·x··+·1652x·x··-·1190x·x··-·1073x·x··+·1470x·x··-·944x··-·93x·x··-·187x·x··-·994x·x··-·1415x·x··-·229x·x··-·796x·x··+·1642x·x··+·1600x··-·344x·x··+·905x·x··+·1032x·x··-·538x·x··-·891x·x··+·1243x·x··+·1290x·x··+·490x·x··-·1148x·,·1613x··+·175x·x··-·1346x··-·1000x·x··-·1217x·x··-·729x··-·1296x·x··+·1456x·x··+·745x·x··+·539x··+·525x·x··-·811x·x··+·753x·x··+·1362x·x··+·1629x··-·840x·x··+·513x·x··+·429x·x··+·842x·x··+·1414x·x··-·308x··+·1415x·x··-·1461x·x··-·1135x·x··+·701x·x··+·766x·x··+·785x·x··+·1503x··+·147x·x··+·929x·x··-·1220x·x··-·853x·x··+·493x·x··+·226x·x··+·1416x·x··+·280x··-·7x·x··+·1632x·x··+·520x·x··+·1259x·x··+·157x·x··+·1596x·x··+·655x·x··-·42x·x··-·586x·})
11 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·····1·3·······2·3········3········0·4········1·4········2·4······3·4·······4·······0·5·······1·5·······2·5········3·5·······4·5········5········0·6········1·6········2·6·······3·6·······4·6·······5·6·······6········0·7·······1·7········2·7········3·7·······4·7········5·7········6·7·······7·······0·8········1·8········2·8········3·8········4·8········5·8·······6·8········7·8········8······0·······0·1········1·······0·2·······1·2········2·······0·3·······1·3·······2·3······3·······0·4·······1·4·······2·4·······3·4········4·······0·5········1·5·······2·5········3·5········4·5·······5······0·6·······1·6·······2·6·······3·6·······4·6········5·6········6········0·7·······1·7·······2·7········3·7·······4·7········5·7·······6·7·······7········0·8·······1·8·······2·8·····3·8········4·8········5·8········6·8········7·8·······8·······0·······0·1········1·······0·2·······1·2········2········0·3·······1·3······2·3······3········0·4········1·4········2·4········3·4·······4·······0·5·······1·5······2·5·······3·5······4·5········5········0·6········1·6········2·6········3·6········4·6········5·6·······6······0·7·······1·7·······2·7········3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8·······4·8········5·8········6·8·······7·8········8·······0·······0·1········1········0·2········1·2·······2········0·3········1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4········4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6········1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7········2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2·8········3·8·······4·8········5·8·······6·8······7·8·······811 ·································0·······0·1········1········0·2········1·2·······2·······0·3········1·3·······2·3········3·······0·4·······1·4········2·4·······3·4·······4······0·5········1·5········2·5·······3·5········4·5·······5·······0·6·······1·6········2·6········3·6·······4·6········5·6········6·······0·7·······1·7········2·7······3·7········4·7········5·7·······6·7······7········0·8········1·8········2·8·······3·8········4·8······5·8········6·8········7·8·······8·········0·······0·1·······1········0·2········1·2·······2·······0·3········1·3·······2·3········3········0·4·······1·4·······2·4······3·4······4········0·5·······1·5········2·5·······3·5·······4·5·······5·······0·6·······1·6·······2·6·······3·6·······4·6·······5·6·······6········0·7········1·7·······2·7·······3·7······4·7········5·7·······6·7·······7·······0·8·······1·8········2·8········3·8·······4·8······5·8········6·8·······7·8········8········0········0·1········1········0·2·······1·2·······2·······0·3········1·3·······2·3·······3········0·4·······1·4········2·4·······3·4········4········0·5·······1·5·······2·5·······3·5·······4·5·······5········0·6········1·6·······2·6·······3·6·······4·6········5·6·······6······0·7·······1·7·······2·7·······3·7········4·7········5·7·······6·7········7······0·8·······1·8········2·8········3·8········4·8········5·8········6·8·······7·8········8········0········0·1········1········0·2········1·2·······2········0·3········1·3·······2·3······3·······0·4········1·4·······2·4·······3·4········4·······0·5·······1·5·······2·5········3·5·······4·5·······5·······0·6·······1·6······2·6········3·6·······4·6········5·6·······6·······0·7·····1·7········2·7········3·7·······4·7········5·7········6·7·······7·······0·8········1·8·······2·8········3·8········4·8·······5·8·······6·8······7·8········8·········0······0·1·······1·······0·2······1·2·······2·······0·3········1·3·······2·3·······3·······0·4········1·4·······2·4········3·4·······4········0·5········1·5·······2·5·······3·5········4·5······5·······0·6·······1·6········2·6········3·6········4·6········5·6·······6······0·7·······1·7·······2·7······3·7········4·7·······5·7·······6·7·······7·······0·8·······1·8········2·8·······3·8········4·8·······5·8·······6·8········7·8·······8········0········0·1·······1·····0·2······1·2········2·······0·3·······1·3········2·3········3········0·4·······1·4·······2·4·······3·4·······4········0·5········1·5·······2·5·······3·5·······4·5·······5········0·6·······1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7·······2·7·······3·7········4·7·······5·7········6·7·······7·······0·8·······1·8·······2·8········3·8·······4·8·······5·8·······6·8········7·8······8·······0·······0·1········1········0·2·······1·2········2······0·3·······1·3·······2·3·······3········0·4······1·4······2·4········3·4········4·······0·5·······1·5·······2·5·······3·5········4·5·······5·······0·6·······1·6·······2·6······3·6·······4·6········5·6········6·······0·7·······1·7·······2·7········3·7········4·7·······5·7········6·7······7······0·8········1·8·······2·8······3·8······4·8·······5·8········6·8········7·8·······8·······0·······0·1·······1······0·2·······1·2········2········0·3·······1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4········4········0·5········1·5·······2·5·······3·5········4·5·······5········0·6········1·6·······2·6········3·6·······4·6········5·6·······6······0·7·······1·7········2·7·······3·7·······4·7········5·7········6·7········7········0·8·······1·8·······2·8·······3·8·······4·8········5·8·······6·8········7·8········8········0········0·1········1········0·2········1·2·······2·······0·3········1·3·······2·3········3·······0·4········1·4········2·4·······3·4········4·······0·5········1·5········2·5········3·5·······4·5·······5·······0·6·······1·6······2·6·······3·6········4·6·······5·6·······6·······0·7·····1·7········2·7·······3·7·······4·7·······5·7·······6·7·······7·······0·8·······1·8·······2·8········3·8········4·8·······5·8········6·8·······7·8·······8······0·······0·1·······1·······0·2········1·2·······2········0·3·······1·3·······2·3········3·······0·4········1·4·······2·4·······3·4·······4·······0·5·······1·5·······2·5·······3·5·······4·5·······5········0·6·······1·6········2·6········3·6········4·6········5·6·······6·······0·7········1·7·······2·7········3·7·······4·7······5·7········6·7·······7·······0·8·······1·8·······2·8········3·8·······4·8·······5·8·······6·8········7·8········8·····0········0·1········1········0·2·······1·2········2·······0·3·······1·3·······2·3········3········0·4········1·4········2·4········3·4·······4·······0·5·······1·5·······2·5·······3·5········4·5·······5·······0·6········1·6········2·6·······3·6········4·6······5·6·······6·······0·7·······1·7········2·7········3·7······4·7········5·7·······6·7·······7·······0·8·······1·8········2·8········3·8·······4·8·······5·8········6·8·······7·8·······8·········0······0·1·······1······0·2········1·2········2········0·3·······1·3·······2·3········3········0·4········1·4········2·4······3·4·······4·······0·5·······1·5·······2·5········3·5·······4·5········5········0·6········1·6········2·6·······3·6·······4·6·······5·6·······6········0·7·······1·7········2·7········3·7·······4·7········5·7········6·7·······7·······0·8········1·8········2·8········3·8········4·8········5·8·······6·8········7·8········8······0·······0·1········1·······0·2·······1·2········2·······0·3·······1·3·······2·3······3·······0·4·······1·4·······2·4·······3·4········4·······0·5········1·5·······2·5········3·5········4·5·······5······0·6·······1·6·······2·6·······3·6·······4·6········5·6········6········0·7·······1·7·······2·7········3·7·······4·7········5·7·······6·7·······7········0·8·······1·8·······2·8·····3·8········4·8········5·8········6·8········7·8·······8·······0·······0·1········1·······0·2·······1·2········2········0·3·······1·3······2·3······3········0·4········1·4········2·4········3·4·······4·······0·5·······1·5······2·5·······3·5······4·5········5········0·6········1·6········2·6········3·6········4·6········5·6·······6······0·7·······1·7·······2·7········3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8·······4·8········5·8········6·8·······7·8········8·······0·······0·1········1········0·2········1·2·······2········0·3········1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4········4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6········1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7········2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2·8········3·8·······4·8········5·8·······6·8······7·8·······8
  
12 o4·:·RingMap·ringP8·<--·ringP1412 o4·:·RingMap·ringP8·<--·ringP14
  
13 i5·:·time·degreeMap·phi13 i5·:·time·degreeMap·phi
14 ·--·used·0.0934991s·(cpu);·0.0516392s·(thread);·0s·(gc)14 ·--·used·0.136903s·(cpu);·0.0693437s·(thread);·0s·(gc)
  
15 o5·=·115 o5·=·1
  
16 i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a·general·subspace·of·P^14·16 i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a·general·subspace·of·P^14·
17 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have·degree·equal·to·deg(G(1,5))=14)17 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have·degree·equal·to·deg(G(1,5))=14)
18 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))18 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))
  
19 ·································2··················2···························2······································2·················································2···························································2···································································2··············································································2··························································································2········2··················2······························2·······································2················································2·····························································2··································································2··············································································2····························································································2········2··················2·····························2·······································2················································2···························································2······································································2··············································································2·························································································2·········2·················2····························2·······································2··················································2·····························································2····································································2················································································2·····························································································2·······2···················2····························2·····································2················································2··························································2··································································2···················································································2····························································································2········2················2···························2······································2··················································2····························································2······································································2·················································································2··························································································2···2···················2···························2·····································2··················································2···························································2····································································2··············································································2·························································································2······2··················2···························2······································2··················································2·····························································2·······································································2··············································································2··························································································2·········2··················2····························2·····································2·················································2······························································2····································································2···············································································2························································································219 ·································2··················2···························2······································2·················································2···························································2···································································2··············································································2··························································································2········2··················2······························2·······································2················································2·····························································2··································································2··············································································2····························································································2········2··················2·····························2·······································2················································2···························································2······································································2··············································································2·························································································2·········2·················2····························2·······································2··················································2·····························································2····································································2················································································2·····························································································2·······2···················2····························2·····································2················································2··························································2··································································2···················································································2····························································································2········2················2···························2······································2··················································2····························································2······································································2·················································································2··························································································2···2···················2···························2·····································2··················································2···························································2····································································2··············································································2·························································································2······2··················2···························2······································2··················································2·····························································2·······································································2··············································································2··························································································2·········2··················2····························2·····································2·················································2······························································2····································································2···············································································2························································································2
20 o6·=·map·(ringP8,·ringP8,·{-·780x··-·506x·x··+·1537x··-·132x·x··-·928x·x··+·386x··-·102x·x··+·422x·x··+·725x·x··-·1073x··-·905x·x··-·830x·x··+·1500x·x··+·276x·x··+·1533x··-·653x·x··+·1558x·x··+·939x·x··-·1432x·x··+·462x·x··-·329x··-·92x·x··+·661x·x··-·1298x·x··-·684x·x··+·70x·x··-·715x·x··+·1093x··+·581x·x··+·329x·x··+·454x·x··-·911x·x··-·84x·x··-·1452x·x··-·809x·x··+·1202x··+·1353x·x··+·1503x·x··+·482x·x··+·893x·x··-·643x·x··+·598x·x··+·110x·x··+·1064x·x··-·472x·,·-·522x··-·583x·x··+·1339x··+·1535x·x··-·1317x·x··+·1113x··-·169x·x··+·1440x·x··-·1657x·x··+·721x··+·40x·x··-·1576x·x··-·367x·x··+·257x·x··-·1454x··+·1612x·x··+·1529x·x··-·1068x·x··+·560x·x··-·1441x·x··+·608x··-·92x·x··-·1006x·x··+·285x·x··+·102x·x··-·397x·x··+·66x·x··-·643x··-·38x·x··+·1380x·x··+·1069x·x··-·426x·x··+·1147x·x··+·982x·x··+·10x·x··-·662x··+·16x·x··+·1561x·x··+·1597x·x··+·512x·x··+·1288x·x··-·1253x·x··+·1317x·x··+·1481x·x··-·354x·,·-·640x··-·1551x·x··+·469x··+·1482x·x··-·1593x·x··-·986x··+·471x·x··+·612x·x··+·1228x·x··+·1156x··-·731x·x··+·1503x·x··-·628x·x··+·674x·x··-·799x··+·1137x·x··+·844x·x··+·589x·x··-·666x·x··+·829x·x··-·1024x··-·170x·x··+·450x·x··+·1497x·x··+·1204x·x··-·907x·x··+·1621x·x··-·417x··+·1297x·x··+·1444x·x··+·4x·x··+·398x·x··+·996x·x··-·1031x·x··+·239x·x··+·303x··+·1215x·x··-·83x·x··+·1571x·x··-·1543x·x··-·925x·x··-·694x·x··+·151x·x··-·520x·x··+·880x·,·-·1210x··-·222x·x··+·185x··+·245x·x··+·1059x·x··-·322x··+·238x·x··+·962x·x··+·1260x·x··-·1581x··+·50x·x··+·1352x·x··-·1465x·x··+·1555x·x··+·1333x··+·1362x·x··+·1365x·x··+·1168x·x··-·1401x·x··+·149x·x··-·652x··+·1378x·x··-·557x·x··-·112x·x··+·26x·x··+·315x·x··+·111x·x··+·1592x··-·283x·x··-·1454x·x··+·907x·x··+·212x·x··+·400x·x··+·1049x·x··-·882x·x··-·1429x··-·183x·x··+·1571x·x··-·1286x·x··-·1179x·x··+·1319x·x··+·240x·x··-·1100x·x··+·1500x·x··-·348x·,·1051x··-·1325x·x··+·1354x··-·346x·x··-·1532x·x··-·466x··+·163x·x··-·659x·x··-·291x·x··+·966x··+·789x·x··+·393x·x··+·403x·x··-·1199x·x··-·570x··-·93x·x··-·492x·x··-·418x·x··+·713x·x··-·1323x·x··-·1384x··-·830x·x··-·54x·x··-·306x·x··+·709x·x··+·421x·x··-·954x·x··-·299x··+·1053x·x··-·1080x·x··+·686x·x··+·170x·x··-·1272x·x··-·1661x·x··+·1235x·x··+·1553x··-·1454x·x··-·1411x·x··-·1195x·x··-·962x·x··+·737x·x··-·390x·x··+·957x·x··+·1538x·x··+·1234x·,·-·509x··+·9x·x··-·1563x··-·710x·x··-·642x·x··+·541x··+·220x·x··-·1214x·x··-·16x·x··+·1008x··-·1088x·x··+·755x·x··-·886x·x··-·1433x·x··+·1154x··+·1627x·x··-·1547x·x··-·951x·x··+·866x·x··+·163x·x··-·1142x··-·668x·x··+·1361x·x··+·1324x·x··-·490x·x··+·282x·x··-·1133x·x··-·612x··+·805x·x··-·126x·x··+·1296x·x··-·973x·x··+·1271x·x··-·1646x·x··+·844x·x··+·1073x··-·1452x·x··-·1112x·x··-·141x·x··+·176x·x··-·1579x·x··-·78x·x··+·848x·x··-·1365x·x··+·711x·,·x··+·1543x·x··-·1076x··+·493x·x··-·526x·x··+·868x··-·582x·x··-·996x·x··+·206x·x··-·419x··+·1258x·x··-·391x·x··+·1002x·x··-·1539x·x··+·931x··-·1504x·x··+·810x·x··+·324x·x··+·1356x·x··+·313x·x··+·772x··+·299x·x··+·1186x·x··+·718x·x··+·407x·x··-·64x·x··-·828x·x··-·1393x··+·94x·x··-·290x·x··-·766x·x··+·950x·x··-·640x·x··+·265x·x··-·1640x·x··-·1403x··-·126x·x··+·891x·x··-·1519x·x··-·927x·x··-·1335x·x··-·1448x·x··-·x·x··-·1103x·x··-·1152x·,·821x··+·558x·x··-·1174x··-·168x·x··+·986x·x··+·790x··+·549x·x··+·817x·x··+·1396x·x··+·695x··+·1211x·x··+·878x·x··-·1061x·x··-·1244x·x··-·880x··+·1409x·x··-·567x·x··+·1240x·x··+·1126x·x··-·1262x·x··+·490x··+·1553x·x··+·1276x·x··+·805x·x··+·576x·x··-·1076x·x··+·1617x·x··-·495x··-·750x·x··-·277x·x··+·544x·x··+·1479x·x··-·784x·x··-·64x·x··-·1203x·x··+·405x··+·1013x·x··+·604x·x··+·1301x·x··+·1003x·x··+·235x·x··+·696x·x··+·939x·x··-·714x·x··-·879x·,·-·1452x··+·727x·x··-·1159x··+·449x·x··-·1169x·x··+·732x··+·575x·x··-·600x·x··+·924x·x··-·837x··+·1298x·x··-·860x·x··+·1010x·x··+·774x·x··+·319x··+·1087x·x··-·1120x·x··+·1439x·x··+·1175x·x··-·1648x·x··+·985x··-·1317x·x··-·878x·x··+·399x·x··-·1339x·x··+·70x·x··-·463x·x··+·470x··-·628x·x··-·907x·x··+·748x·x··+·98x·x··+·1150x·x··+·1140x·x··+·1308x·x··+·621x··+·369x·x··-·991x·x··-·1186x·x··+·61x·x··-·907x·x··-·681x·x··-·1528x·x··+·717x·x··+·854x·})20 o6·=·map·(ringP8,·ringP8,·{-·780x··-·506x·x··+·1537x··-·132x·x··-·928x·x··+·386x··-·102x·x··+·422x·x··+·725x·x··-·1073x··-·905x·x··-·830x·x··+·1500x·x··+·276x·x··+·1533x··-·653x·x··+·1558x·x··+·939x·x··-·1432x·x··+·462x·x··-·329x··-·92x·x··+·661x·x··-·1298x·x··-·684x·x··+·70x·x··-·715x·x··+·1093x··+·581x·x··+·329x·x··+·454x·x··-·911x·x··-·84x·x··-·1452x·x··-·809x·x··+·1202x··+·1353x·x··+·1503x·x··+·482x·x··+·893x·x··-·643x·x··+·598x·x··+·110x·x··+·1064x·x··-·472x·,·-·522x··-·583x·x··+·1339x··+·1535x·x··-·1317x·x··+·1113x··-·169x·x··+·1440x·x··-·1657x·x··+·721x··+·40x·x··-·1576x·x··-·367x·x··+·257x·x··-·1454x··+·1612x·x··+·1529x·x··-·1068x·x··+·560x·x··-·1441x·x··+·608x··-·92x·x··-·1006x·x··+·285x·x··+·102x·x··-·397x·x··+·66x·x··-·643x··-·38x·x··+·1380x·x··+·1069x·x··-·426x·x··+·1147x·x··+·982x·x··+·10x·x··-·662x··+·16x·x··+·1561x·x··+·1597x·x··+·512x·x··+·1288x·x··-·1253x·x··+·1317x·x··+·1481x·x··-·354x·,·-·640x··-·1551x·x··+·469x··+·1482x·x··-·1593x·x··-·986x··+·471x·x··+·612x·x··+·1228x·x··+·1156x··-·731x·x··+·1503x·x··-·628x·x··+·674x·x··-·799x··+·1137x·x··+·844x·x··+·589x·x··-·666x·x··+·829x·x··-·1024x··-·170x·x··+·450x·x··+·1497x·x··+·1204x·x··-·907x·x··+·1621x·x··-·417x··+·1297x·x··+·1444x·x··+·4x·x··+·398x·x··+·996x·x··-·1031x·x··+·239x·x··+·303x··+·1215x·x··-·83x·x··+·1571x·x··-·1543x·x··-·925x·x··-·694x·x··+·151x·x··-·520x·x··+·880x·,·-·1210x··-·222x·x··+·185x··+·245x·x··+·1059x·x··-·322x··+·238x·x··+·962x·x··+·1260x·x··-·1581x··+·50x·x··+·1352x·x··-·1465x·x··+·1555x·x··+·1333x··+·1362x·x··+·1365x·x··+·1168x·x··-·1401x·x··+·149x·x··-·652x··+·1378x·x··-·557x·x··-·112x·x··+·26x·x··+·315x·x··+·111x·x··+·1592x··-·283x·x··-·1454x·x··+·907x·x··+·212x·x··+·400x·x··+·1049x·x··-·882x·x··-·1429x··-·183x·x··+·1571x·x··-·1286x·x··-·1179x·x··+·1319x·x··+·240x·x··-·1100x·x··+·1500x·x··-·348x·,·1051x··-·1325x·x··+·1354x··-·346x·x··-·1532x·x··-·466x··+·163x·x··-·659x·x··-·291x·x··+·966x··+·789x·x··+·393x·x··+·403x·x··-·1199x·x··-·570x··-·93x·x··-·492x·x··-·418x·x··+·713x·x··-·1323x·x··-·1384x··-·830x·x··-·54x·x··-·306x·x··+·709x·x··+·421x·x··-·954x·x··-·299x··+·1053x·x··-·1080x·x··+·686x·x··+·170x·x··-·1272x·x··-·1661x·x··+·1235x·x··+·1553x··-·1454x·x··-·1411x·x··-·1195x·x··-·962x·x··+·737x·x··-·390x·x··+·957x·x··+·1538x·x··+·1234x·,·-·509x··+·9x·x··-·1563x··-·710x·x··-·642x·x··+·541x··+·220x·x··-·1214x·x··-·16x·x··+·1008x··-·1088x·x··+·755x·x··-·886x·x··-·1433x·x··+·1154x··+·1627x·x··-·1547x·x··-·951x·x··+·866x·x··+·163x·x··-·1142x··-·668x·x··+·1361x·x··+·1324x·x··-·490x·x··+·282x·x··-·1133x·x··-·612x··+·805x·x··-·126x·x··+·1296x·x··-·973x·x··+·1271x·x··-·1646x·x··+·844x·x··+·1073x··-·1452x·x··-·1112x·x··-·141x·x··+·176x·x··-·1579x·x··-·78x·x··+·848x·x··-·1365x·x··+·711x·,·x··+·1543x·x··-·1076x··+·493x·x··-·526x·x··+·868x··-·582x·x··-·996x·x··+·206x·x··-·419x··+·1258x·x··-·391x·x··+·1002x·x··-·1539x·x··+·931x··-·1504x·x··+·810x·x··+·324x·x··+·1356x·x··+·313x·x··+·772x··+·299x·x··+·1186x·x··+·718x·x··+·407x·x··-·64x·x··-·828x·x··-·1393x··+·94x·x··-·290x·x··-·766x·x··+·950x·x··-·640x·x··+·265x·x··-·1640x·x··-·1403x··-·126x·x··+·891x·x··-·1519x·x··-·927x·x··-·1335x·x··-·1448x·x··-·x·x··-·1103x·x··-·1152x·,·821x··+·558x·x··-·1174x··-·168x·x··+·986x·x··+·790x··+·549x·x··+·817x·x··+·1396x·x··+·695x··+·1211x·x··+·878x·x··-·1061x·x··-·1244x·x··-·880x··+·1409x·x··-·567x·x··+·1240x·x··+·1126x·x··-·1262x·x··+·490x··+·1553x·x··+·1276x·x··+·805x·x··+·576x·x··-·1076x·x··+·1617x·x··-·495x··-·750x·x··-·277x·x··+·544x·x··+·1479x·x··-·784x·x··-·64x·x··-·1203x·x··+·405x··+·1013x·x··+·604x·x··+·1301x·x··+·1003x·x··+·235x·x··+·696x·x··+·939x·x··-·714x·x··-·879x·,·-·1452x··+·727x·x··-·1159x··+·449x·x··-·1169x·x··+·732x··+·575x·x··-·600x·x··+·924x·x··-·837x··+·1298x·x··-·860x·x··+·1010x·x··+·774x·x··+·319x··+·1087x·x··-·1120x·x··+·1439x·x··+·1175x·x··-·1648x·x··+·985x··-·1317x·x··-·878x·x··+·399x·x··-·1339x·x··+·70x·x··-·463x·x··+·470x··-·628x·x··-·907x·x··+·748x·x··+·98x·x··+·1150x·x··+·1140x·x··+·1308x·x··+·621x··+·369x·x··-·991x·x··-·1186x·x··+·61x·x··-·907x·x··-·681x·x··-·1528x·x··+·717x·x··+·854x·})
21 ·································0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3········3·······0·4·······1·4········2·4·······3·4········4·······0·5········1·5·······2·5········3·5·······4·5·······5······0·6·······1·6········2·6·······3·6······4·6·······5·6········6·······0·7·······1·7·······2·7·······3·7······4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8·······4·8·······5·8·······6·8········7·8·······8········0·······0·1········1········0·2········1·2········2·······0·3········1·3········2·3·······3······0·4········1·4·······2·4·······3·4········4········0·5········1·5········2·5·······3·5········4·5·······5······0·6········1·6·······2·6·······3·6·······4·6······5·6·······6······0·7········1·7········2·7·······3·7········4·7·······5·7······6·7·······7······0·8········1·8········2·8·······3·8········4·8········5·8········6·8········7·8·······8········0········0·1·······1········0·2········1·2·······2·······0·3·······1·3········2·3········3·······0·4········1·4·······2·4·······3·4·······4········0·5·······1·5·······2·5·······3·5·······4·5········5·······0·6·······1·6········2·6········3·6·······4·6········5·6·······6········0·7········1·7·····2·7·······3·7·······4·7········5·7·······6·7·······7········0·8······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1·······1·······0·2········1·2·······2·······0·3·······1·3········2·3········3······0·4········1·4········2·4········3·4········4········0·5········1·5········2·5········3·5·······4·5·······5········0·6·······1·6·······2·6······3·6·······4·6·······5·6········6·······0·7········1·7·······2·7·······3·7·······4·7········5·7·······6·7········7·······0·8········1·8········2·8········3·8········4·8·······5·8········6·8········7·8·······8·······0········0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4·······4······0·5·······1·5·······2·5·······3·5········4·5········5·······0·6······1·6·······2·6·······3·6·······4·6·······5·6·······6········0·7········1·7·······2·7·······3·7········4·7········5·7········6·7········7········0·8········1·8········2·8·······3·8·······4·8·······5·8·······6·8········7·8········8········0·····0·1········1·······0·2·······1·2·······2·······0·3········1·3······2·3········3········0·4·······1·4·······2·4········3·4········4········0·5········1·5·······2·5·······3·5·······4·5········5·······0·6········1·6········2·6·······3·6·······4·6········5·6·······6·······0·7·······1·7········2·7·······3·7········4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8········4·8······5·8·······6·8········7·8·······8···0········0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5·······2·5········3·5·······4·5·······5·······0·6········1·6·······2·6·······3·6······4·6·······5·6········6······0·7·······1·7·······2·7·······3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8········4·8········5·8····6·8········7·8········8······0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3········2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5········2·5········3·5········4·5·······5········0·6········1·6·······2·6·······3·6········4·6········5·6·······6·······0·7·······1·7·······2·7········3·7·······4·7······5·7········6·7·······7········0·8·······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4·······3·4·······4········0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6·······2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3·7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8······3·8·······4·8·······5·8········6·8·······7·8·······821 ·································0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3········3·······0·4·······1·4········2·4·······3·4········4·······0·5········1·5·······2·5········3·5·······4·5·······5······0·6·······1·6········2·6·······3·6······4·6·······5·6········6·······0·7·······1·7·······2·7·······3·7······4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8·······4·8·······5·8·······6·8········7·8·······8········0·······0·1········1········0·2········1·2········2·······0·3········1·3········2·3·······3······0·4········1·4·······2·4·······3·4········4········0·5········1·5········2·5·······3·5········4·5·······5······0·6········1·6·······2·6·······3·6·······4·6······5·6·······6······0·7········1·7········2·7·······3·7········4·7·······5·7······6·7·······7······0·8········1·8········2·8·······3·8········4·8········5·8········6·8········7·8·······8········0········0·1·······1········0·2········1·2·······2·······0·3·······1·3········2·3········3·······0·4········1·4·······2·4·······3·4·······4········0·5·······1·5·······2·5·······3·5·······4·5········5·······0·6·······1·6········2·6········3·6·······4·6········5·6·······6········0·7········1·7·····2·7·······3·7·······4·7········5·7·······6·7·······7········0·8······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1·······1·······0·2········1·2·······2·······0·3·······1·3········2·3········3······0·4········1·4········2·4········3·4········4········0·5········1·5········2·5········3·5·······4·5·······5········0·6·······1·6·······2·6······3·6·······4·6·······5·6········6·······0·7········1·7·······2·7·······3·7·······4·7········5·7·······6·7········7·······0·8········1·8········2·8········3·8········4·8·······5·8········6·8········7·8·······8·······0········0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4·······4······0·5·······1·5·······2·5·······3·5········4·5········5·······0·6······1·6·······2·6·······3·6·······4·6·······5·6·······6········0·7········1·7·······2·7·······3·7········4·7········5·7········6·7········7········0·8········1·8········2·8·······3·8·······4·8·······5·8·······6·8········7·8········8········0·····0·1········1·······0·2·······1·2·······2·······0·3········1·3······2·3········3········0·4·······1·4·······2·4········3·4········4········0·5········1·5·······2·5·······3·5·······4·5········5·······0·6········1·6········2·6·······3·6·······4·6········5·6·······6·······0·7·······1·7········2·7·······3·7········4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8········4·8······5·8·······6·8········7·8·······8···0········0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5·······2·5········3·5·······4·5·······5·······0·6········1·6·······2·6·······3·6······4·6·······5·6········6······0·7·······1·7·······2·7·······3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8········4·8········5·8····6·8········7·8········8······0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3········2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5········2·5········3·5········4·5·······5········0·6········1·6·······2·6·······3·6········4·6········5·6·······6·······0·7·······1·7·······2·7········3·7·······4·7······5·7········6·7·······7········0·8·······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4·······3·4·······4········0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6·······2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3·7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8······3·8·······4·8·······5·8········6·8·······7·8·······8
  
22 o6·:·RingMap·ringP8·<--·ringP822 o6·:·RingMap·ringP8·<--·ringP8
  
23 i7·:·time·degreeMap·phi'23 i7·:·time·degreeMap·phi'
24 ·--·used·0.49013s·(cpu);·0.408322s·(thread);·0s·(gc)24 ·--·used·0.633437s·(cpu);·0.479528s·(thread);·0s·(gc)
  
25 o7·=·1425 o7·=·14
  
26 i8·:·26 i8·:·
⊟
587 B
./usr/share/doc/Macaulay2/Cremona/example-output/_force__Image.out
    
Offset 5, 14 lines modifiedOffset 5, 14 lines modified
5 o2·:·Ideal·of·P65 o2·:·Ideal·of·P6
  
6 i3·:·Phi·=·rationalMap(X,Dominant=>2);6 i3·:·Phi·=·rationalMap(X,Dominant=>2);
  
7 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)7 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)
  
8 i4·:·time·forceImage(Phi,ideal·0_(target·Phi))8 i4·:·time·forceImage(Phi,ideal·0_(target·Phi))
9 ·--·used·0.000437588s·(cpu);·0.000557099s·(thread);·0s·(gc)9 ·--·used·0.000194825s·(cpu);·0.000649497s·(thread);·0s·(gc)
  
10 i5·:·Phi;10 i5·:·Phi;
  
11 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)11 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)
  
12 i6·:·12 i6·:·
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1.13 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_graph.out
    
Offset 35, 15 lines modifiedOffset 35, 15 lines modified
35 ······················-·x··+·x·x35 ······················-·x··+·x·x
36 ·························3····2·436 ·························3····2·4
37 ·····················}37 ·····················}
  
38 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)38 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
  
39 i3·:·time·(p1,p2)·=·graph·phi;39 i3·:·time·(p1,p2)·=·graph·phi;
40 ·--·used·0.279228s·(cpu);·0.0397628s·(thread);·0s·(gc)40 ·--·used·0.263051s·(cpu);·0.0336919s·(thread);·0s·(gc)
  
41 i4·:·p141 i4·:·p1
  
42 o4·=·--·rational·map·--42 o4·=·--·rational·map·--
43 ··································ZZ·································ZZ43 ··································ZZ·································ZZ
44 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y·,·y·,·y·,·y·,·y·])·defined·by44 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y·,·y·,·y·,·y·,·y·])·defined·by
45 ································190181··0···1···2···3···4··········190181··0···1···2···3···4···545 ································190181··0···1···2···3···4··········190181··0···1···2···3···4···5
Offset 173, 15 lines modifiedOffset 173, 15 lines modified
173 i8·:·projectiveDegrees·p2173 i8·:·projectiveDegrees·p2
  
174 o8·=·{51,·28,·14,·6,·2}174 o8·=·{51,·28,·14,·6,·2}
  
175 o8·:·List175 o8·:·List
  
176 i9·:·time·g·=·graph·p2;176 i9·:·time·g·=·graph·p2;
177 ·--·used·0.29238s·(cpu);·0.053494s·(thread);·0s·(gc)177 ·--·used·0.200086s·(cpu);·0.0445791s·(thread);·0s·(gc)
  
178 i10·:·g_0;178 i10·:·g_0;
  
179 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·PP^4)179 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·PP^4)
  
180 i11·:·g_1;180 i11·:·g_1;
  
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1.82 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_ideal_lp__Rational__Map_rp.out
    
Offset 33, 15 lines modifiedOffset 33, 15 lines modified
33 ······················x··-·x·x33 ······················x··-·x·x
34 ·······················1····0·334 ·······················1····0·3
35 ·····················}35 ·····················}
  
36 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)36 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)
  
37 i3·:·time·ideal·phi37 i3·:·time·ideal·phi
38 ·--·used·0.000499582s·(cpu);·0.00281534s·(thread);·0s·(gc)38 ·--·used·0.00399616s·(cpu);·0.00317362s·(thread);·0s·(gc)
  
39 ·············2·····································2······················39 ·············2·····································2······················
40 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+40 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+
41 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·3··41 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·3··
42 ·····------------------------------------------------------------------------42 ·····------------------------------------------------------------------------
43 ············243 ············2
44 ·····x·x·,·x··-·x·x·)44 ·····x·x·,·x··-·x·x·)
Offset 108, 15 lines modifiedOffset 108, 15 lines modified
108 ······················y108 ······················y
109 ·······················4109 ·······················4
110 ·····················}110 ·····················}
  
111 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^5·x·PP^4·to·PP^4)111 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^5·x·PP^4·to·PP^4)
  
112 i6·:·time·ideal·phi'112 i6·:·time·ideal·phi'
113 ·--·used·0.156274s·(cpu);·0.0916209s·(thread);·0s·(gc)113 ·--·used·0.171162s·(cpu);·0.102615s·(thread);·0s·(gc)
  
114 o6·=·ideal·1114 o6·=·ideal·1
  
115 ············································································································QQ[x·..x·,·y·..y·]115 ············································································································QQ[x·..x·,·y·..y·]
116 ················································································································0···5···0···4116 ················································································································0···5···0···4
117 o6·:·Ideal·of·--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------117 o6·:·Ideal·of·--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
118 ·····································································································································································································2118 ·····································································································································································································2
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5.75 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_inverse__Map.out
    
Offset 72, 15 lines modifiedOffset 72, 15 lines modified
72 ······················w·w··-·w·w··+·w·w72 ······················w·w··-·w·w··+·w·w
73 ·······················2·4····1·5····0·673 ·······················2·4····1·5····0·6
74 ·····················}74 ·····················}
  
75 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)75 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)
  
76 i2·:·time·psi·=·inverseMap·phi76 i2·:·time·psi·=·inverseMap·phi
77 ·--·used·0.12278s·(cpu);·0.0826654s·(thread);·0s·(gc)77 ·--·used·0.173236s·(cpu);·0.123646s·(thread);·0s·(gc)
  
78 o2·=·--·rational·map·--78 o2·=·--·rational·map·--
79 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])79 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])
80 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···2080 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20
81 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])81 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])
82 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···2082 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20
83 ·····defining·forms:·{83 ·····defining·forms:·{
Offset 158, 15 lines modifiedOffset 158, 15 lines modified
158 o4·=·map·(QQ[w·..w··],·QQ[w·..w··],·{w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···+·w··w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···+·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···+·w··w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···+·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···-·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w··-·w·w··-·w·w··+·w·w··-·w·w·})158 o4·=·map·(QQ[w·..w··],·QQ[w·..w··],·{w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···-·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···+·w··w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···+·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···+·w··w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···+·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w··w···-·w··w···-·w··w···+·w··w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w···-·w·w···+·w·w···+·w·w···-·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···-·w·w···+·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w···-·w·w···-·w·w···+·w·w···-·w·w··,·w·w··-·w·w··-·w·w··+·w·w··-·w·w·})
159 ··············0···26·······0···26·····21·22····20·23····15·24····10·25····0·26···19·22····18·23····16·24····11·25····1·26···19·20····18·21····17·24····12·25····2·26···15·19····16·21····17·23····13·25····3·26···10·19····11·21····12·23····13·24····4·26···0·19····1·21····2·23····3·24····4·25···15·18····16·20····17·22····14·25····5·26···10·18····11·20····12·22····14·24····6·26···0·18····1·20····2·22····5·24····6·25···12·16····11·17····13·18····14·19····7·26···2·16····1·17····3·18····5·19····7·25···12·15····10·17····13·20····14·21····8·26···11·15····10·16····13·22····14·23····9·26···2·15····0·17····3·20····5·21····8·25···1·15····0·16····3·22····5·23····9·25···5·13····3·14····7·15····8·16····9·17···5·12····2·14····6·17····8·18····7·20···3·12····2·13····4·17····8·19····7·21···5·11····1·14····6·16····9·18····7·22···3·11····1·13····4·16····9·19····7·23···2·11····1·12····4·18····6·19····7·24···7·10····8·11····9·12····6·13····4·14···5·10····0·14····6·15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4·20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7····1·8····2·9159 ··············0···26·······0···26·····21·22····20·23····15·24····10·25····0·26···19·22····18·23····16·24····11·25····1·26···19·20····18·21····17·24····12·25····2·26···15·19····16·21····17·23····13·25····3·26···10·19····11·21····12·23····13·24····4·26···0·19····1·21····2·23····3·24····4·25···15·18····16·20····17·22····14·25····5·26···10·18····11·20····12·22····14·24····6·26···0·18····1·20····2·22····5·24····6·25···12·16····11·17····13·18····14·19····7·26···2·16····1·17····3·18····5·19····7·25···12·15····10·17····13·20····14·21····8·26···11·15····10·16····13·22····14·23····9·26···2·15····0·17····3·20····5·21····8·25···1·15····0·16····3·22····5·23····9·25···5·13····3·14····7·15····8·16····9·17···5·12····2·14····6·17····8·18····7·20···3·12····2·13····4·17····8·19····7·21···5·11····1·14····6·16····9·18····7·22···3·11····1·13····4·16····9·19····7·23···2·11····1·12····4·18····6·19····7·24···7·10····8·11····9·12····6·13····4·14···5·10····0·14····6·15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4·20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7····1·8····2·9
  
160 o4·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]160 o4·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]
161 ·················0···26··········0···26161 ·················0···26··········0···26
  
162 i5·:·time·psi·=·inverseMap·phi162 i5·:·time·psi·=·inverseMap·phi
163 ·--·used·0.24329s·(cpu);·0.149285s·(thread);·0s·(gc)163 ·--·used·0.307946s·(cpu);·0.199633s·(thread);·0s·(gc)
  
164 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·w·w···-·w·w···+·w·w···+·w··w···-·w··w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·w··w···-·w··w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w···+·w·w···-·w·w··,·-·w·w··+·w·w··+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w··+·w·w···-·w·w··})164 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·w·w···-·w·w···+·w·w···+·w··w···-·w··w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·w··w···-·w··w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w···+·w·w···-·w·w··,·-·w·w··+·w·w··+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w··+·w·w···-·w·w··})
165 ··············0···26·······0···26·······5·22····8·23····14·24····13·25····0·26·····5·18····8·19····14·20····10·25····1·26·····5·16····8·17····13·20····10·24····2·26·····5·15····14·17····13·19····10·23····3·26·····5·21····20·23····19·24····17·25····4·26·····8·15····14·16····13·18····10·22····6·26·····8·21····20·22····18·24····16·25····7·26···17·18····16·19····15·20····10·21····9·26·····13·21····17·22····16·23····15·24····11·26·····14·21····19·22····18·23····15·25····12·26···0·21····4·22····7·23····12·24····11·25·····4·18····7·19····12·20····1·21····9·25·····4·16····7·17····11·20····2·21····9·24·····4·15····12·17····11·19····3·21····9·23·····7·15····12·16····11·18····6·21····9·22···12·13····11·14····0·15····3·22····6·23···10·12····9·14····1·15····3·18····6·19···10·11····9·13····2·15····3·16····6·17···8·9····7·10····1·16····2·18····6·20···5·9····4·10····1·17····2·19····3·20···8·11····7·13····0·16····2·22····6·24···5·11····4·13····0·17····2·23····3·24···8·12····7·14····0·18····1·22····6·25···5·12····4·14····0·19····1·23····3·25···5·7····4·8····0·20····1·24····2·25·····5·6····3·8····0·10····1·13····2·14···4·6····3·7····0·9····1·11····2·12165 ··············0···26·······0···26·······5·22····8·23····14·24····13·25····0·26·····5·18····8·19····14·20····10·25····1·26·····5·16····8·17····13·20····10·24····2·26·····5·15····14·17····13·19····10·23····3·26·····5·21····20·23····19·24····17·25····4·26·····8·15····14·16····13·18····10·22····6·26·····8·21····20·22····18·24····16·25····7·26···17·18····16·19····15·20····10·21····9·26·····13·21····17·22····16·23····15·24····11·26·····14·21····19·22····18·23····15·25····12·26···0·21····4·22····7·23····12·24····11·25·····4·18····7·19····12·20····1·21····9·25·····4·16····7·17····11·20····2·21····9·24·····4·15····12·17····11·19····3·21····9·23·····7·15····12·16····11·18····6·21····9·22···12·13····11·14····0·15····3·22····6·23···10·12····9·14····1·15····3·18····6·19···10·11····9·13····2·15····3·16····6·17···8·9····7·10····1·16····2·18····6·20···5·9····4·10····1·17····2·19····3·20···8·11····7·13····0·16····2·22····6·24···5·11····4·13····0·17····2·23····3·24···8·12····7·14····0·18····1·22····6·25···5·12····4·14····0·19····1·23····3·25···5·7····4·8····0·20····1·24····2·25·····5·6····3·8····0·10····1·13····2·14···4·6····3·7····0·9····1·11····2·12
  
166 o5·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]166 o5·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]
167 ·················0···26··········0···26167 ·················0···26··········0···26
  
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3.09 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_inverse_lp__Rational__Map_rp.out
    
Offset 28, 15 lines modifiedOffset 28, 15 lines modified
28 ······················-·-----x··-·---------x·x··+·-----------x·x··+·-----------x·x··-·--------x··-·-------x·x··+·-----------x·x·x··+·-----------x·x·x··+·----------x·x··+·--------x·x··+·--------x·x·x··-·-----------x·x··-·-------x·x··-·----------x·x··-·-------x··-·------x·x··+·----------x·x·x··+·-----------x·x·x··-·----------x·x··+·---------x·x·x··+·----------x·x·x·x··+·-----------x·x·x··+·----------x·x·x··+·------------x·x·x··+·---------x·x··-·---------x·x··-·-----------x·x·x··-·-----------x·x··-·----------x·x·x··+·----------x·x·x··-·---------x·x··+·-------x·x··-·--------x·x··+·----------x·x··+·------x··-·---------x·x··-·-----------x·x·x··+·------------x·x·x··+·----------x·x··-·-----------x·x·x··-·-------------x·x·x·x··+·------------x·x·x··-·----------x·x·x··-·-----------x·x·x··-·-------x·x··+·-----------x·x·x··+·------------x·x·x·x··+·------------x·x·x··+·-----------x·x·x·x··+·--------------x·x·x·x··+·---------x·x·x··-·------------x·x·x··-·------------x·x·x··-·------------x·x·x··+·----------x·x··-·----------x·x··-·------------x·x·x··+·-----------x·x··-·----------x·x·x··-·------------x·x·x··-·---------x·x··+·------------x·x·x··-·--------x·x·x··+·------------x·x·x··+·----------x·x··-·--------x·x··-·---------x·x··-·---------x·x··+·---------x·x··-·-----x28 ······················-·-----x··-·---------x·x··+·-----------x·x··+·-----------x·x··-·--------x··-·-------x·x··+·-----------x·x·x··+·-----------x·x·x··+·----------x·x··+·--------x·x··+·--------x·x·x··-·-----------x·x··-·-------x·x··-·----------x·x··-·-------x··-·------x·x··+·----------x·x·x··+·-----------x·x·x··-·----------x·x··+·---------x·x·x··+·----------x·x·x·x··+·-----------x·x·x··+·----------x·x·x··+·------------x·x·x··+·---------x·x··-·---------x·x··-·-----------x·x·x··-·-----------x·x··-·----------x·x·x··+·----------x·x·x··-·---------x·x··+·-------x·x··-·--------x·x··+·----------x·x··+·------x··-·---------x·x··-·-----------x·x·x··+·------------x·x·x··+·----------x·x··-·-----------x·x·x··-·-------------x·x·x·x··+·------------x·x·x··-·----------x·x·x··-·-----------x·x·x··-·-------x·x··+·-----------x·x·x··+·------------x·x·x·x··+·------------x·x·x··+·-----------x·x·x·x··+·--------------x·x·x·x··+·---------x·x·x··-·------------x·x·x··-·------------x·x·x··-·------------x·x·x··+·----------x·x··-·----------x·x··-·------------x·x·x··+·-----------x·x··-·----------x·x·x··-·------------x·x·x··-·---------x·x··+·------------x·x·x··-·--------x·x·x··+·------------x·x·x··+·----------x·x··-·--------x·x··-·---------x·x··-·---------x·x··+·---------x·x··-·-----x
29 ·························2800·0····6350400··0·1·····50803200··0·1·····33868800··0·1····181440··1····196000·0·2····381024000··0·1·2·····47628000··0·1·2····10160640··1·2····2268000·0·2····2126250·0·1·2····762048000··1·2····992250·0·2····31752000··1·2····529200·2····73500·0·3····28576800··0·1·3·····32659200··0·1·3·····6531840··1·3····15876000·0·2·3····21432600··0·1·2·3····137168640··1·2·3····158760000·0·2·3·····571536000··1·2·3····95256000·2·3····15876000·0·3····228614400··0·1·3·····65318400··1·3····190512000·0·2·3····95256000··1·2·3····31752000·2·3····352800·0·3····604800··1·3····31752000··2·3····15120·3····47628000·0·4····444528000··0·1·4····4267468800··0·1·4····152409600·1·4····714420000··0·2·4····16003008000··0·1·2·4····1524096000··1·2·4····95256000··0·2·4····533433600··1·2·4····211680·2·4····1000188000·0·3·4····2667168000··0·1·3·4·····457228800··1·3·4····240045120··0·2·3·4·····48009024000··1·2·3·4····14817600·2·3·4····4000752000··0·3·4····1524096000··1·3·4····2667168000··2·3·4····27216000··3·4····190512000·0·4····1778112000··0·1·4····304819200··1·4····47628000··0·2·4····1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4····1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2·4····21168000·3·4···28000·429 ·························2800·0····6350400··0·1·····50803200··0·1·····33868800··0·1····181440··1····196000·0·2····381024000··0·1·2·····47628000··0·1·2····10160640··1·2····2268000·0·2····2126250·0·1·2····762048000··1·2····992250·0·2····31752000··1·2····529200·2····73500·0·3····28576800··0·1·3·····32659200··0·1·3·····6531840··1·3····15876000·0·2·3····21432600··0·1·2·3····137168640··1·2·3····158760000·0·2·3·····571536000··1·2·3····95256000·2·3····15876000·0·3····228614400··0·1·3·····65318400··1·3····190512000·0·2·3····95256000··1·2·3····31752000·2·3····352800·0·3····604800··1·3····31752000··2·3····15120·3····47628000·0·4····444528000··0·1·4····4267468800··0·1·4····152409600·1·4····714420000··0·2·4····16003008000··0·1·2·4····1524096000··1·2·4····95256000··0·2·4····533433600··1·2·4····211680·2·4····1000188000·0·3·4····2667168000··0·1·3·4·····457228800··1·3·4····240045120··0·2·3·4·····48009024000··1·2·3·4····14817600·2·3·4····4000752000··0·3·4····1524096000··1·3·4····2667168000··2·3·4····27216000··3·4····190512000·0·4····1778112000··0·1·4····304819200··1·4····47628000··0·2·4····1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4····1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2·4····21168000·3·4···28000·4
30 ·····················}30 ·····················}
  
31 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)31 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)
  
32 i3·:·time·inverse·phi32 i3·:·time·inverse·phi
33 ·--·used·0.104411s·(cpu);·0.0692944s·(thread);·0s·(gc)33 ·--·used·0.129512s·(cpu);·0.0800795s·(thread);·0s·(gc)
  
34 o3·=·--·rational·map·--34 o3·=·--·rational·map·--
35 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])35 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
36 ······················0···1···2···3···436 ······················0···1···2···3···4
37 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])37 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
38 ······················0···1···2···3···438 ······················0···1···2···3···4
39 ·····defining·forms:·{39 ·····defining·forms:·{
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701 B
./usr/share/doc/Macaulay2/Cremona/example-output/_is__Birational.out
    
Offset 40, 18 lines modifiedOffset 40, 18 lines modified
40 ······················-·t··+·t·t40 ······················-·t··+·t·t
41 ·························3····2·441 ·························3····2·4
42 ·····················}42 ·····················}
  
43 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)43 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
  
44 i3·:·time·isBirational·phi44 i3·:·time·isBirational·phi
45 ·--·used·0.0702755s·(cpu);·0.0298302s·(thread);·0s·(gc)45 ·--·used·0.0690809s·(cpu);·0.0277457s·(thread);·0s·(gc)
  
46 o3·=·true46 o3·=·true
  
47 i4·:·time·isBirational(phi,Certify=>true)47 i4·:·time·isBirational(phi,Certify=>true)
48 ·--·used·0.285206s·(cpu);·0.0519896s·(thread);·0s·(gc)48 ·--·used·0.230439s·(cpu);·0.0350467s·(thread);·0s·(gc)
49 Certify:·output·certified!49 Certify:·output·certified!
  
50 o4·=·true50 o4·=·true
  
51 i5·:·51 i5·:·
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938 B
./usr/share/doc/Macaulay2/Cremona/example-output/_is__Dominant.out
    
Offset 3, 15 lines modifiedOffset 3, 15 lines modified
3 i1·:·P8·=·ZZ/101[x_0..x_8];3 i1·:·P8·=·ZZ/101[x_0..x_8];
  
4 i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};4 i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
  
5 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)5 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)
  
6 i3·:·time·isDominant(phi,Certify=>true)6 i3·:·time·isDominant(phi,Certify=>true)
7 ·--·used·1.78811s·(cpu);·1.6522s·(thread);·0s·(gc)7 ·--·used·2.0669s·(cpu);·1.88524s·(thread);·0s·(gc)
8 Certify:·output·certified!8 Certify:·output·certified!
  
9 o3·=·true9 o3·=·true
  
10 i4·:·P7·=·ZZ/101[x_0..x_7];10 i4·:·P7·=·ZZ/101[x_0..x_7];
  
11 i5·:·--·hyperelliptic·curve·of·genus·311 i5·:·--·hyperelliptic·curve·of·genus·3
Offset 20, 13 lines modifiedOffset 20, 13 lines modified
20 o5·:·Ideal·of·P720 o5·:·Ideal·of·P7
  
21 i6·:·phi·=·rationalMap(C,3,2);21 i6·:·phi·=·rationalMap(C,3,2);
  
22 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)22 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)
  
23 i7·:·time·isDominant(phi,Certify=>true)23 i7·:·time·isDominant(phi,Certify=>true)
24 ·--·used·2.43825s·(cpu);·2.01795s·(thread);·0s·(gc)24 ·--·used·2.7713s·(cpu);·2.22914s·(thread);·0s·(gc)
25 Certify:·output·certified!25 Certify:·output·certified!
  
26 o7·=·false26 o7·=·false
  
27 i8·:·27 i8·:·
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3.66 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_kernel_lp__Ring__Map_cm__Z__Z_rp.out
    
Offset 6, 23 lines modifiedOffset 6, 23 lines modified
6 o1·=·map·(QQ[x·..x·],·QQ[y·..y··],·{-·5x·x··+·x·x··+·x·x··+·35x·x··-·7x·x··+·x·x··-·x·x··-·49x··-·5x·x··+·2x·x··-·x·x··+·27x·x··-·4x··+·x·x··-·7x·x··+·2x·x··-·2x·x··+·14x·x··-·4x·x·,·-·x·x··-·6x·x··-·5x·x··+·2x·x··+·x·x··+·x·x··-·5x·x··-·x·x··+·2x·x··+·7x·x··-·2x·x··+·2x·x··-·3x·x·,·-·25x··+·9x·x··+·10x·x··-·2x·x··-·x··+·29x·x··-·x·x··-·7x·x··-·13x·x··+·3x·x··+·x·x··-·x·x··+·2x·x··-·x·x··+·7x·x··-·2x·x··-·8x·x··+·2x·x··-·3x·x·,·x·x··+·x·x··+·x··+·7x·x··-·9x·x··+·12x·x··-·4x··+·2x·x··+·2x·x··-·14x·x··+·4x·x··+·x·x··-·x·x··-·14x·x··+·x·x·,·-·5x·x··+·x·x··-·7x·x··+·8x·x··-·5x·x··+·2x·x··-·x·x··+·x·x··-·x·x··+·7x·x··-·2x·x··-·x·x··+·7x·x··-·2x·x·,·x·x··+·x··-·7x·x··-·8x·x··+·x·x··+·x·x··+·2x·x··-·x·x··+·x·x··-·7x·x··+·2x·x··+·x·x··-·7x·x··+·2x·x·,·x·x··+·x··-·8x·x··+·x·x··+·6x·x··-·2x··+·x·x··+·x·x··-·7x·x··+·2x·x··+·x·x··-·7x·x··+·2x·x·,·x·x··-·7x·x··+·x·x··+·x·x··-·7x·x··+·2x··-·x·x·,·-·4x·x··+·x·x··-·x··-·7x·x··+·8x·x··+·x·x··-·x·x··-·6x·x··+·2x··-·x·x··-·x·x··+·7x·x··-·2x·x··-·x·x··+·7x·x··-·2x·x·,·-·5x·x··+·2x··+·x·x··-·x··-·x·x··+·8x·x··-·10x·x··+·2x·x··+·2x·x··-·2x·x··+·14x·x··-·4x·x··+·5x·x··-·3x·x··-·2x·x··+·7x·x··-·2x·x··-·3x·x·,·-·5x·x··+·x·x··+·x·x··-·4x·x··-·x·x··+·x·x··+·x·x·,·x·x··-·x·x··+·5x·x··+·x·x··-·14x·x··-·x·x··-·8x·x··-·8x·x··+·2x·x··+·4x·x··+·2x·x··+·4x·x··+·3x·x··-·7x·x··+·2x·x··-·3x·x·})6 o1·=·map·(QQ[x·..x·],·QQ[y·..y··],·{-·5x·x··+·x·x··+·x·x··+·35x·x··-·7x·x··+·x·x··-·x·x··-·49x··-·5x·x··+·2x·x··-·x·x··+·27x·x··-·4x··+·x·x··-·7x·x··+·2x·x··-·2x·x··+·14x·x··-·4x·x·,·-·x·x··-·6x·x··-·5x·x··+·2x·x··+·x·x··+·x·x··-·5x·x··-·x·x··+·2x·x··+·7x·x··-·2x·x··+·2x·x··-·3x·x·,·-·25x··+·9x·x··+·10x·x··-·2x·x··-·x··+·29x·x··-·x·x··-·7x·x··-·13x·x··+·3x·x··+·x·x··-·x·x··+·2x·x··-·x·x··+·7x·x··-·2x·x··-·8x·x··+·2x·x··-·3x·x·,·x·x··+·x·x··+·x··+·7x·x··-·9x·x··+·12x·x··-·4x··+·2x·x··+·2x·x··-·14x·x··+·4x·x··+·x·x··-·x·x··-·14x·x··+·x·x·,·-·5x·x··+·x·x··-·7x·x··+·8x·x··-·5x·x··+·2x·x··-·x·x··+·x·x··-·x·x··+·7x·x··-·2x·x··-·x·x··+·7x·x··-·2x·x·,·x·x··+·x··-·7x·x··-·8x·x··+·x·x··+·x·x··+·2x·x··-·x·x··+·x·x··-·7x·x··+·2x·x··+·x·x··-·7x·x··+·2x·x·,·x·x··+·x··-·8x·x··+·x·x··+·6x·x··-·2x··+·x·x··+·x·x··-·7x·x··+·2x·x··+·x·x··-·7x·x··+·2x·x·,·x·x··-·7x·x··+·x·x··+·x·x··-·7x·x··+·2x··-·x·x·,·-·4x·x··+·x·x··-·x··-·7x·x··+·8x·x··+·x·x··-·x·x··-·6x·x··+·2x··-·x·x··-·x·x··+·7x·x··-·2x·x··-·x·x··+·7x·x··-·2x·x·,·-·5x·x··+·2x··+·x·x··-·x··-·x·x··+·8x·x··-·10x·x··+·2x·x··+·2x·x··-·2x·x··+·14x·x··-·4x·x··+·5x·x··-·3x·x··-·2x·x··+·7x·x··-·2x·x··-·3x·x·,·-·5x·x··+·x·x··+·x·x··-·4x·x··-·x·x··+·x·x··+·x·x·,·x·x··-·x·x··+·5x·x··+·x·x··-·14x·x··-·x·x··-·8x·x··-·8x·x··+·2x·x··+·4x·x··+·2x·x··+·4x·x··+·3x·x··-·7x·x··+·2x·x··-·3x·x·})
7 ··············0···8·······0···11········0·3····2·4····3·4······0·5·····2·5····3·5····4·5······5·····0·6·····2·6····4·6······5·6·····6····4·7·····5·7·····6·7·····4·8······5·8·····6·8·····1·2·····1·5·····0·6·····1·6····4·6····5·6·····0·7····1·7·····2·7·····5·7·····6·7·····1·8·····7·8·······0·····0·2······0·4·····2·4····4······0·5····2·5·····4·5······0·6·····4·6····5·6····0·7·····2·7····4·7·····5·7·····6·7·····0·8·····4·8·····7·8···2·4····3·4····4·····2·5·····4·5······5·6·····6·····3·7·····4·7······5·7·····6·7····3·8····4·8······5·8····6·8······0·4····2·4·····2·5·····4·5·····0·6·····2·6····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···0·4····4·····1·5·····4·5····0·6····1·6·····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···2·3····4·····4·5····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8···1·3·····1·5····1·6····4·6·····5·6·····6····3·7······0·3····3·4····4·····0·5·····4·5····0·6····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8······0·2·····2····2·4····4····2·5·····4·5······0·6·····5·6·····2·7·····4·7······5·7·····6·7·····0·8·····2·8·····4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·7···0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····2·7·····0·8·····1·8·····5·8·····6·8·····7·87 ··············0···8·······0···11········0·3····2·4····3·4······0·5·····2·5····3·5····4·5······5·····0·6·····2·6····4·6······5·6·····6····4·7·····5·7·····6·7·····4·8······5·8·····6·8·····1·2·····1·5·····0·6·····1·6····4·6····5·6·····0·7····1·7·····2·7·····5·7·····6·7·····1·8·····7·8·······0·····0·2······0·4·····2·4····4······0·5····2·5·····4·5······0·6·····4·6····5·6····0·7·····2·7····4·7·····5·7·····6·7·····0·8·····4·8·····7·8···2·4····3·4····4·····2·5·····4·5······5·6·····6·····3·7·····4·7······5·7·····6·7····3·8····4·8······5·8····6·8······0·4····2·4·····2·5·····4·5·····0·6·····2·6····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···0·4····4·····1·5·····4·5····0·6····1·6·····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···2·3····4·····4·5····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8···1·3·····1·5····1·6····4·6·····5·6·····6····3·7······0·3····3·4····4·····0·5·····4·5····0·6····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8······0·2·····2····2·4····4····2·5·····4·5······0·6·····5·6·····2·7·····4·7······5·7·····6·7·····0·8·····2·8·····4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·7···0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····2·7·····0·8·····1·8·····5·8·····6·8·····7·8
  
8 o1·:·RingMap·QQ[x·..x·]·<--·QQ[y·..y··]8 o1·:·RingMap·QQ[x·..x·]·<--·QQ[y·..y··]
9 ·················0···8··········0···119 ·················0···8··········0···11
  
10 i2·:·time·kernel(phi,1)10 i2·:·time·kernel(phi,1)
11 ·--·used·0.013838s·(cpu);·0.0137s·(thread);·0s·(gc)11 ·--·used·0.0148932s·(cpu);·0.0163007s·(thread);·0s·(gc)
  
12 o2·=·ideal·()12 o2·=·ideal·()
  
13 o2·:·Ideal·of·QQ[y·..y··]13 o2·:·Ideal·of·QQ[y·..y··]
14 ··················0···1114 ··················0···11
  
15 i3·:·time·kernel(phi,2)15 i3·:·time·kernel(phi,2)
16 ·--·used·0.330787s·(cpu);·0.255611s·(thread);·0s·(gc)16 ·--·used·0.370638s·(cpu);·0.299304s·(thread);·0s·(gc)
  
17 ···························2················································17 ···························2················································
18 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-18 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-
19 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·6··19 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·6··
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ···········································································21 ···········································································
22 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-22 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-
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1.23 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_parametrize_lp__Ideal_rp.out
    
Offset 26, 15 lines modifiedOffset 26, 15 lines modified
26 ··············8···········926 ··············8···········9
  
27 ·················ZZ27 ·················ZZ
28 o2·:·Ideal·of·--------[x·..x·]28 o2·:·Ideal·of·--------[x·..x·]
29 ··············10000019··0···929 ··············10000019··0···9
  
30 i3·:·time·parametrize·L30 i3·:·time·parametrize·L
31 ·--·used·0.00204689s·(cpu);·0.00414759s·(thread);·0s·(gc)31 ·--·used·0.00400104s·(cpu);·0.00500885s·(thread);·0s·(gc)
  
32 o3·=·--·rational·map·--32 o3·=·--·rational·map·--
33 ·····················ZZ33 ·····················ZZ
34 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])34 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])
35 ··················10000019··0···1···2···3···4···535 ··················10000019··0···1···2···3···4···5
36 ·····················ZZ36 ·····················ZZ
37 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])37 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 116, 15 lines modifiedOffset 116, 15 lines modified
116 ·············5·9···········6·9···········7·9···········8·9···········9116 ·············5·9···········6·9···········7·9···········8·9···········9
  
117 ·················ZZ117 ·················ZZ
118 o4·:·Ideal·of·--------[x·..x·]118 o4·:·Ideal·of·--------[x·..x·]
119 ··············10000019··0···9119 ··············10000019··0···9
  
120 i5·:·time·parametrize·Q120 i5·:·time·parametrize·Q
121 ·--·used·0.64793s·(cpu);·0.336382s·(thread);·0s·(gc)121 ·--·used·0.714262s·(cpu);·0.410003s·(thread);·0s·(gc)
  
122 o5·=·--·rational·map·--122 o5·=·--·rational·map·--
123 ·····················ZZ123 ·····················ZZ
124 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])124 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
125 ··················10000019··0···1···2···3···4···5···6125 ··················10000019··0···1···2···3···4···5···6
126 ·····················ZZ126 ·····················ZZ
127 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])127 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
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1.64 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_point_lp__Quotient__Ring_rp.out
    
Offset 1, 15 lines modifiedOffset 1, 15 lines modified
1 --·-*-·M2-comint·-*-·hash:·35605838294899886901 --·-*-·M2-comint·-*-·hash:·3560583829489988690
  
2 i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);2 i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);
  
3 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)3 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)
  
4 i2·:·time·p·=·point·source·f4 i2·:·time·p·=·point·source·f
5 ·--·used·0.159174s·(cpu);·0.117175s·(thread);·0s·(gc)5 ·--·used·0.175983s·(cpu);·0.122205s·(thread);·0s·(gc)
  
6 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+6 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+
7 ·············10········11···9·········11···8········11···7········11···6··7 ·············10········11···9·········11···8········11···7········11···6··
8 ·····------------------------------------------------------------------------8 ·····------------------------------------------------------------------------
9 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-9 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-
10 ···········11···5········11···4········11···3·········11···2········11···1··10 ···········11···5········11···4········11···3·········11···2········11···1··
11 ·····------------------------------------------------------------------------11 ·····------------------------------------------------------------------------
Offset 20, 12 lines modifiedOffset 20, 12 lines modified
20 ···························································-----[y·..y··]20 ···························································-----[y·..y··]
21 ···························································33331··0···1121 ···························································33331··0···11
22 o2·:·Ideal·of·-------------------------------------------------------------------------------------------------------22 o2·:·Ideal·of·-------------------------------------------------------------------------------------------------------
23 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)23 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
24 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·11···3·4····1·6····0·8···2·4····1·5····0·724 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·11···3·4····1·6····0·8···2·4····1·5····0·7
  
25 i3·:·time·p·==·f^*·f·p25 i3·:·time·p·==·f^*·f·p
26 ·--·used·0.152976s·(cpu);·0.109782s·(thread);·0s·(gc)26 ·--·used·0.152868s·(cpu);·0.108276s·(thread);·0s·(gc)
  
27 o3·=·true27 o3·=·true
  
28 i4·:·28 i4·:·
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2.71 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_projective__Degrees.out
    
Offset 7, 15 lines modifiedOffset 7, 15 lines modified
7 o2·=·map·(GF·109561[t·..t·],·GF·109561[x·..x·],·{-·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·-·t·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·a})7 o2·=·map·(GF·109561[t·..t·],·GF·109561[x·..x·],·{-·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·-·t·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·a})
8 ·····················0···4··············0···5·······1····0·2·····1·2····0·3·····2····1·3·····1·3····0·4·····2·3····1·4·····3····2·48 ·····················0···4··············0···5·······1····0·2·····1·2····0·3·····2····1·3·····1·3····0·4·····2·3····1·4·····3····2·4
  
9 o2·:·RingMap·GF·109561[t·..t·]·<--·GF·109561[x·..x·]9 o2·:·RingMap·GF·109561[t·..t·]·<--·GF·109561[x·..x·]
10 ························0···4·················0···510 ························0···4·················0···5
  
11 i3·:·time·projectiveDegrees(phi,Certify=>true)11 i3·:·time·projectiveDegrees(phi,Certify=>true)
12 ·--·used·0.292879s·(cpu);·0.0432214s·(thread);·0s·(gc)12 ·--·used·0.232918s·(cpu);·0.0344567s·(thread);·0s·(gc)
13 Certify:·output·certified!13 Certify:·output·certified!
  
14 o3·=·{1,·2,·4,·4,·2}14 o3·=·{1,·2,·4,·4,·2}
  
15 o3·:·List15 o3·:·List
  
16 i4·:·psi=inverseMap(toMap(phi,Dominant=>infinity))16 i4·:·psi=inverseMap(toMap(phi,Dominant=>infinity))
Offset 29, 15 lines modifiedOffset 29, 15 lines modified
29 ··············GF·109561[x·..x·]29 ··············GF·109561[x·..x·]
30 ·························0···530 ·························0···5
31 o4·:·RingMap·------------------·<--·GF·109561[t·..t·]31 o4·:·RingMap·------------------·<--·GF·109561[t·..t·]
32 ·············x·x··-·x·x··+·x·x·················0···432 ·············x·x··-·x·x··+·x·x·················0···4
33 ··············2·3····1·4····0·533 ··············2·3····1·4····0·5
  
34 i5·:·time·projectiveDegrees(psi,Certify=>true)34 i5·:·time·projectiveDegrees(psi,Certify=>true)
35 ·--·used·0.230338s·(cpu);·0.0297309s·(thread);·0s·(gc)35 ·--·used·0.157863s·(cpu);·0.0409547s·(thread);·0s·(gc)
36 Certify:·output·certified!36 Certify:·output·certified!
  
37 o5·=·{2,·4,·4,·2,·1}37 o5·=·{2,·4,·4,·2,·1}
  
38 o5·:·List38 o5·:·List
  
39 i6·:·--·Cremona·transformation·of·P^6·defined·by·the·quadrics·through·a·rational·octic·surface39 i6·:·--·Cremona·transformation·of·P^6·defined·by·the·quadrics·through·a·rational·octic·surface
Offset 48, 21 lines modifiedOffset 48, 21 lines modified
48 ··········300007··0···6···300007··0···6·····2·4····1·5··········0·4··········1·4··········4·········0·5··········1·5·········2·5··········4·5·········5··········3·6·········4·6·········5·6···2·3····0·5··········1·3··········1·4··········4·········0·5··········1·5·········2·5··········4·5·········5··········3·6·········4·6·········5·6········0·3·········1·4·········3·4·········4··········0·5·········1·5·········2·5··········3·5··········4·5·········5·········3·6··········4·6·········5·6··········0·1··········1·········0·2··········1·2·········2··········1·4··········1·5·········2·5··········0·6·········1·6·········2·6·········0··········1·········0·2·········1·2·········2·········1·4··········4·········0·5·········1·5··········2·5··········4·5·········5·········0·6·········1·6··········2·6··········3·6·········4·6·········5·648 ··········300007··0···6···300007··0···6·····2·4····1·5··········0·4··········1·4··········4·········0·5··········1·5·········2·5··········4·5·········5··········3·6·········4·6·········5·6···2·3····0·5··········1·3··········1·4··········4·········0·5··········1·5·········2·5··········4·5·········5··········3·6·········4·6·········5·6········0·3·········1·4·········3·4·········4··········0·5·········1·5·········2·5··········3·5··········4·5·········5·········3·6··········4·6·········5·6··········0·1··········1·········0·2··········1·2·········2··········1·4··········1·5·········2·5··········0·6·········1·6·········2·6·········0··········1·········0·2·········1·2·········2·········1·4··········4·········0·5·········1·5··········2·5··········4·5·········5·········0·6·········1·6··········2·6··········3·6·········4·6·········5·6
  
49 ···············ZZ·················ZZ49 ···············ZZ·················ZZ
50 o6·:·RingMap·------[x·..x·]·<--·------[x·..x·]50 o6·:·RingMap·------[x·..x·]·<--·------[x·..x·]
51 ·············300007··0···6······300007··0···651 ·············300007··0···6······300007··0···6
  
52 i7·:·time·projectiveDegrees·phi52 i7·:·time·projectiveDegrees·phi
53 ·--·used·0.00292301s·(cpu);·3.7126e-05s·(thread);·0s·(gc)53 ·--·used·0.000623389s·(cpu);·4.3461e-05s·(thread);·0s·(gc)
  
54 o7·=·{1,·2,·4,·8,·8,·4,·1}54 o7·=·{1,·2,·4,·8,·8,·4,·1}
  
55 o7·:·List55 o7·:·List
  
56 i8·:·time·projectiveDegrees(phi,NumDegrees=>1)56 i8·:·time·projectiveDegrees(phi,NumDegrees=>1)
57 ·--·used·8.3025e-05s·(cpu);·1.4762e-05s·(thread);·0s·(gc)57 ·--·used·8.1052e-05s·(cpu);·1.8395e-05s·(thread);·0s·(gc)
  
58 o8·=·{4,·1}58 o8·=·{4,·1}
  
59 o8·:·List59 o8·:·List
  
60 i9·:·60 i9·:·
⊟
841 B
./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.out
    
Offset 3, 15 lines modifiedOffset 3, 15 lines modified
3 i1·:·ZZ/33331[x_0..x_6];·V·=·ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);3 i1·:·ZZ/33331[x_0..x_6];·V·=·ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
  
4 ················ZZ4 ················ZZ
5 o2·:·Ideal·of·-----[x·..x·]5 o2·:·Ideal·of·-----[x·..x·]
6 ··············33331··0···66 ··············33331··0···6
  
7 i3·:·time·phi·=·rationalMap(V,3,2)7 i3·:·time·phi·=·rationalMap(V,3,2)
8 ·--·used·0.0835354s·(cpu);·0.0823515s·(thread);·0s·(gc)8 ·--·used·0.10052s·(cpu);·0.101724s·(thread);·0s·(gc)
  
9 o3·=·--·rational·map·--9 o3·=·--·rational·map·--
10 ····················ZZ10 ····················ZZ
11 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])11 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])
12 ··················33331··0···1···2···3···4···5···612 ··················33331··0···1···2···3···4···5···6
13 ····················ZZ13 ····················ZZ
14 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,·y··])14 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,·y··])
⊟
1.24 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_rational__Map_lp__Ring_cm__Tally_rp.out
    
Offset 18, 15 lines modifiedOffset 18, 15 lines modified
18 ···················0·········1·········2·········3········4·········518 ···················0·········1·········2·········3········4·········5
  
19 o4·:·Ideal·of·X19 o4·:·Ideal·of·X
  
20 i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};20 i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};
  
21 i6·:·time·phi·=·rationalMap·D21 i6·:·time·phi·=·rationalMap·D
22 ·--·used·0.0239816s·(cpu);·0.0242809s·(thread);·0s·(gc)22 ·--·used·0.0367502s·(cpu);·0.0370709s·(thread);·0s·(gc)
  
23 o6·=·--·rational·map·--23 o6·=·--·rational·map·--
24 ··································ZZ24 ··································ZZ
25 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by25 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by
26 ································65521··0···1···2···3···4···526 ································65521··0···1···2···3···4···5
27 ·············{27 ·············{
28 ·················2··················228 ·················2··················2
Offset 123, 13 lines modifiedOffset 123, 13 lines modified
123 ······················x·x·x··+·x·x·x··+·x·x·x··+·x·x··+·x·x·x··-·2x·x·x··+·x·x123 ······················x·x·x··+·x·x·x··+·x·x·x··+·x·x··+·x·x·x··-·2x·x·x··+·x·x
124 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5124 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5
125 ·····················}125 ·····················}
  
126 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)126 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)
  
127 i7·:·time·?·image(phi,"F4")127 i7·:·time·?·image(phi,"F4")
128 ·--·used·0.674053s·(cpu);·0.496989s·(thread);·0s·(gc)128 ·--·used·1.0602s·(cpu);·0.502788s·(thread);·0s·(gc)
  
129 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153129 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153
130 ·····hypersurfaces·of·degree·2130 ·····hypersurfaces·of·degree·2
  
131 i8·:·131 i8·:·
⊟
722 B
./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cremona__Transformation.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·13308466410811 --·-*-·M2-comint·-*-·hash:·1330846641081
  
2 i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))2 i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))
3 ·--·used·0.81958s·(cpu);·0.744604s·(thread);·0s·(gc)3 ·--·used·1.06038s·(cpu);·0.934817s·(thread);·0s·(gc)
  
4 o1·=·(rational·map·defined·by·forms·of·degree·3,4 o1·=·(rational·map·defined·by·forms·of·degree·3,
5 ······source·variety:·PP^3······················5 ······source·variety:·PP^3······················
6 ······target·variety:·PP^3······················6 ······target·variety:·PP^3······················
7 ······dominance:·true···························7 ······dominance:·true···························
8 ······birationality:·true·······················8 ······birationality:·true·······················
9 ······projective·degrees:·{1,·3,·3,·1}··········9 ······projective·degrees:·{1,·3,·3,·1}··········
⊟
2.68 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_special__Cubic__Transformation.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·17300189127154982881 --·-*-·M2-comint·-*-·hash:·1730018912715498288
  
2 i1·:·time·specialCubicTransformation·92 i1·:·time·specialCubicTransformation·9
3 ·--·used·0.0687208s·(cpu);·0.0696801s·(thread);·0s·(gc)3 ·--·used·0.101541s·(cpu);·0.10362s·(thread);·0s·(gc)
  
4 o1·=·--·rational·map·--4 o1·=·--·rational·map·--
5 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])5 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])
6 ······················0···1···2···3···4···5···66 ······················0···1···2···3···4···5···6
7 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])·defined·by7 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])·defined·by
8 ····································0···1···2···3···4···5···6···7···8···98 ····································0···1···2···3···4···5···6···7···8···9
9 ·············{9 ·············{
Offset 62, 15 lines modifiedOffset 62, 15 lines modified
62 ······················8x·x··-·12x·x··+·24x··-·11x·x··+·17x·x·x··-·24x·x··-·10x·x··+·11x·x··-·3x··-·6x·x··+·28x·x·x··-·70x·x··-·21x·x·x··+·47x·x·x··-·13x·x··-·14x·x··+·66x·x··-·22x·x··-·20x··+·2x·x··-·2x·x·x··-·10x·x··-·11x·x·x··+·8x·x·x··-·5x·x··+·3x·x·x··+·23x·x·x··-·11x·x·x··-·12x·x··+·3x·x··-·3x·x··-·2x·x··+·3x·x··+·x··-·11x·x··+·14x·x·x··+·34x·x··-·6x·x·x··-·16x·x·x··+·3x·x··-·15x·x·x··-·66x·x·x··+·12x·x·x··+·30x·x··-·19x·x·x··+·2x·x·x··-·5x·x·x··-·2x·x·x··-·7x·x··+·6x·x··+·21x·x··-·3x·x··-·21x·x··+·x·x··+·5x··-·8x·x··+·7x·x·x··-·32x·x··-·13x·x·x··+·28x·x·x··-·9x·x··+·70x·x·x··-·27x·x·x··-·36x·x··+·x·x·x··+·4x·x·x··-·7x·x·x··-·2x·x·x··+·3x·x··-·25x·x·x··-·23x·x·x··+·4x·x·x··+·27x·x·x··-·14x·x·x··-·9x·x··-·2x·x··+·10x·x··-·6x·x··-·10x·x··+·3x·x··-·2x·x62 ······················8x·x··-·12x·x··+·24x··-·11x·x··+·17x·x·x··-·24x·x··-·10x·x··+·11x·x··-·3x··-·6x·x··+·28x·x·x··-·70x·x··-·21x·x·x··+·47x·x·x··-·13x·x··-·14x·x··+·66x·x··-·22x·x··-·20x··+·2x·x··-·2x·x·x··-·10x·x··-·11x·x·x··+·8x·x·x··-·5x·x··+·3x·x·x··+·23x·x·x··-·11x·x·x··-·12x·x··+·3x·x··-·3x·x··-·2x·x··+·3x·x··+·x··-·11x·x··+·14x·x·x··+·34x·x··-·6x·x·x··-·16x·x·x··+·3x·x··-·15x·x·x··-·66x·x·x··+·12x·x·x··+·30x·x··-·19x·x·x··+·2x·x·x··-·5x·x·x··-·2x·x·x··-·7x·x··+·6x·x··+·21x·x··-·3x·x··-·21x·x··+·x·x··+·5x··-·8x·x··+·7x·x·x··-·32x·x··-·13x·x·x··+·28x·x·x··-·9x·x··+·70x·x·x··-·27x·x·x··-·36x·x··+·x·x·x··+·4x·x·x··-·7x·x·x··-·2x·x·x··+·3x·x··-·25x·x·x··-·23x·x·x··+·4x·x·x··+·27x·x·x··-·14x·x·x··-·9x·x··-·2x·x··+·10x·x··-·6x·x··-·10x·x··+·3x·x··-·2x·x
63 ························0·1······0·1······1······0·2······0·1·2······1·2······0·2······1·2·····2·····0·3······0·1·3······1·3······0·2·3······1·2·3······2·3······0·3······1·3······2·3······3·····0·4·····0·1·4······1·4······0·2·4·····1·2·4·····2·4·····0·3·4······1·3·4······2·3·4······3·4·····0·4·····1·4·····2·4·····3·4····4······0·5······0·1·5······1·5·····0·2·5······1·2·5·····2·5······0·3·5······1·3·5······2·3·5······3·5······0·4·5·····1·4·5·····2·4·5·····3·4·5·····4·5·····0·5······1·5·····2·5······3·5····4·5·····5·····0·6·····0·1·6······1·6······0·2·6······1·2·6·····2·6······1·3·6······2·3·6······3·6····0·4·6·····1·4·6·····2·4·6·····3·4·6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6······1·6·····2·6······3·6·····4·6·····5·663 ························0·1······0·1······1······0·2······0·1·2······1·2······0·2······1·2·····2·····0·3······0·1·3······1·3······0·2·3······1·2·3······2·3······0·3······1·3······2·3······3·····0·4·····0·1·4······1·4······0·2·4·····1·2·4·····2·4·····0·3·4······1·3·4······2·3·4······3·4·····0·4·····1·4·····2·4·····3·4····4······0·5······0·1·5······1·5·····0·2·5······1·2·5·····2·5······0·3·5······1·3·5······2·3·5······3·5······0·4·5·····1·4·5·····2·4·5·····3·4·5·····4·5·····0·5······1·5·····2·5······3·5····4·5·····5·····0·6·····0·1·6······1·6······0·2·6······1·2·6·····2·6······1·3·6······2·3·6······3·6····0·4·6·····1·4·6·····2·4·6·····3·4·6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6······1·6·····2·6······3·6·····4·6·····5·6
64 ·····················}64 ·····················}
  
65 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)65 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)
  
66 i2·:·time·describe·oo66 i2·:·time·describe·oo
67 ·--·used·0.0151069s·(cpu);·0.0154917s·(thread);·0s·(gc)67 ·--·used·0.0164238s·(cpu);·0.0179546s·(thread);·0s·(gc)
  
68 o2·=·rational·map·defined·by·forms·of·degree·368 o2·=·rational·map·defined·by·forms·of·degree·3
69 ·····source·variety:·PP^669 ·····source·variety:·PP^6
70 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^970 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^9
71 ·····dominance:·true71 ·····dominance:·true
72 ·····birationality:·true72 ·····birationality:·true
73 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}73 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}
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1.32 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_special__Quadratic__Transformation.out
    
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1 --·-*-·M2-comint·-*-·hash:·17292005823766787051 --·-*-·M2-comint·-*-·hash:·1729200582376678705
  
2 i1·:·time·specialQuadraticTransformation·42 i1·:·time·specialQuadraticTransformation·4
3 ·--·used·0.050462s·(cpu);·0.0541456s·(thread);·0s·(gc)3 ·--·used·0.0613429s·(cpu);·0.0584016s·(thread);·0s·(gc)
  
4 o1·=·--·rational·map·--4 o1·=·--·rational·map·--
5 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])5 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
6 ······················0···1···2···3···4···5···6···7···86 ······················0···1···2···3···4···5···6···7···8
7 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])·defined·by7 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])·defined·by
8 ····································0···1···2···3···4···5···6···7···8···98 ····································0···1···2···3···4···5···6···7···8···9
9 ·············{9 ·············{
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50 ······················x·x··-·x·x··+·x·x··-·x·x··-·x··-·x·x50 ······················x·x··-·x·x··+·x·x··-·x·x··-·x··-·x·x
51 ·······················0·1····0·4····3·6····4·6····6····5·751 ·······················0·1····0·4····3·6····4·6····6····5·7
52 ·····················}52 ·····················}
  
53 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)53 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)
  
54 i2·:·time·describe·oo54 i2·:·time·describe·oo
55 ·--·used·0.00522602s·(cpu);·0.00520613s·(thread);·0s·(gc)55 ·--·used·0.00841658s·(cpu);·0.00841796s·(thread);·0s·(gc)
  
56 o2·=·rational·map·defined·by·forms·of·degree·256 o2·=·rational·map·defined·by·forms·of·degree·2
57 ·····source·variety:·PP^857 ·····source·variety:·PP^8
58 ·····target·variety:·hypersurface·of·degree·3·in·PP^958 ·····target·variety:·hypersurface·of·degree·3·in·PP^9
59 ·····dominance:·true59 ·····dominance:·true
60 ·····birationality:·true60 ·····birationality:·true
61 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}61 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}
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1.41 KB
./usr/share/doc/Macaulay2/Cremona/example-output/_to__External__String_lp__Rational__Map_rp.out
    
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7 i2·:·str·=·toExternalString·phi;7 i2·:·str·=·toExternalString·phi;
  
8 i3·:·#str8 i3·:·#str
  
9 o3·=·69279 o3·=·6927
  
10 i4·:·time·phi'·=·value·str;10 i4·:·time·phi'·=·value·str;
11 ·--·used·0.0780309s·(cpu);·0.0363949s·(thread);·0s·(gc)11 ·--·used·0.0931018s·(cpu);·0.0443018s·(thread);·0s·(gc)
  
12 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)12 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)
  
13 i5·:·time·describe·phi'13 i5·:·time·describe·phi'
14 ·--·used·0.00119947s·(cpu);·0.0043174s·(thread);·0s·(gc)14 ·--·used·0.00481386s·(cpu);·0.00571627s·(thread);·0s·(gc)
  
15 o5·=·rational·map·defined·by·forms·of·degree·315 o5·=·rational·map·defined·by·forms·of·degree·3
16 ·····source·variety:·PP^316 ·····source·variety:·PP^3
17 ·····target·variety:·smooth·quadric·hypersurface·in·PP^417 ·····target·variety:·smooth·quadric·hypersurface·in·PP^4
18 ·····dominance:·true18 ·····dominance:·true
19 ·····birationality:·true·(the·inverse·map·is·already·calculated)19 ·····birationality:·true·(the·inverse·map·is·already·calculated)
20 ·····projective·degrees:·{1,·3,·4,·2}20 ·····projective·degrees:·{1,·3,·4,·2}
21 ·····number·of·minimal·representatives:·121 ·····number·of·minimal·representatives:·1
22 ·····dimension·base·locus:·122 ·····dimension·base·locus:·1
23 ·····degree·base·locus:·523 ·····degree·base·locus:·5
24 ·····coefficient·ring:·ZZ/3333124 ·····coefficient·ring:·ZZ/33331
  
25 i6·:·time·describe·inverse·phi'25 i6·:·time·describe·inverse·phi'
26 ·--·used·0.000593353s·(cpu);·0.00374519s·(thread);·0s·(gc)26 ·--·used·0.00607502s·(cpu);·0.00601148s·(thread);·0s·(gc)
  
27 o6·=·rational·map·defined·by·forms·of·degree·227 o6·=·rational·map·defined·by·forms·of·degree·2
28 ·····source·variety:·smooth·quadric·hypersurface·in·PP^428 ·····source·variety:·smooth·quadric·hypersurface·in·PP^4
29 ·····target·variety:·PP^329 ·····target·variety:·PP^3
30 ·····dominance:·true30 ·····dominance:·true
31 ·····birationality:·true·(the·inverse·map·is·already·calculated)31 ·····birationality:·true·(the·inverse·map·is·already·calculated)
32 ·····projective·degrees:·{2,·4,·3,·1}32 ·····projective·degrees:·{2,·4,·3,·1}
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3.52 KB
./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
    
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96 ···············1····0·2·····1·2····0·3·····2····1·396 ···············1····0·2·····1·2····0·3·····2····1·3
  
97 o2·:·Ideal·of·GF·78125[x·..x·]97 o2·:·Ideal·of·GF·78125[x·..x·]
98 ························0···4</code></pre>98 ························0···4</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·time·ChernSchwartzMacPherson·C101 <td>··············<pre><code·class="language-macaulay2">i3·:·time·ChernSchwartzMacPherson·C
102 ·--·used·1.18342s·(cpu);·0.815881s·(thread);·0s·(gc)102 ·--·used·1.33858s·(cpu);·0.886344s·(thread);·0s·(gc)
  
103 ·······4·····3·····2103 ·······4·····3·····2
104 o3·=·3H··+·5H··+·3H104 o3·=·3H··+·5H··+·3H
  
105 ·····ZZ[H]105 ·····ZZ[H]
106 o3·:·-----106 o3·:·-----
107 ········5107 ········5
108 ·······H</code></pre>108 ·······H</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)111 <td>··············<pre><code·class="language-macaulay2">i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)
112 ·--·used·2.50688s·(cpu);·0.787772s·(thread);·0s·(gc)112 ·--·used·2.28539s·(cpu);·0.809176s·(thread);·0s·(gc)
113 Certify:·output·certified!113 Certify:·output·certified!
  
114 ·······4·····3·····2114 ·······4·····3·····2
115 o4·=·3H··+·5H··+·3H115 o4·=·3H··+·5H··+·3H
  
116 ·····ZZ[H]116 ·····ZZ[H]
117 o4·:·-----117 o4·:·-----
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154 ················ZZ154 ················ZZ
155 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]155 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]
156 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4</code></pre>156 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i9·:·time·ChernClass·G159 <td>··············<pre><code·class="language-macaulay2">i9·:·time·ChernClass·G
160 ·--·used·0.166395s·(cpu);·0.129677s·(thread);·0s·(gc)160 ·--·used·0.169689s·(cpu);·0.118999s·(thread);·0s·(gc)
  
161 ········9······8······7······6······5······4·····3161 ········9······8······7······6······5······4·····3
162 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H162 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
163 ·····ZZ[H]163 ·····ZZ[H]
164 o9·:·-----164 o9·:·-----
165 ·······10165 ·······10
166 ······H</code></pre>166 ······H</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i10·:·time·ChernClass(G,Certify=>true)169 <td>··············<pre><code·class="language-macaulay2">i10·:·time·ChernClass(G,Certify=>true)
170 ·--·used·0.226004s·(cpu);·0.0496908s·(thread);·0s·(gc)170 ·--·used·0.1636s·(cpu);·0.027524s·(thread);·0s·(gc)
171 Certify:·output·certified!171 Certify:·output·certified!
  
172 ·········9······8······7······6······5······4·····3172 ·········9······8······7······6······5······4·····3
173 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H173 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
174 ······ZZ[H]174 ······ZZ[H]
175 o10·:·-----175 o10·:·-----
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1.55 KB
html2text {}
    
Offset 40, 25 lines modifiedOffset 40, 25 lines modified
40 ···············2···························240 ···············2···························2
41 o2·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)41 o2·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)
42 ···············1····0·2·····1·2····0·3·····2····1·342 ···············1····0·2·····1·2····0·3·····2····1·3
  
43 o2·:·Ideal·of·GF·78125[x·..x·]43 o2·:·Ideal·of·GF·78125[x·..x·]
44 ························0···444 ························0···4
45 i3·:·time·ChernSchwartzMacPherson·C45 i3·:·time·ChernSchwartzMacPherson·C
46 ·--·used·1.18342s·(cpu);·0.815881s·(thread);·0s·(gc)46 ·--·used·1.33858s·(cpu);·0.886344s·(thread);·0s·(gc)
  
47 ·······4·····3·····247 ·······4·····3·····2
48 o3·=·3H··+·5H··+·3H48 o3·=·3H··+·5H··+·3H
  
49 ·····ZZ[H]49 ·····ZZ[H]
50 o3·:·-----50 o3·:·-----
51 ········551 ········5
52 ·······H52 ·······H
53 i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)53 i4·:·time·ChernSchwartzMacPherson(C,Certify=>true)
54 ·--·used·2.50688s·(cpu);·0.787772s·(thread);·0s·(gc)54 ·--·used·2.28539s·(cpu);·0.809176s·(thread);·0s·(gc)
55 Certify:·output·certified!55 Certify:·output·certified!
  
56 ·······4·····3·····256 ·······4·····3·····2
57 o4·=·3H··+·5H··+·3H57 o4·=·3H··+·5H··+·3H
  
58 ·····ZZ[H]58 ·····ZZ[H]
59 o4·:·-----59 o4·:·-----
Offset 89, 25 lines modifiedOffset 89, 25 lines modified
89 ········0,2·1,3····0,1·2,389 ········0,2·1,3····0,1·2,3
  
90 ················ZZ90 ················ZZ
91 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p91 o8·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p
92 ]92 ]
93 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,493 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4
94 i9·:·time·ChernClass·G94 i9·:·time·ChernClass·G
95 ·--·used·0.166395s·(cpu);·0.129677s·(thread);·0s·(gc)95 ·--·used·0.169689s·(cpu);·0.118999s·(thread);·0s·(gc)
  
96 ········9······8······7······6······5······4·····396 ········9······8······7······6······5······4·····3
97 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H97 o9·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
98 ·····ZZ[H]98 ·····ZZ[H]
99 o9·:·-----99 o9·:·-----
100 ·······10100 ·······10
101 ······H101 ······H
102 i10·:·time·ChernClass(G,Certify=>true)102 i10·:·time·ChernClass(G,Certify=>true)
103 ·--·used·0.226004s·(cpu);·0.0496908s·(thread);·0s·(gc)103 ·--·used·0.1636s·(cpu);·0.027524s·(thread);·0s·(gc)
104 Certify:·output·certified!104 Certify:·output·certified!
  
105 ·········9······8······7······6······5······4·····3105 ·········9······8······7······6······5······4·····3
106 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H106 o10·=·10H··+·30H··+·60H··+·75H··+·57H··+·25H··+·5H
  
107 ······ZZ[H]107 ······ZZ[H]
108 o10·:·-----108 o10·:·-----
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Offset 86, 21 lines modifiedOffset 86, 21 lines modified
  
86 ················ZZ86 ················ZZ
87 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]87 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···]
88 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4</code></pre>88 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i2·:·time·EulerCharacteristic·I91 <td>··············<pre><code·class="language-macaulay2">i2·:·time·EulerCharacteristic·I
92 ·--·used·0.20939s·(cpu);·0.13637s·(thread);·0s·(gc)92 ·--·used·0.235163s·(cpu);·0.14315s·(thread);·0s·(gc)
  
93 o2·=·10</code></pre>93 o2·=·10</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i3·:·time·EulerCharacteristic(I,Certify=>true)96 <td>··············<pre><code·class="language-macaulay2">i3·:·time·EulerCharacteristic(I,Certify=>true)
97 ·--·used·0.242261s·(cpu);·0.0362552s·(thread);·0s·(gc)97 ·--·used·0.123625s·(cpu);·0.020704s·(thread);·0s·(gc)
98 Certify:·output·certified!98 Certify:·output·certified!
  
99 o3·=·10</code></pre>99 o3·=·10</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ········</table>101 ········</table>
102 ······</div>102 ······</div>
103 ······<div>103 ······<div>
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Offset 32, 19 lines modifiedOffset 32, 19 lines modified
32 i1·:·I·=·Grassmannian(1,4,CoefficientRing=>ZZ/190181);32 i1·:·I·=·Grassmannian(1,4,CoefficientRing=>ZZ/190181);
  
33 ················ZZ33 ················ZZ
34 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p34 o1·:·Ideal·of·------[p···..p···,·p···,·p···,·p···,·p···,·p···,·p···,·p···,·p
35 ]35 ]
36 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,436 ··············190181··0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4
37 i2·:·time·EulerCharacteristic·I37 i2·:·time·EulerCharacteristic·I
38 ·--·used·0.20939s·(cpu);·0.13637s·(thread);·0s·(gc)38 ·--·used·0.235163s·(cpu);·0.14315s·(thread);·0s·(gc)
  
39 o2·=·1039 o2·=·10
40 i3·:·time·EulerCharacteristic(I,Certify=>true)40 i3·:·time·EulerCharacteristic(I,Certify=>true)
41 ·--·used·0.242261s·(cpu);·0.0362552s·(thread);·0s·(gc)41 ·--·used·0.123625s·(cpu);·0.020704s·(thread);·0s·(gc)
42 Certify:·output·certified!42 Certify:·output·certified!
  
43 o3·=·1043 o3·=·10
44 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*44 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
45 No·test·is·made·to·see·if·the·projective·variety·is·smooth.45 No·test·is·made·to·see·if·the·projective·variety·is·smooth.
46 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*46 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
47 ····*·_\x8e_\x8u_\x8l_\x8e_\x8r_\x8(_\x8P_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·topological·Euler·characteristic·of·a47 ····*·_\x8e_\x8u_\x8l_\x8e_\x8r_\x8(_\x8P_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·topological·Euler·characteristic·of·a
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Offset 82, 15 lines modifiedOffset 82, 15 lines modified
82 o3·=·rational·map·defined·by·forms·of·degree·282 o3·=·rational·map·defined·by·forms·of·degree·2
83 ·····source·variety:·PP^583 ·····source·variety:·PP^5
84 ·····target·variety:·PP^584 ·····target·variety:·PP^5
85 ·····coefficient·ring:·QQ</code></pre>85 ·····coefficient·ring:·QQ</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·time·phi!·;88 <td>··············<pre><code·class="language-macaulay2">i4·:·time·phi!·;
89 ·--·used·0.397259s·(cpu);·0.0971215s·(thread);·0s·(gc)89 ·--·used·0.409125s·(cpu);·0.117687s·(thread);·0s·(gc)
  
90 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))</code></pre>90 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i5·:·describe·phi93 <td>··············<pre><code·class="language-macaulay2">i5·:·describe·phi
  
94 o5·=·rational·map·defined·by·forms·of·degree·294 o5·=·rational·map·defined·by·forms·of·degree·2
Offset 115, 15 lines modifiedOffset 115, 15 lines modified
115 o8·=·rational·map·defined·by·forms·of·degree·2115 o8·=·rational·map·defined·by·forms·of·degree·2
116 ·····source·variety:·PP^4116 ·····source·variety:·PP^4
117 ·····target·variety:·PP^5117 ·····target·variety:·PP^5
118 ·····coefficient·ring:·QQ</code></pre>118 ·····coefficient·ring:·QQ</code></pre>
119 </td>··········</tr>119 </td>··········</tr>
120 ··········<tr>120 ··········<tr>
121 <td>··············<pre><code·class="language-macaulay2">i9·:·time·phi!·;121 <td>··············<pre><code·class="language-macaulay2">i9·:·time·phi!·;
122 ·--·used·0.269141s·(cpu);·0.0534929s·(thread);·0s·(gc)122 ·--·used·0.321513s·(cpu);·0.0796105s·(thread);·0s·(gc)
  
123 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)</code></pre>123 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i10·:·describe·phi126 <td>··············<pre><code·class="language-macaulay2">i10·:·describe·phi
  
127 o10·=·rational·map·defined·by·forms·of·degree·2127 o10·=·rational·map·defined·by·forms·of·degree·2
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Offset 22, 15 lines modifiedOffset 22, 15 lines modified
22 i3·:·describe·phi22 i3·:·describe·phi
  
23 o3·=·rational·map·defined·by·forms·of·degree·223 o3·=·rational·map·defined·by·forms·of·degree·2
24 ·····source·variety:·PP^524 ·····source·variety:·PP^5
25 ·····target·variety:·PP^525 ·····target·variety:·PP^5
26 ·····coefficient·ring:·QQ26 ·····coefficient·ring:·QQ
27 i4·:·time·phi!·;27 i4·:·time·phi!·;
28 ·--·used·0.397259s·(cpu);·0.0971215s·(thread);·0s·(gc)28 ·--·used·0.409125s·(cpu);·0.117687s·(thread);·0s·(gc)
  
29 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))29 o4·:·RationalMap·(Cremona·transformation·of·PP^5·of·type·(2,2))
30 i5·:·describe·phi30 i5·:·describe·phi
  
31 o5·=·rational·map·defined·by·forms·of·degree·231 o5·=·rational·map·defined·by·forms·of·degree·2
32 ·····source·variety:·PP^532 ·····source·variety:·PP^5
33 ·····target·variety:·PP^533 ·····target·variety:·PP^5
Offset 48, 15 lines modifiedOffset 48, 15 lines modified
48 i8·:·describe·phi48 i8·:·describe·phi
  
49 o8·=·rational·map·defined·by·forms·of·degree·249 o8·=·rational·map·defined·by·forms·of·degree·2
50 ·····source·variety:·PP^450 ·····source·variety:·PP^4
51 ·····target·variety:·PP^551 ·····target·variety:·PP^5
52 ·····coefficient·ring:·QQ52 ·····coefficient·ring:·QQ
53 i9·:·time·phi!·;53 i9·:·time·phi!·;
54 ·--·used·0.269141s·(cpu);·0.0534929s·(thread);·0s·(gc)54 ·--·used·0.321513s·(cpu);·0.0796105s·(thread);·0s·(gc)
  
55 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)55 o9·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^5)
56 i10·:·describe·phi56 i10·:·describe·phi
  
57 o10·=·rational·map·defined·by·forms·of·degree·257 o10·=·rational·map·defined·by·forms·of·degree·2
58 ······source·variety:·PP^458 ······source·variety:·PP^4
59 ······target·variety:·PP^559 ······target·variety:·PP^5
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./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
    
Offset 146, 15 lines modifiedOffset 146, 15 lines modified
146 ·····----------*v,·x·+·----------*v)146 ·····----------*v,·x·+·----------*v)
147 ······d*e·-·a*f·········d*e·-·a*f147 ······d*e·-·a*f·········d*e·-·a*f
  
148 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]</code></pre>148 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]</code></pre>
149 </td>··········</tr>149 </td>··········</tr>
150 ··········<tr>150 ··········<tr>
151 <td>··············<pre><code·class="language-macaulay2">i6·:·time·phi^**·q151 <td>··············<pre><code·class="language-macaulay2">i6·:·time·phi^**·q
152 ·--·used·0.172677s·(cpu);·0.134122s·(thread);·0s·(gc)152 ·--·used·0.240581s·(cpu);·0.186416s·(thread);·0s·(gc)
  
153 ················-e········-d········-c········-b········-a153 ················-e········-d········-c········-b········-a
154 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)154 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)
155 ·················f·········f·········f·········f·········f155 ·················f·········f·········f·········f·········f
  
156 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]</code></pre>156 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
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Offset 83, 15 lines modifiedOffset 83, 15 lines modified
83 ··············2·················283 ··············2·················2
84 ·····-·a*c·+·e·········-·b*c·+·f84 ·····-·a*c·+·e·········-·b*c·+·f
85 ·····----------*v,·x·+·----------*v)85 ·····----------*v,·x·+·----------*v)
86 ······d*e·-·a*f·········d*e·-·a*f86 ······d*e·-·a*f·········d*e·-·a*f
  
87 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]87 o5·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]
88 i6·:·time·phi^**·q88 i6·:·time·phi^**·q
89 ·--·used·0.172677s·(cpu);·0.134122s·(thread);·0s·(gc)89 ·--·used·0.240581s·(cpu);·0.186416s·(thread);·0s·(gc)
  
90 ················-e········-d········-c········-b········-a90 ················-e········-d········-c········-b········-a
91 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)91 o6·=·ideal·(u·+·--*v,·t·+·--*v,·z·+·--*v,·y·+·--*v,·x·+·--*v)
92 ·················f·········f·········f·········f·········f92 ·················f·········f·········f·········f·········f
  
93 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]93 o6·:·Ideal·of·frac(QQ[a..f])[x,·y,·z,·t,·u,·v]
94 i7·:·oo·==·p94 i7·:·oo·==·p
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Offset 131, 52 lines modifiedOffset 131, 52 lines modified
131 o3·:·Ideal·of·-------------------------------------------------------------------------------------------------------------------------131 o3·:·Ideal·of·-------------------------------------------------------------------------------------------------------------------------
132 ···············2·2················2·2········································2·2····················································2·2132 ···············2·2················2·2········································2·2····················································2·2
133 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x··+·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x133 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x··+·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x
134 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····1·6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·7</code></pre>134 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····1·6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·7</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i4·:·time·SegreClass·X137 <td>··············<pre><code·class="language-macaulay2">i4·:·time·SegreClass·X
138 ·--·used·0.423006s·(cpu);·0.346287s·(thread);·0s·(gc)138 ·--·used·0.503888s·(cpu);·0.411667s·(thread);·0s·(gc)
  
139 ··········7········6·······5·······4······3139 ··········7········6·······5·······4······3
140 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H140 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
141 ·····ZZ[H]141 ·····ZZ[H]
142 o4·:·-----142 o4·:·-----
143 ········8143 ········8
144 ·······H</code></pre>144 ·······H</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i5·:·time·SegreClass·lift(X,P7)147 <td>··············<pre><code·class="language-macaulay2">i5·:·time·SegreClass·lift(X,P7)
148 ·--·used·0.253089s·(cpu);·0.216709s·(thread);·0s·(gc)148 ·--·used·0.2606s·(cpu);·0.211402s·(thread);·0s·(gc)
  
149 ··········7········6·······5······4······3149 ··········7········6·······5······4······3
150 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H150 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
151 ·····ZZ[H]151 ·····ZZ[H]
152 o5·:·-----152 o5·:·-----
153 ········8153 ········8
154 ·······H</code></pre>154 ·······H</code></pre>
155 </td>··········</tr>155 </td>··········</tr>
156 ··········<tr>156 ··········<tr>
157 <td>··············<pre><code·class="language-macaulay2">i6·:·time·SegreClass(X,Certify=>true)157 <td>··············<pre><code·class="language-macaulay2">i6·:·time·SegreClass(X,Certify=>true)
158 ·--·used·0.294334s·(cpu);·0.0596635s·(thread);·0s·(gc)158 ·--·used·0.230385s·(cpu);·0.035137s·(thread);·0s·(gc)
159 Certify:·output·certified!159 Certify:·output·certified!
  
160 ··········7········6·······5·······4······3160 ··········7········6·······5·······4······3
161 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H161 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
162 ·····ZZ[H]162 ·····ZZ[H]
163 o6·:·-----163 o6·:·-----
164 ········8164 ········8
165 ·······H</code></pre>165 ·······H</code></pre>
166 </td>··········</tr>166 </td>··········</tr>
167 ··········<tr>167 ··········<tr>
168 <td>··············<pre><code·class="language-macaulay2">i7·:·time·SegreClass(lift(X,P7),Certify=>true)168 <td>··············<pre><code·class="language-macaulay2">i7·:·time·SegreClass(lift(X,P7),Certify=>true)
169 ·--·used·0.38928s·(cpu);·0.138091s·(thread);·0s·(gc)169 ·--·used·0.341144s·(cpu);·0.121427s·(thread);·0s·(gc)
170 Certify:·output·certified!170 Certify:·output·certified!
  
171 ··········7········6·······5······4······3171 ··········7········6·······5······4······3
172 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H172 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
173 ·····ZZ[H]173 ·····ZZ[H]
174 o7·:·-----174 o7·:·-----
Offset 198, 15 lines modifiedOffset 198, 15 lines modified
198 o9·=·------[x·..x·]198 o9·=·------[x·..x·]
199 ·····100003··0···6199 ·····100003··0···6
  
200 o9·:·PolynomialRing</code></pre>200 o9·:·PolynomialRing</code></pre>
201 </td>··········</tr>201 </td>··········</tr>
202 ··········<tr>202 ··········<tr>
203 <td>··············<pre><code·class="language-macaulay2">i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)203 <td>··············<pre><code·class="language-macaulay2">i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
204 ·--·used·0.0455862s·(cpu);·0.0477137s·(thread);·0s·(gc)204 ·--·used·0.0599694s·(cpu);·0.0579296s·(thread);·0s·(gc)
  
205 ························································ZZ205 ························································ZZ
206 ······················································------[y·..y·]206 ······················································------[y·..y·]
207 ······················································100003··0···9················································ZZ··············2······························2207 ······················································100003··0···9················································ZZ··············2······························2
208 o10·=·map·(----------------------------------------------------------------------------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y·,·y··-·y·y··-·y·y·,·y·y··+·y·y··-·y·y·,·y·y··-·y·y·,·y·y··-·y·y··-·y·y·,·y·y··-·y·y··-·y·y·})208 o10·=·map·(----------------------------------------------------------------------------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y·,·y··-·y·y··-·y·y·,·y·y··+·y·y··-·y·y·,·y·y··-·y·y·,·y·y··-·y·y··-·y·y·,·y·y··-·y·y··-·y·y·})
209 ···········(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)··100003··0···6·····3····0·5····1·6···3·4····1·7···4····2·7····0·9···2·5····3·5····1·8···4·5····1·9···4·8····2·9····3·9···7·8····4·9····6·9209 ···········(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)··100003··0···6·····3····0·5····1·6···3·4····1·7···4····2·7····0·9···2·5····3·5····1·8···4·5····1·9···4·8····2·9····3·9···7·8····4·9····6·9
210 ·············5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5210 ·············5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5
Offset 216, 15 lines modifiedOffset 216, 15 lines modified
216 ·························································100003··0···9···················································ZZ216 ·························································100003··0···9···················································ZZ
217 o10·:·RingMap·----------------------------------------------------------------------------------------------------·&lt;--·------[x·..x·]217 o10·:·RingMap·----------------------------------------------------------------------------------------------------·&lt;--·------[x·..x·]
218 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6218 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6
219 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5</code></pre>219 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5</code></pre>
220 </td>··········</tr>220 </td>··········</tr>
221 ··········<tr>221 ··········<tr>
222 <td>··············<pre><code·class="language-macaulay2">i11·:·time·SegreClass·phi222 <td>··············<pre><code·class="language-macaulay2">i11·:·time·SegreClass·phi
223 ·--·used·0.269223s·(cpu);·0.19767s·(thread);·0s·(gc)223 ·--·used·0.305646s·(cpu);·0.212694s·(thread);·0s·(gc)
  
224 ·········9······8······7······6·····5224 ·········9······8······7······6·····5
225 o11·=·23H··-·42H··+·36H··-·22H··+·9H225 o11·=·23H··-·42H··+·36H··-·22H··+·9H
  
226 ······ZZ[H]226 ······ZZ[H]
227 o11·:·-----227 o11·:·-----
228 ········10228 ········10
Offset 246, 28 lines modifiedOffset 246, 28 lines modified
246 o12·:·Ideal·of·----------------------------------------------------------------------------------------------------246 o12·:·Ideal·of·----------------------------------------------------------------------------------------------------
247 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)247 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
248 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5</code></pre>248 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6····1·7····0·8···2·3····1·4····0·5</code></pre>
249 </td>··········</tr>249 </td>··········</tr>
250 ··········<tr>250 ··········<tr>
251 <td>··············<pre><code·class="language-macaulay2">i13·:·--·Segre·class·of·B·in·G(1,4)251 <td>··············<pre><code·class="language-macaulay2">i13·:·--·Segre·class·of·B·in·G(1,4)
252 ······time·SegreClass·B252 ······time·SegreClass·B
253 ·--·used·0.241335s·(cpu);·0.200347s·(thread);·0s·(gc)253 ·--·used·0.260493s·(cpu);·0.213822s·(thread);·0s·(gc)
  
254 ·········9······8······7······6·····5254 ·········9······8······7······6·····5
255 o13·=·23H··-·42H··+·36H··-·22H··+·9H255 o13·=·23H··-·42H··+·36H··-·22H··+·9H
  
256 ······ZZ[H]256 ······ZZ[H]
257 o13·:·-----257 o13·:·-----
258 ········10258 ········10
259 ·······H</code></pre>259 ·······H</code></pre>
260 </td>··········</tr>260 </td>··········</tr>
261 ··········<tr>261 ··········<tr>
262 <td>··············<pre><code·class="language-macaulay2">i14·:·--·Segre·class·of·B·in·P^9262 <td>··············<pre><code·class="language-macaulay2">i14·:·--·Segre·class·of·B·in·P^9
263 ······time·SegreClass·lift(B,ambient·ring·B)263 ······time·SegreClass·lift(B,ambient·ring·B)
264 ·--·used·0.767014s·(cpu);·0.613042s·(thread);·0s·(gc)264 ·--·used·0.883692s·(cpu);·0.687767s·(thread);·0s·(gc)
  
265 ···········9·······8·······7······6·····5265 ···········9·······8·······7······6·····5
266 o14·=·2764H··-·984H··+·294H··-·67H··+·9H266 o14·=·2764H··-·984H··+·294H··-·67H··+·9H
  
267 ······ZZ[H]267 ······ZZ[H]
268 o14·:·-----268 o14·:·-----
269 ········10269 ········10
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Offset 82, 46 lines modifiedOffset 82, 46 lines modified
82 ···············2·2················2·2········································282 ···············2·2················2·2········································2
83 2····················································2·283 2····················································2·2
84 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x84 ··············x·x··-·2x·x·x·x··+·x·x··-·2x·x·x·x··-·2x·x·x·x··+·4x·x·x·x··+·x·x
85 +·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x85 +·4x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··-·2x·x·x·x··+·x·x
86 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····186 ···············3·4·····2·3·4·5····2·5·····1·3·4·6·····1·2·5·6·····0·3·5·6····1
87 6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·787 6·····1·2·4·7·····0·3·4·7·····0·2·5·7·····0·1·6·7····0·7
88 i4·:·time·SegreClass·X88 i4·:·time·SegreClass·X
89 ·--·used·0.423006s·(cpu);·0.346287s·(thread);·0s·(gc)89 ·--·used·0.503888s·(cpu);·0.411667s·(thread);·0s·(gc)
  
90 ··········7········6·······5·······4······390 ··········7········6·······5·······4······3
91 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H91 o4·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
92 ·····ZZ[H]92 ·····ZZ[H]
93 o4·:·-----93 o4·:·-----
94 ········894 ········8
95 ·······H95 ·······H
96 i5·:·time·SegreClass·lift(X,P7)96 i5·:·time·SegreClass·lift(X,P7)
97 ·--·used·0.253089s·(cpu);·0.216709s·(thread);·0s·(gc)97 ·--·used·0.2606s·(cpu);·0.211402s·(thread);·0s·(gc)
  
98 ··········7········6·······5······4······398 ··········7········6·······5······4······3
99 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H99 o5·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
100 ·····ZZ[H]100 ·····ZZ[H]
101 o5·:·-----101 o5·:·-----
102 ········8102 ········8
103 ·······H103 ·······H
104 i6·:·time·SegreClass(X,Certify=>true)104 i6·:·time·SegreClass(X,Certify=>true)
105 ·--·used·0.294334s·(cpu);·0.0596635s·(thread);·0s·(gc)105 ·--·used·0.230385s·(cpu);·0.035137s·(thread);·0s·(gc)
106 Certify:·output·certified!106 Certify:·output·certified!
  
107 ··········7········6·······5·······4······3107 ··········7········6·······5·······4······3
108 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H108 o6·=·3240H··-·1188H··+·396H··-·114H··+·24H
  
109 ·····ZZ[H]109 ·····ZZ[H]
110 o6·:·-----110 o6·:·-----
111 ········8111 ········8
112 ·······H112 ·······H
113 i7·:·time·SegreClass(lift(X,P7),Certify=>true)113 i7·:·time·SegreClass(lift(X,P7),Certify=>true)
114 ·--·used·0.38928s·(cpu);·0.138091s·(thread);·0s·(gc)114 ·--·used·0.341144s·(cpu);·0.121427s·(thread);·0s·(gc)
115 Certify:·output·certified!115 Certify:·output·certified!
  
116 ··········7········6·······5······4······3116 ··········7········6·······5······4······3
117 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H117 o7·=·2816H··-·1056H··+·324H··-·78H··+·12H
  
118 ·····ZZ[H]118 ·····ZZ[H]
119 o7·:·-----119 o7·:·-----
Offset 142, 15 lines modifiedOffset 142, 15 lines modified
142 ·······ZZ142 ·······ZZ
143 o9·=·------[x·..x·]143 o9·=·------[x·..x·]
144 ·····100003··0···6144 ·····100003··0···6
  
145 o9·:·PolynomialRing145 o9·:·PolynomialRing
146 i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},146 i10·:·time·phi·=·inverseMap·toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
147 {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)147 {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
148 ·--·used·0.0455862s·(cpu);·0.0477137s·(thread);·0s·(gc)148 ·--·used·0.0599694s·(cpu);·0.0579296s·(thread);·0s·(gc)
  
149 ························································ZZ149 ························································ZZ
150 ······················································------[y·..y·]150 ······················································------[y·..y·]
151 ······················································100003··0···9151 ······················································100003··0···9
152 ZZ··············2······························2152 ZZ··············2······························2
153 o10·=·map·(--------------------------------------------------------------------153 o10·=·map·(--------------------------------------------------------------------
154 --------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y154 --------------------------------,·------[x·..x·],·{y··-·y·y··-·y·y·,·y·y··-·y·y
Offset 170, 15 lines modifiedOffset 170, 15 lines modified
170 o10·:·RingMap·-----------------------------------------------------------------170 o10·:·RingMap·-----------------------------------------------------------------
171 -----------------------------------·<--·------[x·..x·]171 -----------------------------------·<--·------[x·..x·]
172 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y172 ··············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y
173 -·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6173 -·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)·····100003··0···6
174 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6174 ················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2·6
175 1·7····0·8···2·3····1·4····0·5175 1·7····0·8···2·3····1·4····0·5
176 i11·:·time·SegreClass·phi176 i11·:·time·SegreClass·phi
177 ·--·used·0.269223s·(cpu);·0.19767s·(thread);·0s·(gc)177 ·--·used·0.305646s·(cpu);·0.212694s·(thread);·0s·(gc)
  
178 ·········9······8······7······6·····5178 ·········9······8······7······6·····5
179 o11·=·23H··-·42H··+·36H··-·22H··+·9H179 o11·=·23H··-·42H··+·36H··-·22H··+·9H
  
180 ······ZZ[H]180 ······ZZ[H]
181 o11·:·-----181 o11·:·-----
182 ········10182 ········10
Offset 199, 26 lines modifiedOffset 199, 26 lines modified
199 ------------------------------------199 ------------------------------------
200 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y200 ···············(y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·,·y·y
201 -·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)201 -·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
202 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2202 ·················5·7····4·8····2·9···5·6····3·8····1·9···4·6····3·7····0·9···2
203 6····1·7····0·8···2·3····1·4····0·5203 6····1·7····0·8···2·3····1·4····0·5
204 i13·:·--·Segre·class·of·B·in·G(1,4)204 i13·:·--·Segre·class·of·B·in·G(1,4)
205 ······time·SegreClass·B205 ······time·SegreClass·B
206 ·--·used·0.241335s·(cpu);·0.200347s·(thread);·0s·(gc)206 ·--·used·0.260493s·(cpu);·0.213822s·(thread);·0s·(gc)
  
207 ·········9······8······7······6·····5207 ·········9······8······7······6·····5
208 o13·=·23H··-·42H··+·36H··-·22H··+·9H208 o13·=·23H··-·42H··+·36H··-·22H··+·9H
  
209 ······ZZ[H]209 ······ZZ[H]
210 o13·:·-----210 o13·:·-----
211 ········10211 ········10
212 ·······H212 ·······H
213 i14·:·--·Segre·class·of·B·in·P^9213 i14·:·--·Segre·class·of·B·in·P^9
214 ······time·SegreClass·lift(B,ambient·ring·B)214 ······time·SegreClass·lift(B,ambient·ring·B)
215 ·--·used·0.767014s·(cpu);·0.613042s·(thread);·0s·(gc)215 ·--·used·0.883692s·(cpu);·0.687767s·(thread);·0s·(gc)
  
216 ···········9·······8·······7······6·····5216 ···········9·······8·······7······6·····5
217 o14·=·2764H··-·984H··+·294H··-·67H··+·9H217 o14·=·2764H··-·984H··+·294H··-·67H··+·9H
  
218 ······ZZ[H]218 ······ZZ[H]
219 o14·:·-----219 o14·:·-----
220 ········10220 ········10
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./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
    
Offset 97, 15 lines modifiedOffset 97, 15 lines modified
97 o3·=·QQ[u·..u·]97 o3·=·QQ[u·..u·]
98 ·········0···598 ·········0···5
  
99 o3·:·PolynomialRing</code></pre>99 o3·:·PolynomialRing</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i4·:·time·psi·=·abstractRationalMap(P4,P5,f)102 <td>··············<pre><code·class="language-macaulay2">i4·:·time·psi·=·abstractRationalMap(P4,P5,f)
103 ·--·used·0.0039265s·(cpu);·0.000510879s·(thread);·0s·(gc)103 ·--·used·1.6471e-05s·(cpu);·0.000453851s·(thread);·0s·(gc)
  
104 o4·=·--·rational·map·--104 o4·=·--·rational·map·--
105 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])105 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
106 ······················0···1···2···3···4106 ······················0···1···2···3···4
107 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])107 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
108 ······················0···1···2···3···4···5108 ······················0···1···2···3···4···5
109 ·····defining·forms:·given·by·a·function109 ·····defining·forms:·given·by·a·function
Offset 113, 21 lines modifiedOffset 113, 21 lines modified
113 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)</code></pre>113 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ········</table>115 ········</table>
116 ········<p>Now·we·compute·first·the·degree·of·the·forms·defining·the·abstract·map·<span·class="tt">psi</span>·and·then·the·corresponding·concrete·rational·map.</p>116 ········<p>Now·we·compute·first·the·degree·of·the·forms·defining·the·abstract·map·<span·class="tt">psi</span>·and·then·the·corresponding·concrete·rational·map.</p>
117 ········<table·class="examples">117 ········<table·class="examples">
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees(psi,3)119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees(psi,3)
120 ·--·used·0.160223s·(cpu);·0.120764s·(thread);·0s·(gc)120 ·--·used·0.192038s·(cpu);·0.123195s·(thread);·0s·(gc)
  
121 o5·=·2</code></pre>121 o5·=·2</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i6·:·time·rationalMap·psi124 <td>··············<pre><code·class="language-macaulay2">i6·:·time·rationalMap·psi
125 ·--·used·0.32006s·(cpu);·0.279947s·(thread);·0s·(gc)125 ·--·used·0.336602s·(cpu);·0.298803s·(thread);·0s·(gc)
  
126 o6·=·--·rational·map·--126 o6·=·--·rational·map·--
127 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])127 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
128 ······················0···1···2···3···4128 ······················0···1···2···3···4
129 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])129 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
130 ······················0···1···2···3···4···5130 ······················0···1···2···3···4···5
131 ·····defining·forms:·{131 ·····defining·forms:·{
Offset 211, 15 lines modifiedOffset 211, 15 lines modified
  
211 ·················ZZ211 ·················ZZ
212 o13·:·Ideal·of·-----[x·..x·]212 o13·:·Ideal·of·-----[x·..x·]
213 ···············65521··0···3</code></pre>213 ···············65521··0···3</code></pre>
214 </td>··········</tr>214 </td>··········</tr>
215 ··········<tr>215 ··········<tr>
216 <td>··············<pre><code·class="language-macaulay2">i14·:·time·T·=·abstractRationalMap(I,&quot;OADP&quot;)216 <td>··············<pre><code·class="language-macaulay2">i14·:·time·T·=·abstractRationalMap(I,&quot;OADP&quot;)
217 ·--·used·0.0344235s·(cpu);·0.0350582s·(thread);·0s·(gc)217 ·--·used·0.0408175s·(cpu);·0.0442475s·(thread);·0s·(gc)
  
218 o14·=·--·rational·map·--218 o14·=·--·rational·map·--
219 ·····················ZZ219 ·····················ZZ
220 ······source:·Proj(-----[x·,·x·,·x·,·x·])220 ······source:·Proj(-----[x·,·x·,·x·,·x·])
221 ···················65521··0···1···2···3221 ···················65521··0···1···2···3
222 ·····················ZZ222 ·····················ZZ
223 ······target:·Proj(-----[x·,·x·,·x·,·x·])223 ······target:·Proj(-----[x·,·x·,·x·,·x·])
Offset 229, 39 lines modifiedOffset 229, 39 lines modified
229 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)</code></pre>229 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)</code></pre>
230 </td>··········</tr>230 </td>··········</tr>
231 ········</table>231 ········</table>
232 ········<p>The·degree·of·the·forms·defining·the·abstract·map·<span·class="tt">T</span>·can·be·obtained·by·the·following·command:</p>232 ········<p>The·degree·of·the·forms·defining·the·abstract·map·<span·class="tt">T</span>·can·be·obtained·by·the·following·command:</p>
233 ········<table·class="examples">233 ········<table·class="examples">
234 ··········<tr>234 ··········<tr>
235 <td>··············<pre><code·class="language-macaulay2">i15·:·time·projectiveDegrees(T,2)235 <td>··············<pre><code·class="language-macaulay2">i15·:·time·projectiveDegrees(T,2)
236 ·--·used·2.42403s·(cpu);·1.58854s·(thread);·0s·(gc)236 ·--·used·2.9282s·(cpu);·1.83756s·(thread);·0s·(gc)
  
237 o15·=·3</code></pre>237 o15·=·3</code></pre>
238 </td>··········</tr>238 </td>··········</tr>
239 ········</table>239 ········</table>
240 ········<p>We·verify·that·the·composition·of·<span·class="tt">T</span>·with·itself·is·defined·by·linear·forms:</p>240 ········<p>We·verify·that·the·composition·of·<span·class="tt">T</span>·with·itself·is·defined·by·linear·forms:</p>
241 ········<table·class="examples">241 ········<table·class="examples">
242 ··········<tr>242 ··········<tr>
243 <td>··············<pre><code·class="language-macaulay2">i16·:·time·T2·=·T·*·T243 <td>··············<pre><code·class="language-macaulay2">i16·:·time·T2·=·T·*·T
244 ·--·used·0.000178328s·(cpu);·1.8668e-05s·(thread);·0s·(gc)244 ·--·used·0.000163336s·(cpu);·1.8605e-05s·(thread);·0s·(gc)
  
245 o16·=·--·rational·map·--245 o16·=·--·rational·map·--
246 ·····················ZZ246 ·····················ZZ
247 ······source:·Proj(-----[x·,·x·,·x·,·x·])247 ······source:·Proj(-----[x·,·x·,·x·,·x·])
248 ···················65521··0···1···2···3248 ···················65521··0···1···2···3
249 ·····················ZZ249 ·····················ZZ
250 ······target:·Proj(-----[x·,·x·,·x·,·x·])250 ······target:·Proj(-----[x·,·x·,·x·,·x·])
251 ···················65521··0···1···2···3251 ···················65521··0···1···2···3
252 ······defining·forms:·given·by·a·function252 ······defining·forms:·given·by·a·function
  
253 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)</code></pre>253 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)</code></pre>
254 </td>··········</tr>254 </td>··········</tr>
255 ··········<tr>255 ··········<tr>
256 <td>··············<pre><code·class="language-macaulay2">i17·:·time·projectiveDegrees(T2,2)256 <td>··············<pre><code·class="language-macaulay2">i17·:·time·projectiveDegrees(T2,2)
257 ·--·used·3.82737s·(cpu);·2.55243s·(thread);·0s·(gc)257 ·--·used·4.03348s·(cpu);·2.4503s·(thread);·0s·(gc)
  
258 o17·=·1</code></pre>258 o17·=·1</code></pre>
259 </td>··········</tr>259 </td>··········</tr>
260 ········</table>260 ········</table>
261 ········<p>We·verify·that·the·composition·of·<span·class="tt">T</span>·with·itself·leaves·a·random·point·fixed:</p>261 ········<p>We·verify·that·the·composition·of·<span·class="tt">T</span>·with·itself·leaves·a·random·point·fixed:</p>
262 ········<table·class="examples">262 ········<table·class="examples">
263 ··········<tr>263 ··········<tr>
Offset 286, 15 lines modifiedOffset 286, 15 lines modified
286 o20·:·List</code></pre>286 o20·:·List</code></pre>
287 </td>··········</tr>287 </td>··········</tr>
288 ········</table>288 ········</table>
289 ········<p>We·now·compute·the·concrete·rational·map·corresponding·to·<span·class="tt">T</span>:</p>289 ········<p>We·now·compute·the·concrete·rational·map·corresponding·to·<span·class="tt">T</span>:</p>
290 ········<table·class="examples">290 ········<table·class="examples">
291 ··········<tr>291 ··········<tr>
292 <td>··············<pre><code·class="language-macaulay2">i21·:·time·f·=·rationalMap·T292 <td>··············<pre><code·class="language-macaulay2">i21·:·time·f·=·rationalMap·T
293 ·--·used·3.17764s·(cpu);·2.16018s·(thread);·0s·(gc)293 ·--·used·3.35844s·(cpu);·2.13863s·(thread);·0s·(gc)
  
294 o21·=·--·rational·map·--294 o21·=·--·rational·map·--
295 ·····················ZZ295 ·····················ZZ
296 ······source:·Proj(-----[x·,·x·,·x·,·x·])296 ······source:·Proj(-----[x·,·x·,·x·,·x·])
297 ···················65521··0···1···2···3297 ···················65521··0···1···2···3
298 ·····················ZZ298 ·····················ZZ
299 ······target:·Proj(-----[x·,·x·,·x·,·x·])299 ······target:·Proj(-----[x·,·x·,·x·,·x·])
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Offset 36, 32 lines modifiedOffset 36, 32 lines modified
36 i3·:·P5·:=·QQ[u_0..u_5]36 i3·:·P5·:=·QQ[u_0..u_5]
  
37 o3·=·QQ[u·..u·]37 o3·=·QQ[u·..u·]
38 ·········0···538 ·········0···5
  
39 o3·:·PolynomialRing39 o3·:·PolynomialRing
40 i4·:·time·psi·=·abstractRationalMap(P4,P5,f)40 i4·:·time·psi·=·abstractRationalMap(P4,P5,f)
41 ·--·used·0.0039265s·(cpu);·0.000510879s·(thread);·0s·(gc)41 ·--·used·1.6471e-05s·(cpu);·0.000453851s·(thread);·0s·(gc)
  
42 o4·=·--·rational·map·--42 o4·=·--·rational·map·--
43 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])43 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
44 ······················0···1···2···3···444 ······················0···1···2···3···4
45 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])45 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
46 ······················0···1···2···3···4···546 ······················0···1···2···3···4···5
47 ·····defining·forms:·given·by·a·function47 ·····defining·forms:·given·by·a·function
  
48 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)48 o4·:·AbstractRationalMap·(rational·map·from·PP^4·to·PP^5)
49 Now·we·compute·first·the·degree·of·the·forms·defining·the·abstract·map·psi·and49 Now·we·compute·first·the·degree·of·the·forms·defining·the·abstract·map·psi·and
50 then·the·corresponding·concrete·rational·map.50 then·the·corresponding·concrete·rational·map.
51 i5·:·time·projectiveDegrees(psi,3)51 i5·:·time·projectiveDegrees(psi,3)
52 ·--·used·0.160223s·(cpu);·0.120764s·(thread);·0s·(gc)52 ·--·used·0.192038s·(cpu);·0.123195s·(thread);·0s·(gc)
  
53 o5·=·253 o5·=·2
54 i6·:·time·rationalMap·psi54 i6·:·time·rationalMap·psi
55 ·--·used·0.32006s·(cpu);·0.279947s·(thread);·0s·(gc)55 ·--·used·0.336602s·(cpu);·0.298803s·(thread);·0s·(gc)
  
56 o6·=·--·rational·map·--56 o6·=·--·rational·map·--
57 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])57 ·····source:·Proj(QQ[t·,·t·,·t·,·t·,·t·])
58 ······················0···1···2···3···458 ······················0···1···2···3···4
59 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])59 ·····target:·Proj(QQ[u·,·u·,·u·,·u·,·u·,·u·])
60 ······················0···1···2···3···4···560 ······················0···1···2···3···4···5
61 ·····defining·forms:·{61 ·····defining·forms:·{
Offset 140, 48 lines modifiedOffset 140, 48 lines modified
140 o13·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)140 o13·=·ideal·(-·x··+·x·x·,·-·x·x··+·x·x·,·-·x··+·x·x·)
141 ················1····0·2·····1·2····0·3·····2····1·3141 ················1····0·2·····1·2····0·3·····2····1·3
  
142 ·················ZZ142 ·················ZZ
143 o13·:·Ideal·of·-----[x·..x·]143 o13·:·Ideal·of·-----[x·..x·]
144 ···············65521··0···3144 ···············65521··0···3
145 i14·:·time·T·=·abstractRationalMap(I,"OADP")145 i14·:·time·T·=·abstractRationalMap(I,"OADP")
146 ·--·used·0.0344235s·(cpu);·0.0350582s·(thread);·0s·(gc)146 ·--·used·0.0408175s·(cpu);·0.0442475s·(thread);·0s·(gc)
  
147 o14·=·--·rational·map·--147 o14·=·--·rational·map·--
148 ·····················ZZ148 ·····················ZZ
149 ······source:·Proj(-----[x·,·x·,·x·,·x·])149 ······source:·Proj(-----[x·,·x·,·x·,·x·])
150 ···················65521··0···1···2···3150 ···················65521··0···1···2···3
151 ·····················ZZ151 ·····················ZZ
152 ······target:·Proj(-----[x·,·x·,·x·,·x·])152 ······target:·Proj(-----[x·,·x·,·x·,·x·])
153 ···················65521··0···1···2···3153 ···················65521··0···1···2···3
154 ······defining·forms:·given·by·a·function154 ······defining·forms:·given·by·a·function
  
155 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)155 o14·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)
156 The·degree·of·the·forms·defining·the·abstract·map·T·can·be·obtained·by·the156 The·degree·of·the·forms·defining·the·abstract·map·T·can·be·obtained·by·the
157 following·command:157 following·command:
158 i15·:·time·projectiveDegrees(T,2)158 i15·:·time·projectiveDegrees(T,2)
159 ·--·used·2.42403s·(cpu);·1.58854s·(thread);·0s·(gc)159 ·--·used·2.9282s·(cpu);·1.83756s·(thread);·0s·(gc)
  
160 o15·=·3160 o15·=·3
161 We·verify·that·the·composition·of·T·with·itself·is·defined·by·linear·forms:161 We·verify·that·the·composition·of·T·with·itself·is·defined·by·linear·forms:
162 i16·:·time·T2·=·T·*·T162 i16·:·time·T2·=·T·*·T
163 ·--·used·0.000178328s·(cpu);·1.8668e-05s·(thread);·0s·(gc)163 ·--·used·0.000163336s·(cpu);·1.8605e-05s·(thread);·0s·(gc)
  
164 o16·=·--·rational·map·--164 o16·=·--·rational·map·--
165 ·····················ZZ165 ·····················ZZ
166 ······source:·Proj(-----[x·,·x·,·x·,·x·])166 ······source:·Proj(-----[x·,·x·,·x·,·x·])
167 ···················65521··0···1···2···3167 ···················65521··0···1···2···3
168 ·····················ZZ168 ·····················ZZ
169 ······target:·Proj(-----[x·,·x·,·x·,·x·])169 ······target:·Proj(-----[x·,·x·,·x·,·x·])
170 ···················65521··0···1···2···3170 ···················65521··0···1···2···3
171 ······defining·forms:·given·by·a·function171 ······defining·forms:·given·by·a·function
  
172 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)172 o16·:·AbstractRationalMap·(rational·map·from·PP^3·to·PP^3)
173 i17·:·time·projectiveDegrees(T2,2)173 i17·:·time·projectiveDegrees(T2,2)
174 ·--·used·3.82737s·(cpu);·2.55243s·(thread);·0s·(gc)174 ·--·used·4.03348s·(cpu);·2.4503s·(thread);·0s·(gc)
  
175 o17·=·1175 o17·=·1
176 We·verify·that·the·composition·of·T·with·itself·leaves·a·random·point·fixed:176 We·verify·that·the·composition·of·T·with·itself·leaves·a·random·point·fixed:
177 i18·:·p·=·apply(3,i->random(ZZ/65521))|{1}177 i18·:·p·=·apply(3,i->random(ZZ/65521))|{1}
  
178 o18·=·{28963,·31975,·-30172,·1}178 o18·=·{28963,·31975,·-30172,·1}
  
Offset 194, 15 lines modifiedOffset 194, 15 lines modified
194 i20·:·T·q194 i20·:·T·q
  
195 o20·=·{28963,·31975,·-30172,·1}195 o20·=·{28963,·31975,·-30172,·1}
  
196 o20·:·List196 o20·:·List
197 We·now·compute·the·concrete·rational·map·corresponding·to·T:197 We·now·compute·the·concrete·rational·map·corresponding·to·T:
198 i21·:·time·f·=·rationalMap·T198 i21·:·time·f·=·rationalMap·T
199 ·--·used·3.17764s·(cpu);·2.16018s·(thread);·0s·(gc)199 ·--·used·3.35844s·(cpu);·2.13863s·(thread);·0s·(gc)
  
200 o21·=·--·rational·map·--200 o21·=·--·rational·map·--
201 ·····················ZZ201 ·····················ZZ
202 ······source:·Proj(-----[x·,·x·,·x·,·x·])202 ······source:·Proj(-----[x·,·x·,·x·,·x·])
203 ···················65521··0···1···2···3203 ···················65521··0···1···2···3
204 ·····················ZZ204 ·····················ZZ
205 ······target:·Proj(-----[x·,·x·,·x·,·x·])205 ······target:·Proj(-----[x·,·x·,·x·,·x·])
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129 ·······················1·2····0·4129 ·······················1·2····0·4
130 ·····················}130 ·····················}
  
131 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)</code></pre>131 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i3·:·time·psi·=·approximateInverseMap·phi134 <td>··············<pre><code·class="language-macaulay2">i3·:·time·psi·=·approximateInverseMap·phi
135 ·--·used·0.211611s·(cpu);·0.16808s·(thread);·0s·(gc)135 ·--·used·0.256069s·(cpu);·0.198245s·(thread);·0s·(gc)
136 --·approximateInverseMap:·step·1·of·10136 --·approximateInverseMap:·step·1·of·10
137 --·approximateInverseMap:·step·2·of·10137 --·approximateInverseMap:·step·2·of·10
138 --·approximateInverseMap:·step·3·of·10138 --·approximateInverseMap:·step·3·of·10
139 --·approximateInverseMap:·step·4·of·10139 --·approximateInverseMap:·step·4·of·10
140 --·approximateInverseMap:·step·5·of·10140 --·approximateInverseMap:·step·5·of·10
141 --·approximateInverseMap:·step·6·of·10141 --·approximateInverseMap:·step·6·of·10
142 --·approximateInverseMap:·step·7·of·10142 --·approximateInverseMap:·step·7·of·10
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193 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>193 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>
194 </td>··········</tr>194 </td>··········</tr>
195 ··········<tr>195 ··········<tr>
196 <td>··············<pre><code·class="language-macaulay2">i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)</code></pre>196 <td>··············<pre><code·class="language-macaulay2">i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)</code></pre>
197 </td>··········</tr>197 </td>··········</tr>
198 ··········<tr>198 ··········<tr>
199 <td>··············<pre><code·class="language-macaulay2">i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);199 <td>··············<pre><code·class="language-macaulay2">i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);
200 ·--·used·0.182534s·(cpu);·0.139433s·(thread);·0s·(gc)200 ·--·used·0.218309s·(cpu);·0.148777s·(thread);·0s·(gc)
201 --·approximateInverseMap:·step·1·of·3201 --·approximateInverseMap:·step·1·of·3
202 --·approximateInverseMap:·step·2·of·3202 --·approximateInverseMap:·step·2·of·3
203 --·approximateInverseMap:·step·3·of·3203 --·approximateInverseMap:·step·3·of·3
  
204 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>204 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>
205 </td>··········</tr>205 </td>··········</tr>
206 ··········<tr>206 ··········<tr>
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282 ·····················}282 ·····················}
  
283 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)</code></pre>283 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)</code></pre>
284 </td>··········</tr>284 </td>··········</tr>
285 ··········<tr>285 ··········<tr>
286 <td>··············<pre><code·class="language-macaulay2">i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time·286 <td>··············<pre><code·class="language-macaulay2">i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time·
287 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)287 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)
288 ·--·used·1.59839s·(cpu);·1.39582s·(thread);·0s·(gc)288 ·--·used·2.30278s·(cpu);·1.94875s·(thread);·0s·(gc)
289 --·approximateInverseMap:·step·1·of·3289 --·approximateInverseMap:·step·1·of·3
290 --·approximateInverseMap:·step·2·of·3290 --·approximateInverseMap:·step·2·of·3
291 --·approximateInverseMap:·step·3·of·3291 --·approximateInverseMap:·step·3·of·3
  
292 o8·=·--·rational·map·--292 o8·=·--·rational·map·--
293 ································ZZ293 ································ZZ
294 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by294 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by
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349 ·····phi·*·psi·==·1349 ·····phi·*·psi·==·1
  
350 o9·=·false</code></pre>350 o9·=·false</code></pre>
351 </td>··········</tr>351 </td>··········</tr>
352 ··········<tr>352 ··········<tr>
353 <td>··············<pre><code·class="language-macaulay2">i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify353 <td>··············<pre><code·class="language-macaulay2">i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify
354 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)354 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
355 ·--·used·2.43036s·(cpu);·2.12624s·(thread);·0s·(gc)355 ·--·used·2.9462s·(cpu);·2.54577s·(thread);·0s·(gc)
356 --·approximateInverseMap:·step·1·of·3356 --·approximateInverseMap:·step·1·of·3
357 --·approximateInverseMap:·step·2·of·3357 --·approximateInverseMap:·step·2·of·3
358 --·approximateInverseMap:·step·3·of·3358 --·approximateInverseMap:·step·3·of·3
359 Certify:·output·certified!359 Certify:·output·certified!
  
360 o10·=·--·rational·map·--360 o10·=·--·rational·map·--
361 ·································ZZ361 ·································ZZ
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126 ······················x·x··-·x·x126 ······················x·x··-·x·x
127 ·······················1·2····0·4127 ·······················1·2····0·4
128 ·····················}128 ·····················}
  
129 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)129 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^9·to·PP^8)
130 i3·:·time·psi·=·approximateInverseMap·phi130 i3·:·time·psi·=·approximateInverseMap·phi
131 ·--·used·0.211611s·(cpu);·0.16808s·(thread);·0s·(gc)131 ·--·used·0.256069s·(cpu);·0.198245s·(thread);·0s·(gc)
132 --·approximateInverseMap:·step·1·of·10132 --·approximateInverseMap:·step·1·of·10
133 --·approximateInverseMap:·step·2·of·10133 --·approximateInverseMap:·step·2·of·10
134 --·approximateInverseMap:·step·3·of·10134 --·approximateInverseMap:·step·3·of·10
135 --·approximateInverseMap:·step·4·of·10135 --·approximateInverseMap:·step·4·of·10
136 --·approximateInverseMap:·step·5·of·10136 --·approximateInverseMap:·step·5·of·10
137 --·approximateInverseMap:·step·6·of·10137 --·approximateInverseMap:·step·6·of·10
138 --·approximateInverseMap:·step·7·of·10138 --·approximateInverseMap:·step·7·of·10
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250 0·6·····3·6······6······0·7······1·7······3·7······4·7······6·7······7······0·8250 0·6·····3·6······6······0·7······1·7······3·7······4·7······6·7······7······0·8
251 1·8······2·8······3·8······4·8······5·8······6·8······7·8·····8251 1·8······2·8······3·8······4·8······5·8······6·8······7·8·····8
252 ·····················}252 ·····················}
  
253 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)253 o3·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)
254 i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)254 i4·:·assert(phi·*·psi·==·1·and·psi·*·phi·==·1)
255 i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);255 i5·:·time·psi'·=·approximateInverseMap(phi,CodimBsInv=>5);
256 ·--·used·0.182534s·(cpu);·0.139433s·(thread);·0s·(gc)256 ·--·used·0.218309s·(cpu);·0.148777s·(thread);·0s·(gc)
257 --·approximateInverseMap:·step·1·of·3257 --·approximateInverseMap:·step·1·of·3
258 --·approximateInverseMap:·step·2·of·3258 --·approximateInverseMap:·step·2·of·3
259 --·approximateInverseMap:·step·3·of·3259 --·approximateInverseMap:·step·3·of·3
  
260 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)260 o5·:·RationalMap·(quadratic·rational·map·from·PP^8·to·hypersurface·in·PP^9)
261 i6·:·assert(psi·==·psi')261 i6·:·assert(psi·==·psi')
262 A·more·complicated·example·is·the·following·(here·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8M_\x8a_\x8p·takes·a·lot·of262 A·more·complicated·example·is·the·following·(here·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8M_\x8a_\x8p·takes·a·lot·of
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416 4·6······5·6······6······0·7······1·7······2·7·····3·7······4·7······5·7······6·7·····0·8······1·8······3·8·····4·8416 4·6······5·6······6······0·7······1·7······2·7·····3·7······4·7······5·7······6·7·····0·8······1·8······3·8·····4·8
417 5·8······6·8·····7·8417 5·8······6·8·····7·8
418 ·····················}418 ·····················}
  
419 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)419 o7·:·RationalMap·(quadratic·rational·map·from·PP^8·to·8-dimensional·subvariety·of·PP^11)
420 i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time420 i8·:·--·without·the·option·'CodimBsInv=>4',·it·takes·about·triple·time
421 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)421 ·····time·psi=approximateInverseMap(phi,CodimBsInv=>4)
422 ·--·used·1.59839s·(cpu);·1.39582s·(thread);·0s·(gc)422 ·--·used·2.30278s·(cpu);·1.94875s·(thread);·0s·(gc)
423 --·approximateInverseMap:·step·1·of·3423 --·approximateInverseMap:·step·1·of·3
424 --·approximateInverseMap:·step·2·of·3424 --·approximateInverseMap:·step·2·of·3
425 --·approximateInverseMap:·step·3·of·3425 --·approximateInverseMap:·step·3·of·3
  
426 o8·=·--·rational·map·--426 o8·=·--·rational·map·--
427 ································ZZ427 ································ZZ
428 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by428 ·····source:·subvariety·of·Proj(--[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x··,·x··])·defined·by
Offset 523, 15 lines modifiedOffset 523, 15 lines modified
523 o8·:·RationalMap·(quadratic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)523 o8·:·RationalMap·(quadratic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)
524 i9·:·--·but...524 i9·:·--·but...
525 ·····phi·*·psi·==·1525 ·····phi·*·psi·==·1
  
526 o9·=·false526 o9·=·false
527 i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify527 i10·:·--·in·this·case·we·can·remedy·enabling·the·option·Certify
528 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)528 ······time·psi·=·approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
529 ·--·used·2.43036s·(cpu);·2.12624s·(thread);·0s·(gc)529 ·--·used·2.9462s·(cpu);·2.54577s·(thread);·0s·(gc)
530 --·approximateInverseMap:·step·1·of·3530 --·approximateInverseMap:·step·1·of·3
531 --·approximateInverseMap:·step·2·of·3531 --·approximateInverseMap:·step·2·of·3
532 --·approximateInverseMap:·step·3·of·3532 --·approximateInverseMap:·step·3·of·3
533 Certify:·output·certified!533 Certify:·output·certified!
  
534 o10·=·--·rational·map·--534 o10·=·--·rational·map·--
535 ·································ZZ535 ·································ZZ
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./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
    
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o4·=·map·(ringP8,·ringP14,·{-·95x··+·181x·x··+·1028x··-·1384x·x··-·1455x·x··+·559x··-·502x·x··+·1264x·x··-·162x·x··+·1209x··-·180x·x··-·504x·x··-·1168x·x··-·676x·x··+·501x··+·73x·x··+·1263x·x··+·1035x·x··+·844x·x··+·1593x·x··+·785x··+·982x·x··-·412x·x··+·1335x·x··+·1136x·x··+·826x·x··+·1078x·x··+·1158x··+·335x·x··-·982x·x··-·1479x·x··-·15x·x··+·1363x·x··+·1397x·x··-·575x·x··-·71x··+·1255x·x··-·1138x·x··-·1590x·x··+·604x·x··+·1182x·x··-·63x·x··-·1382x·x··-·1255x·x··-·613x·,·-·1444x··+·575x·x··+·767x··-·1495x·x··+·1631x·x··-·217x··-·294x·x··-·1511x·x··-·504x·x··-·1284x··-·1459x·x··+·152x·x··+·141x·x··-·10x·x··-·95x··+·1056x·x··+·654x·x··+·1397x·x··-·930x·x··+·578x·x··-·696x··+·759x·x··+·733x·x··+·505x·x··-·609x·x··+·526x·x··-·659x·x··+·846x··+·1253x·x··-·1519x·x··+·635x·x··+·576x·x··+·54x·x··-·1261x·x··-·822x·x··-·257x··-·986x·x··+·356x·x··-·1488x·x··-·1561x·x··-·850x·x··-·85x·x··-·1350x·x··-·783x·x··-·1335x·,·-·871x··+·1006x·x··-·1399x··-·1636x·x··-·699x·x··-·769x··-·307x·x··-·1645x·x··-·502x·x··-·719x··+·1405x·x··+·870x·x··-·1133x·x··+·425x·x··-·1203x··-·1601x·x··+·117x·x··-·382x·x··+·318x·x··-·117x·x··-·560x··+·1135x·x··+·1468x·x··+·869x·x··-·943x·x··-·335x·x··-·1218x·x··+·201x··-·11x·x··+·540x·x··-·710x·x··-·489x·x··+·1605x·x··+·1663x·x··-·423x·x··+·1246x··+·97x·x··-·644x·x··+·1655x·x··+·1219x·x··+·1476x·x··+·1355x·x··+·1594x·x··+·893x·x··+·1150x·,·-·143x··+·1240x·x··-·1042x··+·1649x·x··+·1024x·x··+·794x··+·1442x·x··-·1263x·x··+·537x·x··-·82x··-·734x·x··-·1569x·x··-·798x·x··-·366x·x··+·1289x··-·569x·x··-·254x·x··+·237x·x··-·1234x·x··-·807x·x··+·264x··-·202x·x··-·616x·x··+·44x·x··+·1465x·x··+·685x·x··+·1630x·x··-·406x··-·123x·x··-·4x·x··+·1583x·x··+·1235x·x··+·162x·x··+·1034x·x··-·1035x·x··+·737x··+·660x·x··+·1459x·x··-·359x·x··-·1291x·x··+·1638x·x··-·325x·x··-·631x·x··+·73x·x··-·1471x·,·-·1340x··+·31x·x··-·994x··-·880x·x··-·89x·x··+·574x··+·760x·x··-·1054x·x··+·772x·x··-·239x··-·443x·x··+·1240x·x··+·637x·x··-·1423x·x··+·320x··-·1363x·x··-·1139x·x··-·158x·x··-·325x·x··-·1578x·x··+·32x··+·695x·x··+·305x·x··+·1012x·x··+·1492x·x··+·1290x·x··+·1579x·x··-·342x··-·83x·x··-·104x·x··+·998x·x··-·92x·x··+·1554x·x··+·201x·x··-·237x·x··+·160x··-·228x·x··-·543x·x··-·1147x·x··-·376x·x··+·1313x·x··+·603x·x··+·106x·x··-·1361x·x··+·699x·,·-·228x··-·1510x·x··+·277x··-·4x·x··-·22x·x··-·1526x··+·234x·x··+·969x·x··+·1253x·x··-·1426x··-·1474x·x··+·947x·x··+·194x·x··-·316x·x··-·988x··-·1211x·x··+·1087x·x··+·536x·x··-·491x·x··+·870x·x··-·659x··+·1490x·x··-·469x·x··+·1190x·x··+·807x·x··+·650x·x··+·448x·x··-·1353x··-·218x·x··+·759x·x··-·253x·x··+·830x·x··-·1080x·x··-·143x·x··-·1313x·x··-·374x··-·180x·x··+·741x·x··+·742x·x··-·1254x·x··+·458x·x··-·345x·x··+·597x·x··+·1567x·x··-·31x·,·1120x··+·709x·x··-·1538x··-·1048x·x··-·162x·x··-·1518x··-·73x·x··+·380x·x··+·533x·x··-·286x··+·1374x·x··-·74x·x··-·22x·x··+·1535x·x··-·1071x··-·839x·x··-·560x·x··+·928x·x··+·335x·x··-·1008x·x··+·810x··-·448x·x··-·357x·x··-·107x·x··+·40x·x··+·784x·x··-·1423x·x··+·1276x··+·147x·x··+·443x·x··-·598x·x··-·1077x·x··-·1214x·x··+·322x·x··-·1408x·x··+·72x··-·63x·x··-·1513x·x··-·791x·x··+·11x·x··+·77x·x··+·836x·x··-·1100x·x··+·1637x·x··-·788x·,·1331x··+·318x·x··-·704x··+·51x·x··+·275x·x··+·1149x··+·1526x·x··+·768x·x··+·414x·x··-·782x··-·262x·x··+·686x·x··-·380x·x··+·1377x·x··+·1077x··+·1650x·x··-·1129x·x··-·508x·x··+·846x·x··+·1513x·x··+·460x··-·1626x·x··-·1024x·x··+·862x·x··+·1352x·x··-·188x·x··-·1382x·x··-·650x··+·55x·x··-·326x·x··+·1037x·x··+·705x·x··-·667x·x··+·1483x·x··+·1661x·x··-·1652x··-·1052x·x··-·692x·x··-·542x·x··+·162x·x··+·582x·x··-·1369x·x··+·934x·x··+·1392x·x··+·1227x·,·-·346x··+·1408x·x··-·1225x··-·1536x·x··-·1028x·x··-·985x··-·210x·x··-·1312x·x··+·915x·x··+·1633x··-·202x·x··-·1636x·x··-·1653x·x··-·480x·x··-·1260x··-·813x·x··-·1623x·x··-·1429x·x··+·1094x·x··-·747x·x··+·955x··+·898x·x··-·795x·x··-·35x·x··-·566x·x··+·1631x·x··-·324x·x··+·926x··-·132x·x··-·9x·x··-·1290x·x··-·543x·x··+·902x·x··+·735x·x··-·342x·x··-·400x··+·900x·x··-·463x·x··+·694x·x··-·1262x·x··-·1449x·x··-·448x·x··-·1402x·x··-·731x·x··-·996x·,·301x··+·166x·x··-·955x··-·739x·x··-·1199x·x··-·319x··+·1047x·x··-·532x·x··+·902x·x··+·1195x··-·663x·x··+·1215x·x··-·534x·x··-·332x·x··-·973x··+·772x·x··-·308x·x··+·315x·x··-·454x·x··-·483x·x··-·239x··-·1313x·x··-·419x·x··-·1340x·x··-·1388x·x··-·1340x·x··-·1665x·x··-·333x··-·465x·x··-·1084x·x··+·676x·x··-·1612x·x··-·288x·x··+·11x·x··-·1170x·x··-·189x··+·498x·x··-·889x·x··+·693x·x··+·1460x·x··-·473x·x··-·414x·x··-·122x·x··-·1659x·x··-·1421x·,·14x··-·1049x·x··+·1506x··+·1235x·x··+·642x·x··-·1034x··+·460x·x··+·150x·x··+·760x·x··-·1246x··-·1407x·x··+·1570x·x··+·1403x·x··-·1610x·x··-·431x··+·574x·x··+·893x·x··-·657x·x··+·417x·x··+·1362x·x··+·224x··+·268x·x··+·1097x·x··+·1132x·x··+·148x·x··+·1331x·x··-·77x·x··-·756x··+·228x·x··+·136x·x··-·1484x·x··-·1478x·x··-·13x·x··+·1620x·x··-·701x·x··-·769x··-·760x·x··-·492x·x··-·1077x·x··-·1249x·x··-·834x·x··-·395x·x··-·1358x·x··-·988x·x··+·113x·,·-·1634x··-·13x·x··+·805x··-·21x·x··-·1655x·x··+·1479x··-·1510x·x··-·646x·x··+·225x·x··-·1411x··+·1227x·x··-·1108x·x··+·1291x·x··-·59x·x··-·142x··+·586x·x··-·676x·x··+·655x·x··-·1476x·x··+·453x·x··-·1076x··-·1152x·x··+·1373x·x··-·1191x·x··-·416x·x··+·699x·x··+·317x·x··+·825x··-·1560x·x··-·488x·x··-·1035x·x··-·1561x·x··-·644x·x··-·1178x·x··-·1320x·x··+·158x··+·889x·x··+·1444x·x··-·1486x·x··-·1211x·x··+·1269x·x··-·1228x·x··+·568x·x··+·1591x·x··+·1207x·,·105x··-·538x·x··-·1222x··-·277x·x··+·716x·x··-·1067x··-·428x·x··+·154x·x··-·469x·x··+·77x··+·538x·x··-·179x·x··+·921x·x··-·223x·x··+·1093x··-·262x·x··+·1299x·x··+·631x·x··+·1486x·x··-·1280x·x··-·121x··-·50x·x··-·978x·x··-·694x·x··-·531x·x··+·505x·x··+·1412x·x··-·1061x··+·1202x·x··+·448x·x··-·187x·x··+·1276x·x··-·121x·x··+·1361x·x··+·697x·x··+·682x··+·1592x·x··+·705x·x··-·227x·x··-·7x·x··-·1423x·x··-·1446x·x··-·1578x·x··+·1511x·x··+·917x·,·1270x··-·391x·x··-·1116x··-·287x·x··+·653x·x··+·1643x··+·1623x·x··+·514x·x··-·14x·x··-·90x··+·1232x·x··-·1434x·x··+·1296x·x··+·1522x·x··+·136x··-·623x·x··-·607x·x··+·18x·x··+·896x·x··-·29x·x··+·1059x··-·1053x·x··+·1643x·x··+·1652x·x··-·1190x·x··-·1073x·x··+·1470x·x··-·944x··-·93x·x··-·187x·x··-·994x·x··-·1415x·x··-·229x·x··-·796x·x··+·1642x·x··+·1600x··-·344x·x··+·905x·x··+·1032x·x··-·538x·x··-·891x·x··+·1243x·x··+·1290x·x··+·490x·x··-·1148x·,·1613x··+·175x·x··-·1346x··-·1000x·x··-·1217x·x··-·729x··-·1296x·x··+·1456x·x··+·745x·x··+·539x··+·525x·x··-·811x·x··+·753x·x··+·1362x·x··+·1629x··-·840x·x··+·513x·x··+·429x·x··+·842x·x··+·1414x·x··-·308x··+·1415x·x··-·1461x·x··-·1135x·x··+·701x·x··+·766x·x··+·785x·x··+·1503x··+·147x·x··+·929x·x··-·1220x·x··-·853x·x··+·493x·x··+·226x·x··+·1416x·x··+·280x··-·7x·x··+·1632x·x··+·520x·x··+·1259x·x··+·157x·x··+·1596x·x··+·655x·x··-·42x·x··-·586x·})
92 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·····1·3·······2·3········3········0·4········1·4········2·4······3·4·······4·······0·5·······1·5·······2·5········3·5·······4·5········5········0·6········1·6········2·6·······3·6·······4·6·······5·6·······6········0·7·······1·7········2·7········3·7·······4·7········5·7········6·7·······7·······0·8········1·8········2·8········3·8········4·8········5·8·······6·8········7·8········8······0·······0·1········1·······0·2·······1·2········2·······0·3·······1·3·······2·3······3·······0·4·······1·4·······2·4·······3·4········4·······0·5········1·5·······2·5········3·5········4·5·······5······0·6·······1·6·······2·6·······3·6·······4·6········5·6········6········0·7·······1·7·······2·7········3·7·······4·7········5·7·······6·7·······7········0·8·······1·8·······2·8·····3·8········4·8········5·8········6·8········7·8·······8·······0·······0·1········1·······0·2·······1·2········2········0·3·······1·3······2·3······3········0·4········1·4········2·4········3·4·······4·······0·5·······1·5······2·5·······3·5······4·5········5········0·6········1·6········2·6········3·6········4·6········5·6·······6······0·7·······1·7·······2·7········3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8·······4·8········5·8········6·8·······7·8········8·······0·······0·1········1········0·2········1·2·······2········0·3········1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4········4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6········1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7········2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2·8········3·8·······4·8········5·8·······6·8······7·8·······892 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·····1·3·······2·3········3········0·4········1·4········2·4······3·4·······4·······0·5·······1·5·······2·5········3·5·······4·5········5········0·6········1·6········2·6·······3·6·······4·6·······5·6·······6········0·7·······1·7········2·7········3·7·······4·7········5·7········6·7·······7·······0·8········1·8········2·8········3·8········4·8········5·8·······6·8········7·8········8······0·······0·1········1·······0·2·······1·2········2·······0·3·······1·3·······2·3······3·······0·4·······1·4·······2·4·······3·4········4·······0·5········1·5·······2·5········3·5········4·5·······5······0·6·······1·6·······2·6·······3·6·······4·6········5·6········6········0·7·······1·7·······2·7········3·7·······4·7········5·7·······6·7·······7········0·8·······1·8·······2·8·····3·8········4·8········5·8········6·8········7·8·······8·······0·······0·1········1·······0·2·······1·2········2········0·3·······1·3······2·3······3········0·4········1·4········2·4········3·4·······4·······0·5·······1·5······2·5·······3·5······4·5········5········0·6········1·6········2·6········3·6········4·6········5·6·······6······0·7·······1·7·······2·7········3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8·······4·8········5·8········6·8·······7·8········8·······0·······0·1········1········0·2········1·2·······2········0·3········1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4········4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6········1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7········2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2·8········3·8·······4·8········5·8·······6·8······7·8·······8
  
93 o4·:·RingMap·ringP8·&lt;--·ringP14</code></pre>93 o4·:·RingMap·ringP8·&lt;--·ringP14</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i5·:·time·degreeMap·phi96 <td>··············<pre><code·class="language-macaulay2">i5·:·time·degreeMap·phi
97 ·--·used·0.0934991s·(cpu);·0.0516392s·(thread);·0s·(gc)97 ·--·used·0.136903s·(cpu);·0.0693437s·(thread);·0s·(gc)
  
98 o5·=·1</code></pre>98 o5·=·1</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a·general·subspace·of·P^14·101 <td>··············<pre><code·class="language-macaulay2">i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a·general·subspace·of·P^14·
102 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have·degree·equal·to·deg(G(1,5))=14)102 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have·degree·equal·to·deg(G(1,5))=14)
103 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))103 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))
Offset 108, 15 lines modifiedOffset 108, 15 lines modified
108 o6·=·map·(ringP8,·ringP8,·{-·780x··-·506x·x··+·1537x··-·132x·x··-·928x·x··+·386x··-·102x·x··+·422x·x··+·725x·x··-·1073x··-·905x·x··-·830x·x··+·1500x·x··+·276x·x··+·1533x··-·653x·x··+·1558x·x··+·939x·x··-·1432x·x··+·462x·x··-·329x··-·92x·x··+·661x·x··-·1298x·x··-·684x·x··+·70x·x··-·715x·x··+·1093x··+·581x·x··+·329x·x··+·454x·x··-·911x·x··-·84x·x··-·1452x·x··-·809x·x··+·1202x··+·1353x·x··+·1503x·x··+·482x·x··+·893x·x··-·643x·x··+·598x·x··+·110x·x··+·1064x·x··-·472x·,·-·522x··-·583x·x··+·1339x··+·1535x·x··-·1317x·x··+·1113x··-·169x·x··+·1440x·x··-·1657x·x··+·721x··+·40x·x··-·1576x·x··-·367x·x··+·257x·x··-·1454x··+·1612x·x··+·1529x·x··-·1068x·x··+·560x·x··-·1441x·x··+·608x··-·92x·x··-·1006x·x··+·285x·x··+·102x·x··-·397x·x··+·66x·x··-·643x··-·38x·x··+·1380x·x··+·1069x·x··-·426x·x··+·1147x·x··+·982x·x··+·10x·x··-·662x··+·16x·x··+·1561x·x··+·1597x·x··+·512x·x··+·1288x·x··-·1253x·x··+·1317x·x··+·1481x·x··-·354x·,·-·640x··-·1551x·x··+·469x··+·1482x·x··-·1593x·x··-·986x··+·471x·x··+·612x·x··+·1228x·x··+·1156x··-·731x·x··+·1503x·x··-·628x·x··+·674x·x··-·799x··+·1137x·x··+·844x·x··+·589x·x··-·666x·x··+·829x·x··-·1024x··-·170x·x··+·450x·x··+·1497x·x··+·1204x·x··-·907x·x··+·1621x·x··-·417x··+·1297x·x··+·1444x·x··+·4x·x··+·398x·x··+·996x·x··-·1031x·x··+·239x·x··+·303x··+·1215x·x··-·83x·x··+·1571x·x··-·1543x·x··-·925x·x··-·694x·x··+·151x·x··-·520x·x··+·880x·,·-·1210x··-·222x·x··+·185x··+·245x·x··+·1059x·x··-·322x··+·238x·x··+·962x·x··+·1260x·x··-·1581x··+·50x·x··+·1352x·x··-·1465x·x··+·1555x·x··+·1333x··+·1362x·x··+·1365x·x··+·1168x·x··-·1401x·x··+·149x·x··-·652x··+·1378x·x··-·557x·x··-·112x·x··+·26x·x··+·315x·x··+·111x·x··+·1592x··-·283x·x··-·1454x·x··+·907x·x··+·212x·x··+·400x·x··+·1049x·x··-·882x·x··-·1429x··-·183x·x··+·1571x·x··-·1286x·x··-·1179x·x··+·1319x·x··+·240x·x··-·1100x·x··+·1500x·x··-·348x·,·1051x··-·1325x·x··+·1354x··-·346x·x··-·1532x·x··-·466x··+·163x·x··-·659x·x··-·291x·x··+·966x··+·789x·x··+·393x·x··+·403x·x··-·1199x·x··-·570x··-·93x·x··-·492x·x··-·418x·x··+·713x·x··-·1323x·x··-·1384x··-·830x·x··-·54x·x··-·306x·x··+·709x·x··+·421x·x··-·954x·x··-·299x··+·1053x·x··-·1080x·x··+·686x·x··+·170x·x··-·1272x·x··-·1661x·x··+·1235x·x··+·1553x··-·1454x·x··-·1411x·x··-·1195x·x··-·962x·x··+·737x·x··-·390x·x··+·957x·x··+·1538x·x··+·1234x·,·-·509x··+·9x·x··-·1563x··-·710x·x··-·642x·x··+·541x··+·220x·x··-·1214x·x··-·16x·x··+·1008x··-·1088x·x··+·755x·x··-·886x·x··-·1433x·x··+·1154x··+·1627x·x··-·1547x·x··-·951x·x··+·866x·x··+·163x·x··-·1142x··-·668x·x··+·1361x·x··+·1324x·x··-·490x·x··+·282x·x··-·1133x·x··-·612x··+·805x·x··-·126x·x··+·1296x·x··-·973x·x··+·1271x·x··-·1646x·x··+·844x·x··+·1073x··-·1452x·x··-·1112x·x··-·141x·x··+·176x·x··-·1579x·x··-·78x·x··+·848x·x··-·1365x·x··+·711x·,·x··+·1543x·x··-·1076x··+·493x·x··-·526x·x··+·868x··-·582x·x··-·996x·x··+·206x·x··-·419x··+·1258x·x··-·391x·x··+·1002x·x··-·1539x·x··+·931x··-·1504x·x··+·810x·x··+·324x·x··+·1356x·x··+·313x·x··+·772x··+·299x·x··+·1186x·x··+·718x·x··+·407x·x··-·64x·x··-·828x·x··-·1393x··+·94x·x··-·290x·x··-·766x·x··+·950x·x··-·640x·x··+·265x·x··-·1640x·x··-·1403x··-·126x·x··+·891x·x··-·1519x·x··-·927x·x··-·1335x·x··-·1448x·x··-·x·x··-·1103x·x··-·1152x·,·821x··+·558x·x··-·1174x··-·168x·x··+·986x·x··+·790x··+·549x·x··+·817x·x··+·1396x·x··+·695x··+·1211x·x··+·878x·x··-·1061x·x··-·1244x·x··-·880x··+·1409x·x··-·567x·x··+·1240x·x··+·1126x·x··-·1262x·x··+·490x··+·1553x·x··+·1276x·x··+·805x·x··+·576x·x··-·1076x·x··+·1617x·x··-·495x··-·750x·x··-·277x·x··+·544x·x··+·1479x·x··-·784x·x··-·64x·x··-·1203x·x··+·405x··+·1013x·x··+·604x·x··+·1301x·x··+·1003x·x··+·235x·x··+·696x·x··+·939x·x··-·714x·x··-·879x·,·-·1452x··+·727x·x··-·1159x··+·449x·x··-·1169x·x··+·732x··+·575x·x··-·600x·x··+·924x·x··-·837x··+·1298x·x··-·860x·x··+·1010x·x··+·774x·x··+·319x··+·1087x·x··-·1120x·x··+·1439x·x··+·1175x·x··-·1648x·x··+·985x··-·1317x·x··-·878x·x··+·399x·x··-·1339x·x··+·70x·x··-·463x·x··+·470x··-·628x·x··-·907x·x··+·748x·x··+·98x·x··+·1150x·x··+·1140x·x··+·1308x·x··+·621x··+·369x·x··-·991x·x··-·1186x·x··+·61x·x··-·907x·x··-·681x·x··-·1528x·x··+·717x·x··+·854x·})108 o6·=·map·(ringP8,·ringP8,·{-·780x··-·506x·x··+·1537x··-·132x·x··-·928x·x··+·386x··-·102x·x··+·422x·x··+·725x·x··-·1073x··-·905x·x··-·830x·x··+·1500x·x··+·276x·x··+·1533x··-·653x·x··+·1558x·x··+·939x·x··-·1432x·x··+·462x·x··-·329x··-·92x·x··+·661x·x··-·1298x·x··-·684x·x··+·70x·x··-·715x·x··+·1093x··+·581x·x··+·329x·x··+·454x·x··-·911x·x··-·84x·x··-·1452x·x··-·809x·x··+·1202x··+·1353x·x··+·1503x·x··+·482x·x··+·893x·x··-·643x·x··+·598x·x··+·110x·x··+·1064x·x··-·472x·,·-·522x··-·583x·x··+·1339x··+·1535x·x··-·1317x·x··+·1113x··-·169x·x··+·1440x·x··-·1657x·x··+·721x··+·40x·x··-·1576x·x··-·367x·x··+·257x·x··-·1454x··+·1612x·x··+·1529x·x··-·1068x·x··+·560x·x··-·1441x·x··+·608x··-·92x·x··-·1006x·x··+·285x·x··+·102x·x··-·397x·x··+·66x·x··-·643x··-·38x·x··+·1380x·x··+·1069x·x··-·426x·x··+·1147x·x··+·982x·x··+·10x·x··-·662x··+·16x·x··+·1561x·x··+·1597x·x··+·512x·x··+·1288x·x··-·1253x·x··+·1317x·x··+·1481x·x··-·354x·,·-·640x··-·1551x·x··+·469x··+·1482x·x··-·1593x·x··-·986x··+·471x·x··+·612x·x··+·1228x·x··+·1156x··-·731x·x··+·1503x·x··-·628x·x··+·674x·x··-·799x··+·1137x·x··+·844x·x··+·589x·x··-·666x·x··+·829x·x··-·1024x··-·170x·x··+·450x·x··+·1497x·x··+·1204x·x··-·907x·x··+·1621x·x··-·417x··+·1297x·x··+·1444x·x··+·4x·x··+·398x·x··+·996x·x··-·1031x·x··+·239x·x··+·303x··+·1215x·x··-·83x·x··+·1571x·x··-·1543x·x··-·925x·x··-·694x·x··+·151x·x··-·520x·x··+·880x·,·-·1210x··-·222x·x··+·185x··+·245x·x··+·1059x·x··-·322x··+·238x·x··+·962x·x··+·1260x·x··-·1581x··+·50x·x··+·1352x·x··-·1465x·x··+·1555x·x··+·1333x··+·1362x·x··+·1365x·x··+·1168x·x··-·1401x·x··+·149x·x··-·652x··+·1378x·x··-·557x·x··-·112x·x··+·26x·x··+·315x·x··+·111x·x··+·1592x··-·283x·x··-·1454x·x··+·907x·x··+·212x·x··+·400x·x··+·1049x·x··-·882x·x··-·1429x··-·183x·x··+·1571x·x··-·1286x·x··-·1179x·x··+·1319x·x··+·240x·x··-·1100x·x··+·1500x·x··-·348x·,·1051x··-·1325x·x··+·1354x··-·346x·x··-·1532x·x··-·466x··+·163x·x··-·659x·x··-·291x·x··+·966x··+·789x·x··+·393x·x··+·403x·x··-·1199x·x··-·570x··-·93x·x··-·492x·x··-·418x·x··+·713x·x··-·1323x·x··-·1384x··-·830x·x··-·54x·x··-·306x·x··+·709x·x··+·421x·x··-·954x·x··-·299x··+·1053x·x··-·1080x·x··+·686x·x··+·170x·x··-·1272x·x··-·1661x·x··+·1235x·x··+·1553x··-·1454x·x··-·1411x·x··-·1195x·x··-·962x·x··+·737x·x··-·390x·x··+·957x·x··+·1538x·x··+·1234x·,·-·509x··+·9x·x··-·1563x··-·710x·x··-·642x·x··+·541x··+·220x·x··-·1214x·x··-·16x·x··+·1008x··-·1088x·x··+·755x·x··-·886x·x··-·1433x·x··+·1154x··+·1627x·x··-·1547x·x··-·951x·x··+·866x·x··+·163x·x··-·1142x··-·668x·x··+·1361x·x··+·1324x·x··-·490x·x··+·282x·x··-·1133x·x··-·612x··+·805x·x··-·126x·x··+·1296x·x··-·973x·x··+·1271x·x··-·1646x·x··+·844x·x··+·1073x··-·1452x·x··-·1112x·x··-·141x·x··+·176x·x··-·1579x·x··-·78x·x··+·848x·x··-·1365x·x··+·711x·,·x··+·1543x·x··-·1076x··+·493x·x··-·526x·x··+·868x··-·582x·x··-·996x·x··+·206x·x··-·419x··+·1258x·x··-·391x·x··+·1002x·x··-·1539x·x··+·931x··-·1504x·x··+·810x·x··+·324x·x··+·1356x·x··+·313x·x··+·772x··+·299x·x··+·1186x·x··+·718x·x··+·407x·x··-·64x·x··-·828x·x··-·1393x··+·94x·x··-·290x·x··-·766x·x··+·950x·x··-·640x·x··+·265x·x··-·1640x·x··-·1403x··-·126x·x··+·891x·x··-·1519x·x··-·927x·x··-·1335x·x··-·1448x·x··-·x·x··-·1103x·x··-·1152x·,·821x··+·558x·x··-·1174x··-·168x·x··+·986x·x··+·790x··+·549x·x··+·817x·x··+·1396x·x··+·695x··+·1211x·x··+·878x·x··-·1061x·x··-·1244x·x··-·880x··+·1409x·x··-·567x·x··+·1240x·x··+·1126x·x··-·1262x·x··+·490x··+·1553x·x··+·1276x·x··+·805x·x··+·576x·x··-·1076x·x··+·1617x·x··-·495x··-·750x·x··-·277x·x··+·544x·x··+·1479x·x··-·784x·x··-·64x·x··-·1203x·x··+·405x··+·1013x·x··+·604x·x··+·1301x·x··+·1003x·x··+·235x·x··+·696x·x··+·939x·x··-·714x·x··-·879x·,·-·1452x··+·727x·x··-·1159x··+·449x·x··-·1169x·x··+·732x··+·575x·x··-·600x·x··+·924x·x··-·837x··+·1298x·x··-·860x·x··+·1010x·x··+·774x·x··+·319x··+·1087x·x··-·1120x·x··+·1439x·x··+·1175x·x··-·1648x·x··+·985x··-·1317x·x··-·878x·x··+·399x·x··-·1339x·x··+·70x·x··-·463x·x··+·470x··-·628x·x··-·907x·x··+·748x·x··+·98x·x··+·1150x·x··+·1140x·x··+·1308x·x··+·621x··+·369x·x··-·991x·x··-·1186x·x··+·61x·x··-·907x·x··-·681x·x··-·1528x·x··+·717x·x··+·854x·})
109 ·································0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3········3·······0·4·······1·4········2·4·······3·4········4·······0·5········1·5·······2·5········3·5·······4·5·······5······0·6·······1·6········2·6·······3·6······4·6·······5·6········6·······0·7·······1·7·······2·7·······3·7······4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8·······4·8·······5·8·······6·8········7·8·······8········0·······0·1········1········0·2········1·2········2·······0·3········1·3········2·3·······3······0·4········1·4·······2·4·······3·4········4········0·5········1·5········2·5·······3·5········4·5·······5······0·6········1·6·······2·6·······3·6·······4·6······5·6·······6······0·7········1·7········2·7·······3·7········4·7·······5·7······6·7·······7······0·8········1·8········2·8·······3·8········4·8········5·8········6·8········7·8·······8········0········0·1·······1········0·2········1·2·······2·······0·3·······1·3········2·3········3·······0·4········1·4·······2·4·······3·4·······4········0·5·······1·5·······2·5·······3·5·······4·5········5·······0·6·······1·6········2·6········3·6·······4·6········5·6·······6········0·7········1·7·····2·7·······3·7·······4·7········5·7·······6·7·······7········0·8······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1·······1·······0·2········1·2·······2·······0·3·······1·3········2·3········3······0·4········1·4········2·4········3·4········4········0·5········1·5········2·5········3·5·······4·5·······5········0·6·······1·6·······2·6······3·6·······4·6·······5·6········6·······0·7········1·7·······2·7·······3·7·······4·7········5·7·······6·7········7·······0·8········1·8········2·8········3·8········4·8·······5·8········6·8········7·8·······8·······0········0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4·······4······0·5·······1·5·······2·5·······3·5········4·5········5·······0·6······1·6·······2·6·······3·6·······4·6·······5·6·······6········0·7········1·7·······2·7·······3·7········4·7········5·7········6·7········7········0·8········1·8········2·8·······3·8·······4·8·······5·8·······6·8········7·8········8········0·····0·1········1·······0·2·······1·2·······2·······0·3········1·3······2·3········3········0·4·······1·4·······2·4········3·4········4········0·5········1·5·······2·5·······3·5·······4·5········5·······0·6········1·6········2·6·······3·6·······4·6········5·6·······6·······0·7·······1·7········2·7·······3·7········4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8········4·8······5·8·······6·8········7·8·······8···0········0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5·······2·5········3·5·······4·5·······5·······0·6········1·6·······2·6·······3·6······4·6·······5·6········6······0·7·······1·7·······2·7·······3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8········4·8········5·8····6·8········7·8········8······0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3········2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5········2·5········3·5········4·5·······5········0·6········1·6·······2·6·······3·6········4·6········5·6·······6·······0·7·······1·7·······2·7········3·7·······4·7······5·7········6·7·······7········0·8·······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4·······3·4·······4········0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6·······2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3·7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8······3·8·······4·8·······5·8········6·8·······7·8·······8109 ·································0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3········3·······0·4·······1·4········2·4·······3·4········4·······0·5········1·5·······2·5········3·5·······4·5·······5······0·6·······1·6········2·6·······3·6······4·6·······5·6········6·······0·7·······1·7·······2·7·······3·7······4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8·······4·8·······5·8·······6·8········7·8·······8········0·······0·1········1········0·2········1·2········2·······0·3········1·3········2·3·······3······0·4········1·4·······2·4·······3·4········4········0·5········1·5········2·5·······3·5········4·5·······5······0·6········1·6·······2·6·······3·6·······4·6······5·6·······6······0·7········1·7········2·7·······3·7········4·7·······5·7······6·7·······7······0·8········1·8········2·8·······3·8········4·8········5·8········6·8········7·8·······8········0········0·1·······1········0·2········1·2·······2·······0·3·······1·3········2·3········3·······0·4········1·4·······2·4·······3·4·······4········0·5·······1·5·······2·5·······3·5·······4·5········5·······0·6·······1·6········2·6········3·6·······4·6········5·6·······6········0·7········1·7·····2·7·······3·7·······4·7········5·7·······6·7·······7········0·8······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1·······1·······0·2········1·2·······2·······0·3·······1·3········2·3········3······0·4········1·4········2·4········3·4········4········0·5········1·5········2·5········3·5·······4·5·······5········0·6·······1·6·······2·6······3·6·······4·6·······5·6········6·······0·7········1·7·······2·7·······3·7·······4·7········5·7·······6·7········7·······0·8········1·8········2·8········3·8········4·8·······5·8········6·8········7·8·······8·······0········0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3·······0·4·······1·4·······2·4········3·4·······4······0·5·······1·5·······2·5·······3·5········4·5········5·······0·6······1·6·······2·6·······3·6·······4·6·······5·6·······6········0·7········1·7·······2·7·······3·7········4·7········5·7········6·7········7········0·8········1·8········2·8·······3·8·······4·8·······5·8·······6·8········7·8········8········0·····0·1········1·······0·2·······1·2·······2·······0·3········1·3······2·3········3········0·4·······1·4·······2·4········3·4········4········0·5········1·5·······2·5·······3·5·······4·5········5·······0·6········1·6········2·6·······3·6·······4·6········5·6·······6·······0·7·······1·7········2·7·······3·7········4·7········5·7·······6·7········7········0·8········1·8·······2·8·······3·8········4·8······5·8·······6·8········7·8·······8···0········0·1········1·······0·2·······1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5·······2·5········3·5·······4·5·······5·······0·6········1·6·······2·6·······3·6······4·6·······5·6········6······0·7·······1·7·······2·7·······3·7·······4·7·······5·7········6·7········7·······0·8·······1·8········2·8·······3·8········4·8········5·8····6·8········7·8········8······0·······0·1········1·······0·2·······1·2·······2·······0·3·······1·3········2·3·······3········0·4·······1·4········2·4········3·4·······4········0·5·······1·5········2·5········3·5········4·5·······5········0·6········1·6·······2·6·······3·6········4·6········5·6·······6·······0·7·······1·7·······2·7········3·7·······4·7······5·7········6·7·······7········0·8·······1·8········2·8········3·8·······4·8·······5·8·······6·8·······7·8·······8·········0·······0·1········1·······0·2········1·2·······2·······0·3·······1·3·······2·3·······3········0·4·······1·4········2·4·······3·4·······4········0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6·······2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3·7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8······3·8·······4·8·······5·8········6·8·······7·8·······8
  
110 o6·:·RingMap·ringP8·&lt;--·ringP8</code></pre>110 o6·:·RingMap·ringP8·&lt;--·ringP8</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i7·:·time·degreeMap·phi'113 <td>··············<pre><code·class="language-macaulay2">i7·:·time·degreeMap·phi'
114 ·--·used·0.49013s·(cpu);·0.408322s·(thread);·0s·(gc)114 ·--·used·0.633437s·(cpu);·0.479528s·(thread);·0s·(gc)
  
115 o7·=·14</code></pre>115 o7·=·14</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ········</table>117 ········</table>
118 ······</div>118 ······</div>
119 ······<div>119 ······<div>
120 ········<h2>See·also</h2>120 ········<h2>See·also</h2>
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267 4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6267 4·······0·5·······1·5·······2·5·······3·5········4·5·······5········0·6
268 1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7268 1·6········2·6·······3·6·······4·6·······5·6········6·······0·7·······1·7
269 2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2269 2·7·······3·7·······4·7·······5·7········6·7·······7·····0·8········1·8·······2
270 8········3·8·······4·8········5·8·······6·8······7·8·······8270 8········3·8·······4·8········5·8·······6·8······7·8·······8
  
271 o4·:·RingMap·ringP8·<--·ringP14271 o4·:·RingMap·ringP8·<--·ringP14
272 i5·:·time·degreeMap·phi272 i5·:·time·degreeMap·phi
273 ·--·used·0.0934991s·(cpu);·0.0516392s·(thread);·0s·(gc)273 ·--·used·0.136903s·(cpu);·0.0693437s·(thread);·0s·(gc)
  
274 o5·=·1274 o5·=·1
275 i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a275 i6·:·--·Compose·phi:P^8--->P^14·with·a·linear·projection·P^14--->P^8·from·a
276 general·subspace·of·P^14276 general·subspace·of·P^14
277 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have277 ·····--·of·dimension·5·(so·that·the·composition·phi':P^8--->P^8·must·have
278 degree·equal·to·deg(G(1,5))=14)278 degree·equal·to·deg(G(1,5))=14)
279 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))279 ·····phi'=phi*map(ringP14,ringP8,for·i·to·8·list·random(1,ringP14))
Offset 419, 15 lines modifiedOffset 419, 15 lines modified
419 0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6419 0·5········1·5········2·5········3·5········4·5·······5········0·6·······1·6
420 2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3420 2·6········3·6······4·6·······5·6·······6·······0·7·······1·7·······2·7······3
421 7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8421 7········4·7········5·7········6·7·······7·······0·8·······1·8········2·8
422 3·8·······4·8·······5·8········6·8·······7·8·······8422 3·8·······4·8·······5·8········6·8·······7·8·······8
  
423 o6·:·RingMap·ringP8·<--·ringP8423 o6·:·RingMap·ringP8·<--·ringP8
424 i7·:·time·degreeMap·phi'424 i7·:·time·degreeMap·phi'
425 ·--·used·0.49013s·(cpu);·0.408322s·(thread);·0s·(gc)425 ·--·used·0.633437s·(cpu);·0.479528s·(thread);·0s·(gc)
  
426 o7·=·14426 o7·=·14
427 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*427 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
428 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map428 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map
429 ····*·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8s·--·projective·degrees·of·a·rational·map·between429 ····*·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8s·--·projective·degrees·of·a·rational·map·between
430 ······projective·varieties430 ······projective·varieties
431 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·d\x8de\x8eg\x8gr\x8re\x8ee\x8eM\x8Ma\x8ap\x8p:\x8:·*\x8**\x8**\x8**\x8**\x8*431 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·d\x8de\x8eg\x8gr\x8re\x8ee\x8eM\x8Ma\x8ap\x8p:\x8:·*\x8**\x8**\x8**\x8**\x8*
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79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi·=·rationalMap(X,Dominant=>2);80 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi·=·rationalMap(X,Dominant=>2);
  
81 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>81 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·time·forceImage(Phi,ideal·0_(target·Phi))84 <td>··············<pre><code·class="language-macaulay2">i4·:·time·forceImage(Phi,ideal·0_(target·Phi))
85 ·--·used·0.000437588s·(cpu);·0.000557099s·(thread);·0s·(gc)</code></pre>85 ·--·used·0.000194825s·(cpu);·0.000649497s·(thread);·0s·(gc)</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi;88 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi;
  
89 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>89 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ········</table>91 ········</table>
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20 o2·:·Ideal·of·P620 o2·:·Ideal·of·P6
21 i3·:·Phi·=·rationalMap(X,Dominant=>2);21 i3·:·Phi·=·rationalMap(X,Dominant=>2);
  
22 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of22 o3·:·RationalMap·(cubic·rational·map·from·PP^6·to·6-dimensional·subvariety·of
23 PP^9)23 PP^9)
24 i4·:·time·forceImage(Phi,ideal·0_(target·Phi))24 i4·:·time·forceImage(Phi,ideal·0_(target·Phi))
25 ·--·used·0.000437588s·(cpu);·0.000557099s·(thread);·0s·(gc)25 ·--·used·0.000194825s·(cpu);·0.000649497s·(thread);·0s·(gc)
26 i5·:·Phi;26 i5·:·Phi;
  
27 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional27 o5·:·RationalMap·(cubic·dominant·rational·map·from·PP^6·to·6-dimensional
28 subvariety·of·PP^9)28 subvariety·of·PP^9)
29 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
30 If·the·declaration·is·false,·nonsensical·answers·may·result.30 If·the·declaration·is·false,·nonsensical·answers·may·result.
31 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*31 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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113 ·························3····2·4113 ·························3····2·4
114 ·····················}114 ·····················}
  
115 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>115 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(p1,p2)·=·graph·phi;118 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(p1,p2)·=·graph·phi;
119 ·--·used·0.279228s·(cpu);·0.0397628s·(thread);·0s·(gc)</code></pre>119 ·--·used·0.263051s·(cpu);·0.0336919s·(thread);·0s·(gc)</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i4·:·p1122 <td>··············<pre><code·class="language-macaulay2">i4·:·p1
  
123 o4·=·--·rational·map·--123 o4·=·--·rational·map·--
124 ··································ZZ·································ZZ124 ··································ZZ·································ZZ
125 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y·,·y·,·y·,·y·,·y·])·defined·by125 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y·,·y·,·y·,·y·,·y·])·defined·by
Offset 260, 15 lines modifiedOffset 260, 15 lines modified
260 o8·:·List</code></pre>260 o8·:·List</code></pre>
261 </td>··········</tr>261 </td>··········</tr>
262 ········</table>262 ········</table>
263 ········<p>When·the·source·of·the·rational·map·is·a·multi-projective·variety,·the·method·returns·all·the·projections.</p>263 ········<p>When·the·source·of·the·rational·map·is·a·multi-projective·variety,·the·method·returns·all·the·projections.</p>
264 ········<table·class="examples">264 ········<table·class="examples">
265 ··········<tr>265 ··········<tr>
266 <td>··············<pre><code·class="language-macaulay2">i9·:·time·g·=·graph·p2;266 <td>··············<pre><code·class="language-macaulay2">i9·:·time·g·=·graph·p2;
267 ·--·used·0.29238s·(cpu);·0.053494s·(thread);·0s·(gc)</code></pre>267 ·--·used·0.200086s·(cpu);·0.0445791s·(thread);·0s·(gc)</code></pre>
268 </td>··········</tr>268 </td>··········</tr>
269 ··········<tr>269 ··········<tr>
270 <td>··············<pre><code·class="language-macaulay2">i10·:·g_0;270 <td>··············<pre><code·class="language-macaulay2">i10·:·g_0;
  
271 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·PP^4)</code></pre>271 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·PP^4)</code></pre>
272 </td>··········</tr>272 </td>··········</tr>
273 ··········<tr>273 ··········<tr>
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51 ······················-·x··+·x·x51 ······················-·x··+·x·x
52 ·························3····2·452 ·························3····2·4
53 ·····················}53 ·····················}
  
54 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in54 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in
55 PP^5)55 PP^5)
56 i3·:·time·(p1,p2)·=·graph·phi;56 i3·:·time·(p1,p2)·=·graph·phi;
57 ·--·used·0.279228s·(cpu);·0.0397628s·(thread);·0s·(gc)57 ·--·used·0.263051s·(cpu);·0.0336919s·(thread);·0s·(gc)
58 i4·:·p158 i4·:·p1
  
59 o4·=·--·rational·map·--59 o4·=·--·rational·map·--
60 ··································ZZ·································ZZ60 ··································ZZ·································ZZ
61 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y61 ·····source:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·])·x·Proj(------[y·,·y
62 ,·y·,·y·,·y·,·y·])·defined·by62 ,·y·,·y·,·y·,·y·])·defined·by
63 ································190181··0···1···2···3···4··········190181··063 ································190181··0···1···2···3···4··········190181··0
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193 o8·=·{51,·28,·14,·6,·2}193 o8·=·{51,·28,·14,·6,·2}
  
194 o8·:·List194 o8·:·List
195 When·the·source·of·the·rational·map·is·a·multi-projective·variety,·the·method195 When·the·source·of·the·rational·map·is·a·multi-projective·variety,·the·method
196 returns·all·the·projections.196 returns·all·the·projections.
197 i9·:·time·g·=·graph·p2;197 i9·:·time·g·=·graph·p2;
198 ·--·used·0.29238s·(cpu);·0.053494s·(thread);·0s·(gc)198 ·--·used·0.200086s·(cpu);·0.0445791s·(thread);·0s·(gc)
199 i10·:·g_0;199 i10·:·g_0;
  
200 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety200 o10·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety
201 of·PP^4·x·PP^5·x·PP^5·to·PP^4)201 of·PP^4·x·PP^5·x·PP^5·to·PP^4)
202 i11·:·g_1;202 i11·:·g_1;
  
203 o11·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety203 o11·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety
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107 ·······················1····0·3107 ·······················1····0·3
108 ·····················}108 ·····················}
  
109 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)</code></pre>109 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i3·:·time·ideal·phi112 <td>··············<pre><code·class="language-macaulay2">i3·:·time·ideal·phi
113 ·--·used·0.000499582s·(cpu);·0.00281534s·(thread);·0s·(gc)113 ·--·used·0.00399616s·(cpu);·0.00317362s·(thread);·0s·(gc)
  
114 ·············2·····································2······················114 ·············2·····································2······················
115 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+115 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+
116 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·3··116 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·3··
117 ·····------------------------------------------------------------------------117 ·····------------------------------------------------------------------------
118 ············2118 ············2
119 ·····x·x·,·x··-·x·x·)119 ·····x·x·,·x··-·x·x·)
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185 ·······················4185 ·······················4
186 ·····················}186 ·····················}
  
187 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^5·x·PP^4·to·PP^4)</code></pre>187 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^5·x·PP^4·to·PP^4)</code></pre>
188 </td>··········</tr>188 </td>··········</tr>
189 ··········<tr>189 ··········<tr>
190 <td>··············<pre><code·class="language-macaulay2">i6·:·time·ideal·phi'190 <td>··············<pre><code·class="language-macaulay2">i6·:·time·ideal·phi'
191 ·--·used·0.156274s·(cpu);·0.0916209s·(thread);·0s·(gc)191 ·--·used·0.171162s·(cpu);·0.102615s·(thread);·0s·(gc)
  
192 o6·=·ideal·1192 o6·=·ideal·1
  
193 ············································································································QQ[x·..x·,·y·..y·]193 ············································································································QQ[x·..x·,·y·..y·]
194 ················································································································0···5···0···4194 ················································································································0···5···0···4
195 o6·:·Ideal·of·--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------195 o6·:·Ideal·of·--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
196 ·····································································································································································································2196 ·····································································································································································································2
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47 ·······················247 ·······················2
48 ······················x··-·x·x48 ······················x··-·x·x
49 ·······················1····0·349 ·······················1····0·3
50 ·····················}50 ·····················}
  
51 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)51 o2·:·RationalMap·(quadratic·rational·map·from·hypersurface·in·PP^5·to·PP^4)
52 i3·:·time·ideal·phi52 i3·:·time·ideal·phi
53 ·--·used·0.000499582s·(cpu);·0.00281534s·(thread);·0s·(gc)53 ·--·used·0.00399616s·(cpu);·0.00317362s·(thread);·0s·(gc)
  
54 ·············2·····································254 ·············2·····································2
55 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+55 o3·=·ideal·(x··-·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x··+·x·x·,·x·x··-·x·x··+
56 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·356 ·············4····3·5···2·4····3·4····1·5···2·3····3····1·4···1·2····1·3
57 ·····------------------------------------------------------------------------57 ·····------------------------------------------------------------------------
58 ············258 ············2
59 ·····x·x·,·x··-·x·x·)59 ·····x·x·,·x··-·x·x·)
Offset 122, 15 lines modifiedOffset 122, 15 lines modified
122 ······················y122 ······················y
123 ·······················4123 ·······················4
124 ·····················}124 ·····················}
  
125 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of125 o5·:·MultihomogeneousRationalMap·(rational·map·from·4-dimensional·subvariety·of
126 PP^5·x·PP^4·to·PP^4)126 PP^5·x·PP^4·to·PP^4)
127 i6·:·time·ideal·phi'127 i6·:·time·ideal·phi'
128 ·--·used·0.156274s·(cpu);·0.0916209s·(thread);·0s·(gc)128 ·--·used·0.171162s·(cpu);·0.102615s·(thread);·0s·(gc)
  
129 o6·=·ideal·1129 o6·=·ideal·1
  
  
130 QQ[x·..x·,·y·..y·]130 QQ[x·..x·,·y·..y·]
  
131 0···5···0···4131 0···5···0···4
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./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
    
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154 ·······················2·4····1·5····0·6154 ·······················2·4····1·5····0·6
155 ·····················}155 ·····················}
  
156 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)</code></pre>156 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i2·:·time·psi·=·inverseMap·phi159 <td>··············<pre><code·class="language-macaulay2">i2·:·time·psi·=·inverseMap·phi
160 ·--·used·0.12278s·(cpu);·0.0826654s·(thread);·0s·(gc)160 ·--·used·0.173236s·(cpu);·0.123646s·(thread);·0s·(gc)
  
161 o2·=·--·rational·map·--161 o2·=·--·rational·map·--
162 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])162 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])
163 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20163 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20
164 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])164 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··,·w··])
165 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20165 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13···14···15···16···17···18···19···20
166 ·····defining·forms:·{166 ·····defining·forms:·{
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246 ··············0···26·······0···26·····21·22····20·23····15·24····10·25····0·26···19·22····18·23····16·24····11·25····1·26···19·20····18·21····17·24····12·25····2·26···15·19····16·21····17·23····13·25····3·26···10·19····11·21····12·23····13·24····4·26···0·19····1·21····2·23····3·24····4·25···15·18····16·20····17·22····14·25····5·26···10·18····11·20····12·22····14·24····6·26···0·18····1·20····2·22····5·24····6·25···12·16····11·17····13·18····14·19····7·26···2·16····1·17····3·18····5·19····7·25···12·15····10·17····13·20····14·21····8·26···11·15····10·16····13·22····14·23····9·26···2·15····0·17····3·20····5·21····8·25···1·15····0·16····3·22····5·23····9·25···5·13····3·14····7·15····8·16····9·17···5·12····2·14····6·17····8·18····7·20···3·12····2·13····4·17····8·19····7·21···5·11····1·14····6·16····9·18····7·22···3·11····1·13····4·16····9·19····7·23···2·11····1·12····4·18····6·19····7·24···7·10····8·11····9·12····6·13····4·14···5·10····0·14····6·15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4·20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7····1·8····2·9246 ··············0···26·······0···26·····21·22····20·23····15·24····10·25····0·26···19·22····18·23····16·24····11·25····1·26···19·20····18·21····17·24····12·25····2·26···15·19····16·21····17·23····13·25····3·26···10·19····11·21····12·23····13·24····4·26···0·19····1·21····2·23····3·24····4·25···15·18····16·20····17·22····14·25····5·26···10·18····11·20····12·22····14·24····6·26···0·18····1·20····2·22····5·24····6·25···12·16····11·17····13·18····14·19····7·26···2·16····1·17····3·18····5·19····7·25···12·15····10·17····13·20····14·21····8·26···11·15····10·16····13·22····14·23····9·26···2·15····0·17····3·20····5·21····8·25···1·15····0·16····3·22····5·23····9·25···5·13····3·14····7·15····8·16····9·17···5·12····2·14····6·17····8·18····7·20···3·12····2·13····4·17····8·19····7·21···5·11····1·14····6·16····9·18····7·22···3·11····1·13····4·16····9·19····7·23···2·11····1·12····4·18····6·19····7·24···7·10····8·11····9·12····6·13····4·14···5·10····0·14····6·15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4·20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7····1·8····2·9
  
247 o4·:·RingMap·QQ[w·..w··]·&lt;--·QQ[w·..w··]247 o4·:·RingMap·QQ[w·..w··]·&lt;--·QQ[w·..w··]
248 ·················0···26··········0···26</code></pre>248 ·················0···26··········0···26</code></pre>
249 </td>··········</tr>249 </td>··········</tr>
250 ··········<tr>250 ··········<tr>
251 <td>··············<pre><code·class="language-macaulay2">i5·:·time·psi·=·inverseMap·phi251 <td>··············<pre><code·class="language-macaulay2">i5·:·time·psi·=·inverseMap·phi
252 ·--·used·0.24329s·(cpu);·0.149285s·(thread);·0s·(gc)252 ·--·used·0.307946s·(cpu);·0.199633s·(thread);·0s·(gc)
  
253 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·w·w···-·w·w···+·w·w···+·w··w···-·w··w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·w··w···-·w··w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w···+·w·w···-·w·w··,·-·w·w··+·w·w··+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w··+·w·w···-·w·w··})253 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w··,·w·w···-·w·w···+·w·w···+·w··w···-·w··w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···+·w·w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·-·w·w···-·w··w···+·w··w···+·w·w···-·w·w··,·w··w···-·w··w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w··w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w···-·w·w···+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w···+·w·w···-·w·w··,·-·w·w··+·w·w··+·w·w···-·w·w···+·w·w··,·w·w··-·w·w··-·w·w··+·w·w···-·w·w··})
254 ··············0···26·······0···26·······5·22····8·23····14·24····13·25····0·26·····5·18····8·19····14·20····10·25····1·26·····5·16····8·17····13·20····10·24····2·26·····5·15····14·17····13·19····10·23····3·26·····5·21····20·23····19·24····17·25····4·26·····8·15····14·16····13·18····10·22····6·26·····8·21····20·22····18·24····16·25····7·26···17·18····16·19····15·20····10·21····9·26·····13·21····17·22····16·23····15·24····11·26·····14·21····19·22····18·23····15·25····12·26···0·21····4·22····7·23····12·24····11·25·····4·18····7·19····12·20····1·21····9·25·····4·16····7·17····11·20····2·21····9·24·····4·15····12·17····11·19····3·21····9·23·····7·15····12·16····11·18····6·21····9·22···12·13····11·14····0·15····3·22····6·23···10·12····9·14····1·15····3·18····6·19···10·11····9·13····2·15····3·16····6·17···8·9····7·10····1·16····2·18····6·20···5·9····4·10····1·17····2·19····3·20···8·11····7·13····0·16····2·22····6·24···5·11····4·13····0·17····2·23····3·24···8·12····7·14····0·18····1·22····6·25···5·12····4·14····0·19····1·23····3·25···5·7····4·8····0·20····1·24····2·25·····5·6····3·8····0·10····1·13····2·14···4·6····3·7····0·9····1·11····2·12254 ··············0···26·······0···26·······5·22····8·23····14·24····13·25····0·26·····5·18····8·19····14·20····10·25····1·26·····5·16····8·17····13·20····10·24····2·26·····5·15····14·17····13·19····10·23····3·26·····5·21····20·23····19·24····17·25····4·26·····8·15····14·16····13·18····10·22····6·26·····8·21····20·22····18·24····16·25····7·26···17·18····16·19····15·20····10·21····9·26·····13·21····17·22····16·23····15·24····11·26·····14·21····19·22····18·23····15·25····12·26···0·21····4·22····7·23····12·24····11·25·····4·18····7·19····12·20····1·21····9·25·····4·16····7·17····11·20····2·21····9·24·····4·15····12·17····11·19····3·21····9·23·····7·15····12·16····11·18····6·21····9·22···12·13····11·14····0·15····3·22····6·23···10·12····9·14····1·15····3·18····6·19···10·11····9·13····2·15····3·16····6·17···8·9····7·10····1·16····2·18····6·20···5·9····4·10····1·17····2·19····3·20···8·11····7·13····0·16····2·22····6·24···5·11····4·13····0·17····2·23····3·24···8·12····7·14····0·18····1·22····6·25···5·12····4·14····0·19····1·23····3·25···5·7····4·8····0·20····1·24····2·25·····5·6····3·8····0·10····1·13····2·14···4·6····3·7····0·9····1·11····2·12
  
255 o5·:·RingMap·QQ[w·..w··]·&lt;--·QQ[w·..w··]255 o5·:·RingMap·QQ[w·..w··]·&lt;--·QQ[w·..w··]
256 ·················0···26··········0···26</code></pre>256 ·················0···26··········0···26</code></pre>
257 </td>··········</tr>257 </td>··········</tr>
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html2text {}
    
Offset 99, 15 lines modifiedOffset 99, 15 lines modified
  
99 ······················w·w··-·w·w··+·w·w99 ······················w·w··-·w·w··+·w·w
100 ·······················2·4····1·5····0·6100 ·······················2·4····1·5····0·6
101 ·····················}101 ·····················}
  
102 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)102 o1·:·RationalMap·(quadratic·Cremona·transformation·of·PP^20)
103 i2·:·time·psi·=·inverseMap·phi103 i2·:·time·psi·=·inverseMap·phi
104 ·--·used·0.12278s·(cpu);·0.0826654s·(thread);·0s·(gc)104 ·--·used·0.173236s·(cpu);·0.123646s·(thread);·0s·(gc)
  
105 o2·=·--·rational·map·--105 o2·=·--·rational·map·--
106 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w106 ·····source:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w
107 ,·w··,·w··,·w··,·w··,·w··,·w··,·w··])107 ,·w··,·w··,·w··,·w··,·w··,·w··,·w··])
108 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13108 ······················0···1···2···3···4···5···6···7···8···9···10···11···12···13
109 14···15···16···17···18···19···20109 14···15···16···17···18···19···20
110 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w110 ·····target:·Proj(QQ[w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w·,·w··,·w··,·w··,·w
Offset 217, 15 lines modifiedOffset 217, 15 lines modified
217 15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4217 15····9·20····8·22···3·10····0·13····4·15····9·21····8·23···2·10····0·12····4
218 20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7218 20····6·21····8·24···1·10····0·11····4·22····6·23····9·24···4·5····3·6····0·7
219 1·8····2·9219 1·8····2·9
  
220 o4·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]220 o4·:·RingMap·QQ[w·..w··]·<--·QQ[w·..w··]
221 ·················0···26··········0···26221 ·················0···26··········0···26
222 i5·:·time·psi·=·inverseMap·phi222 i5·:·time·psi·=·inverseMap·phi
223 ·--·used·0.24329s·(cpu);·0.149285s·(thread);·0s·(gc)223 ·--·used·0.307946s·(cpu);·0.199633s·(thread);·0s·(gc)
  
224 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,224 o5·=·map·(QQ[w·..w··],·QQ[w·..w··],·{-·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,
225 -·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-225 -·w·w···+·w·w···+·w··w···-·w··w···-·w·w··,·-·w·w···+·w·w···+·w··w···-·w··w···-
226 w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-226 w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-
227 w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+227 w··w···-·w·w··,·-·w·w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w·w···-·w··w···+
228 w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+228 w··w···-·w··w···-·w·w··,·w··w···-·w··w···+·w··w···-·w··w···-·w·w··,·-·w··w···+
229 w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w229 w··w···-·w··w···+·w··w···-·w··w··,·-·w··w···+·w··w···-·w··w···+·w··w···-·w··w
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102 ·························2800·0····6350400··0·1·····50803200··0·1·····33868800··0·1····181440··1····196000·0·2····381024000··0·1·2·····47628000··0·1·2····10160640··1·2····2268000·0·2····2126250·0·1·2····762048000··1·2····992250·0·2····31752000··1·2····529200·2····73500·0·3····28576800··0·1·3·····32659200··0·1·3·····6531840··1·3····15876000·0·2·3····21432600··0·1·2·3····137168640··1·2·3····158760000·0·2·3·····571536000··1·2·3····95256000·2·3····15876000·0·3····228614400··0·1·3·····65318400··1·3····190512000·0·2·3····95256000··1·2·3····31752000·2·3····352800·0·3····604800··1·3····31752000··2·3····15120·3····47628000·0·4····444528000··0·1·4····4267468800··0·1·4····152409600·1·4····714420000··0·2·4····16003008000··0·1·2·4····1524096000··1·2·4····95256000··0·2·4····533433600··1·2·4····211680·2·4····1000188000·0·3·4····2667168000··0·1·3·4·····457228800··1·3·4····240045120··0·2·3·4·····48009024000··1·2·3·4····14817600·2·3·4····4000752000··0·3·4····1524096000··1·3·4····2667168000··2·3·4····27216000··3·4····190512000·0·4····1778112000··0·1·4····304819200··1·4····47628000··0·2·4····1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4····1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2·4····21168000·3·4···28000·4102 ·························2800·0····6350400··0·1·····50803200··0·1·····33868800··0·1····181440··1····196000·0·2····381024000··0·1·2·····47628000··0·1·2····10160640··1·2····2268000·0·2····2126250·0·1·2····762048000··1·2····992250·0·2····31752000··1·2····529200·2····73500·0·3····28576800··0·1·3·····32659200··0·1·3·····6531840··1·3····15876000·0·2·3····21432600··0·1·2·3····137168640··1·2·3····158760000·0·2·3·····571536000··1·2·3····95256000·2·3····15876000·0·3····228614400··0·1·3·····65318400··1·3····190512000·0·2·3····95256000··1·2·3····31752000·2·3····352800·0·3····604800··1·3····31752000··2·3····15120·3····47628000·0·4····444528000··0·1·4····4267468800··0·1·4····152409600·1·4····714420000··0·2·4····16003008000··0·1·2·4····1524096000··1·2·4····95256000··0·2·4····533433600··1·2·4····211680·2·4····1000188000·0·3·4····2667168000··0·1·3·4·····457228800··1·3·4····240045120··0·2·3·4·····48009024000··1·2·3·4····14817600·2·3·4····4000752000··0·3·4····1524096000··1·3·4····2667168000··2·3·4····27216000··3·4····190512000·0·4····1778112000··0·1·4····304819200··1·4····47628000··0·2·4····1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4····1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2·4····21168000·3·4···28000·4
103 ·····················}103 ·····················}
  
104 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)</code></pre>104 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i3·:·time·inverse·phi107 <td>··············<pre><code·class="language-macaulay2">i3·:·time·inverse·phi
108 ·--·used·0.104411s·(cpu);·0.0692944s·(thread);·0s·(gc)108 ·--·used·0.129512s·(cpu);·0.0800795s·(thread);·0s·(gc)
  
109 o3·=·--·rational·map·--109 o3·=·--·rational·map·--
110 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])110 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
111 ······················0···1···2···3···4111 ······················0···1···2···3···4
112 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])112 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
113 ······················0···1···2···3···4113 ······················0···1···2···3···4
114 ·····defining·forms:·{114 ·····defining·forms:·{
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282 1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4282 1333584000··1·2·4····5292000··2·4····4000752000··0·3·4···71442000·1·3·4
283 1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2283 1333584000··2·3·4····95256000··3·4····3969000·0·4···127008000·1·4····5292000··2
284 4····21168000·3·4···28000·4284 4····21168000·3·4···28000·4
285 ·····················}285 ·····················}
  
286 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)286 o2·:·RationalMap·(rational·map·from·PP^4·to·PP^4)
287 i3·:·time·inverse·phi287 i3·:·time·inverse·phi
288 ·--·used·0.104411s·(cpu);·0.0692944s·(thread);·0s·(gc)288 ·--·used·0.129512s·(cpu);·0.0800795s·(thread);·0s·(gc)
  
289 o3·=·--·rational·map·--289 o3·=·--·rational·map·--
290 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])290 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
291 ······················0···1···2···3···4291 ······················0···1···2···3···4
292 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])292 ·····target:·Proj(QQ[x·,·x·,·x·,·x·,·x·])
293 ······················0···1···2···3···4293 ······················0···1···2···3···4
294 ·····defining·forms:·{294 ·····defining·forms:·{
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122 ·························3····2·4122 ·························3····2·4
123 ·····················}123 ·····················}
  
124 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>124 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ··········<tr>126 ··········<tr>
127 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isBirational·phi127 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isBirational·phi
128 ·--·used·0.0702755s·(cpu);·0.0298302s·(thread);·0s·(gc)128 ·--·used·0.0690809s·(cpu);·0.0277457s·(thread);·0s·(gc)
  
129 o3·=·true</code></pre>129 o3·=·true</code></pre>
130 </td>··········</tr>130 </td>··········</tr>
131 ··········<tr>131 ··········<tr>
132 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isBirational(phi,Certify=>true)132 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isBirational(phi,Certify=>true)
133 ·--·used·0.285206s·(cpu);·0.0519896s·(thread);·0s·(gc)133 ·--·used·0.230439s·(cpu);·0.0350467s·(thread);·0s·(gc)
134 Certify:·output·certified!134 Certify:·output·certified!
  
135 o4·=·true</code></pre>135 o4·=·true</code></pre>
136 </td>··········</tr>136 </td>··········</tr>
137 ········</table>137 ········</table>
138 ······</div>138 ······</div>
139 ······<div>139 ······<div>
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59 ······················-·t··+·t·t59 ······················-·t··+·t·t
60 ·························3····2·460 ·························3····2·4
61 ·····················}61 ·····················}
  
62 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in62 o2·:·RationalMap·(quadratic·dominant·rational·map·from·PP^4·to·hypersurface·in
63 PP^5)63 PP^5)
64 i3·:·time·isBirational·phi64 i3·:·time·isBirational·phi
65 ·--·used·0.0702755s·(cpu);·0.0298302s·(thread);·0s·(gc)65 ·--·used·0.0690809s·(cpu);·0.0277457s·(thread);·0s·(gc)
  
66 o3·=·true66 o3·=·true
67 i4·:·time·isBirational(phi,Certify=>true)67 i4·:·time·isBirational(phi,Certify=>true)
68 ·--·used·0.285206s·(cpu);·0.0519896s·(thread);·0s·(gc)68 ·--·used·0.230439s·(cpu);·0.0350467s·(thread);·0s·(gc)
69 Certify:·output·certified!69 Certify:·output·certified!
  
70 o4·=·true70 o4·=·true
71 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*71 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
72 ····*·_\x8i_\x8s_\x8D_\x8o_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·whether·a·rational·map·is·dominant72 ····*·_\x8i_\x8s_\x8D_\x8o_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·whether·a·rational·map·is·dominant
73 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sB\x8Bi\x8ir\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*73 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sB\x8Bi\x8ir\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*
74 ····*·isBirational(RationalMap)74 ····*·isBirational(RationalMap)
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83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};84 <td>··············<pre><code·class="language-macaulay2">i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},{x_4..x_8}};
  
85 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)</code></pre>85 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isDominant(phi,Certify=>true)88 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isDominant(phi,Certify=>true)
89 ·--·used·1.78811s·(cpu);·1.6522s·(thread);·0s·(gc)89 ·--·used·2.0669s·(cpu);·1.88524s·(thread);·0s·(gc)
90 Certify:·output·certified!90 Certify:·output·certified!
  
91 o3·=·true</code></pre>91 o3·=·true</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·P7·=·ZZ/101[x_0..x_7];</code></pre>94 <td>··············<pre><code·class="language-macaulay2">i4·:·P7·=·ZZ/101[x_0..x_7];</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
Offset 104, 15 lines modifiedOffset 104, 15 lines modified
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i6·:·phi·=·rationalMap(C,3,2);105 <td>··············<pre><code·class="language-macaulay2">i6·:·phi·=·rationalMap(C,3,2);
  
106 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)</code></pre>106 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isDominant(phi,Certify=>true)109 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isDominant(phi,Certify=>true)
110 ·--·used·2.43825s·(cpu);·2.01795s·(thread);·0s·(gc)110 ·--·used·2.7713s·(cpu);·2.22914s·(thread);·0s·(gc)
111 Certify:·output·certified!111 Certify:·output·certified!
  
112 o7·=·false</code></pre>112 o7·=·false</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ········</table>114 ········</table>
115 ······</div>115 ······</div>
116 ······<div>116 ······<div>
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20 be·to·perform·the·command·kernel·map·phi·==·0.20 be·to·perform·the·command·kernel·map·phi·==·0.
21 i1·:·P8·=·ZZ/101[x_0..x_8];21 i1·:·P8·=·ZZ/101[x_0..x_8];
22 i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},22 i2·:·phi·=·rationalMap·ideal·jacobian·ideal·det·matrix{{x_0..x_4},{x_1..x_5},{x_2..x_6},{x_3..x_7},
23 {x_4..x_8}};23 {x_4..x_8}};
  
24 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)24 o2·:·RationalMap·(rational·map·from·PP^8·to·PP^8)
25 i3·:·time·isDominant(phi,Certify=>true)25 i3·:·time·isDominant(phi,Certify=>true)
26 ·--·used·1.78811s·(cpu);·1.6522s·(thread);·0s·(gc)26 ·--·used·2.0669s·(cpu);·1.88524s·(thread);·0s·(gc)
27 Certify:·output·certified!27 Certify:·output·certified!
  
28 o3·=·true28 o3·=·true
29 i4·:·P7·=·ZZ/101[x_0..x_7];29 i4·:·P7·=·ZZ/101[x_0..x_7];
30 i5·:·--·hyperelliptic·curve·of·genus·330 i5·:·--·hyperelliptic·curve·of·genus·3
31 ·····C·=·ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-31 ·····C·=·ideal(x_4*x_5+23*x_5^2-23*x_0*x_6-18*x_1*x_6+6*x_2*x_6+37*x_3*x_6+23*x_4*x_6-
32 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-32 26*x_5*x_6+2*x_6^2-25*x_0*x_7+45*x_1*x_7+30*x_2*x_7-49*x_3*x_7-49*x_4*x_7+50*x_5*x_7,x_3*x_5-
Offset 65, 15 lines modifiedOffset 65, 15 lines modified
65 47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);65 47*x_1*x_7-19*x_2*x_7+25*x_3*x_7+28*x_4*x_7+34*x_5*x_7);
  
66 o5·:·Ideal·of·P766 o5·:·Ideal·of·P7
67 i6·:·phi·=·rationalMap(C,3,2);67 i6·:·phi·=·rationalMap(C,3,2);
  
68 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)68 o6·:·RationalMap·(cubic·rational·map·from·PP^7·to·PP^7)
69 i7·:·time·isDominant(phi,Certify=>true)69 i7·:·time·isDominant(phi,Certify=>true)
70 ·--·used·2.43825s·(cpu);·2.01795s·(thread);·0s·(gc)70 ·--·used·2.7713s·(cpu);·2.22914s·(thread);·0s·(gc)
71 Certify:·output·certified!71 Certify:·output·certified!
  
72 o7·=·false72 o7·=·false
73 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*73 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
74 ····*·_\x8i_\x8s_\x8B_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l·--·whether·a·rational·map·is·birational74 ····*·_\x8i_\x8s_\x8B_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l·--·whether·a·rational·map·is·birational
75 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sD\x8Do\x8om\x8mi\x8in\x8na\x8an\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*75 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sD\x8Do\x8om\x8mi\x8in\x8na\x8an\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
76 ····*·isDominant(RationalMap)76 ····*·isDominant(RationalMap)
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87 ··············0···8·······0···11········0·3····2·4····3·4······0·5·····2·5····3·5····4·5······5·····0·6·····2·6····4·6······5·6·····6····4·7·····5·7·····6·7·····4·8······5·8·····6·8·····1·2·····1·5·····0·6·····1·6····4·6····5·6·····0·7····1·7·····2·7·····5·7·····6·7·····1·8·····7·8·······0·····0·2······0·4·····2·4····4······0·5····2·5·····4·5······0·6·····4·6····5·6····0·7·····2·7····4·7·····5·7·····6·7·····0·8·····4·8·····7·8···2·4····3·4····4·····2·5·····4·5······5·6·····6·····3·7·····4·7······5·7·····6·7····3·8····4·8······5·8····6·8······0·4····2·4·····2·5·····4·5·····0·6·····2·6····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···0·4····4·····1·5·····4·5····0·6····1·6·····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···2·3····4·····4·5····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8···1·3·····1·5····1·6····4·6·····5·6·····6····3·7······0·3····3·4····4·····0·5·····4·5····0·6····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8······0·2·····2····2·4····4····2·5·····4·5······0·6·····5·6·····2·7·····4·7······5·7·····6·7·····0·8·····2·8·····4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·7···0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····2·7·····0·8·····1·8·····5·8·····6·8·····7·887 ··············0···8·······0···11········0·3····2·4····3·4······0·5·····2·5····3·5····4·5······5·····0·6·····2·6····4·6······5·6·····6····4·7·····5·7·····6·7·····4·8······5·8·····6·8·····1·2·····1·5·····0·6·····1·6····4·6····5·6·····0·7····1·7·····2·7·····5·7·····6·7·····1·8·····7·8·······0·····0·2······0·4·····2·4····4······0·5····2·5·····4·5······0·6·····4·6····5·6····0·7·····2·7····4·7·····5·7·····6·7·····0·8·····4·8·····7·8···2·4····3·4····4·····2·5·····4·5······5·6·····6·····3·7·····4·7······5·7·····6·7····3·8····4·8······5·8····6·8······0·4····2·4·····2·5·····4·5·····0·6·····2·6····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···0·4····4·····1·5·····4·5····0·6····1·6·····4·6····5·6····4·7·····5·7·····6·7····4·8·····5·8·····6·8···2·3····4·····4·5····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8···1·3·····1·5····1·6····4·6·····5·6·····6····3·7······0·3····3·4····4·····0·5·····4·5····0·6····4·6·····5·6·····6····3·7····4·7·····5·7·····6·7····4·8·····5·8·····6·8······0·2·····2····2·4····4····2·5·····4·5······0·6·····5·6·····2·7·····4·7······5·7·····6·7·····0·8·····2·8·····4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·7···0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····2·7·····0·8·····1·8·····5·8·····6·8·····7·8
  
88 o1·:·RingMap·QQ[x·..x·]·&lt;--·QQ[y·..y··]88 o1·:·RingMap·QQ[x·..x·]·&lt;--·QQ[y·..y··]
89 ·················0···8··········0···11</code></pre>89 ·················0···8··········0···11</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i2·:·time·kernel(phi,1)92 <td>··············<pre><code·class="language-macaulay2">i2·:·time·kernel(phi,1)
93 ·--·used·0.013838s·(cpu);·0.0137s·(thread);·0s·(gc)93 ·--·used·0.0148932s·(cpu);·0.0163007s·(thread);·0s·(gc)
  
94 o2·=·ideal·()94 o2·=·ideal·()
  
95 o2·:·Ideal·of·QQ[y·..y··]95 o2·:·Ideal·of·QQ[y·..y··]
96 ··················0···11</code></pre>96 ··················0···11</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·kernel(phi,2)99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·kernel(phi,2)
100 ·--·used·0.330787s·(cpu);·0.255611s·(thread);·0s·(gc)100 ·--·used·0.370638s·(cpu);·0.299304s·(thread);·0s·(gc)
  
101 ···························2················································101 ···························2················································
102 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-102 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-
103 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·6··103 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·6··
104 ·····------------------------------------------------------------------------104 ·····------------------------------------------------------------------------
105 ···········································································105 ···········································································
106 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-106 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-
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68 4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·768 4·8·····5·8·····6·8·····7·8······0·1····1·2····1·4·····0·6····1·6····4·6····0·7
69 0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····269 0·2····1·2·····0·4····1·4······1·5····2·5·····4·5·····0·6·····1·6·····4·6·····2
70 7·····0·8·····1·8·····5·8·····6·8·····7·870 7·····0·8·····1·8·····5·8·····6·8·····7·8
  
71 o1·:·RingMap·QQ[x·..x·]·<--·QQ[y·..y··]71 o1·:·RingMap·QQ[x·..x·]·<--·QQ[y·..y··]
72 ·················0···8··········0···1172 ·················0···8··········0···11
73 i2·:·time·kernel(phi,1)73 i2·:·time·kernel(phi,1)
74 ·--·used·0.013838s·(cpu);·0.0137s·(thread);·0s·(gc)74 ·--·used·0.0148932s·(cpu);·0.0163007s·(thread);·0s·(gc)
  
75 o2·=·ideal·()75 o2·=·ideal·()
  
76 o2·:·Ideal·of·QQ[y·..y··]76 o2·:·Ideal·of·QQ[y·..y··]
77 ··················0···1177 ··················0···11
78 i3·:·time·kernel(phi,2)78 i3·:·time·kernel(phi,2)
79 ·--·used·0.330787s·(cpu);·0.255611s·(thread);·0s·(gc)79 ·--·used·0.370638s·(cpu);·0.299304s·(thread);·0s·(gc)
  
80 ···························280 ···························2
81 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-81 o3·=·ideal·(y·y··+·y·y··+·y··+·5y·y··+·y·y··+·5y·y··-·y·y··-·4y·y··-·5y·y··-
82 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·682 ·············2·4····3·4····4·····2·5····3·5·····4·5····1·6·····2·6·····5·6
83 ·····------------------------------------------------------------------------83 ·····------------------------------------------------------------------------
  
84 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-84 ·····4y·y··-·2y·y··-·y·y··+·4y·y··-·5y·y··-·4y·y··+·3y·y··-·4y·y··-·y·y···-
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101 ·················ZZ101 ·················ZZ
102 o2·:·Ideal·of·--------[x·..x·]102 o2·:·Ideal·of·--------[x·..x·]
103 ··············10000019··0···9</code></pre>103 ··············10000019··0···9</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i3·:·time·parametrize·L106 <td>··············<pre><code·class="language-macaulay2">i3·:·time·parametrize·L
107 ·--·used·0.00204689s·(cpu);·0.00414759s·(thread);·0s·(gc)107 ·--·used·0.00400104s·(cpu);·0.00500885s·(thread);·0s·(gc)
  
108 o3·=·--·rational·map·--108 o3·=·--·rational·map·--
109 ·····················ZZ109 ·····················ZZ
110 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])110 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])
111 ··················10000019··0···1···2···3···4···5111 ··················10000019··0···1···2···3···4···5
112 ·····················ZZ112 ·····················ZZ
113 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])113 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 193, 15 lines modifiedOffset 193, 15 lines modified
  
193 ·················ZZ193 ·················ZZ
194 o4·:·Ideal·of·--------[x·..x·]194 o4·:·Ideal·of·--------[x·..x·]
195 ··············10000019··0···9</code></pre>195 ··············10000019··0···9</code></pre>
196 </td>··········</tr>196 </td>··········</tr>
197 ··········<tr>197 ··········<tr>
198 <td>··············<pre><code·class="language-macaulay2">i5·:·time·parametrize·Q198 <td>··············<pre><code·class="language-macaulay2">i5·:·time·parametrize·Q
199 ·--·used·0.64793s·(cpu);·0.336382s·(thread);·0s·(gc)199 ·--·used·0.714262s·(cpu);·0.410003s·(thread);·0s·(gc)
  
200 o5·=·--·rational·map·--200 o5·=·--·rational·map·--
201 ·····················ZZ201 ·····················ZZ
202 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])202 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
203 ··················10000019··0···1···2···3···4···5···6203 ··················10000019··0···1···2···3···4···5···6
204 ·····················ZZ204 ·····················ZZ
205 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])205 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
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41 ·····-·849671x··+·3034137x·)41 ·····-·849671x··+·3034137x·)
42 ··············8···········942 ··············8···········9
  
43 ·················ZZ43 ·················ZZ
44 o2·:·Ideal·of·--------[x·..x·]44 o2·:·Ideal·of·--------[x·..x·]
45 ··············10000019··0···945 ··············10000019··0···9
46 i3·:·time·parametrize·L46 i3·:·time·parametrize·L
47 ·--·used·0.00204689s·(cpu);·0.00414759s·(thread);·0s·(gc)47 ·--·used·0.00400104s·(cpu);·0.00500885s·(thread);·0s·(gc)
  
48 o3·=·--·rational·map·--48 o3·=·--·rational·map·--
49 ·····················ZZ49 ·····················ZZ
50 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])50 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·])
51 ··················10000019··0···1···2···3···4···551 ··················10000019··0···1···2···3···4···5
52 ·····················ZZ52 ·····················ZZ
53 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])53 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 137, 15 lines modifiedOffset 137, 15 lines modified
137 ·····1211601x·x··-·2168594x·x··-·1801762x·x··+·3022242x·x··+·3618789x·)137 ·····1211601x·x··-·2168594x·x··-·1801762x·x··+·3022242x·x··+·3618789x·)
138 ·············5·9···········6·9···········7·9···········8·9···········9138 ·············5·9···········6·9···········7·9···········8·9···········9
  
139 ·················ZZ139 ·················ZZ
140 o4·:·Ideal·of·--------[x·..x·]140 o4·:·Ideal·of·--------[x·..x·]
141 ··············10000019··0···9141 ··············10000019··0···9
142 i5·:·time·parametrize·Q142 i5·:·time·parametrize·Q
143 ·--·used·0.64793s·(cpu);·0.336382s·(thread);·0s·(gc)143 ·--·used·0.714262s·(cpu);·0.410003s·(thread);·0s·(gc)
  
144 o5·=·--·rational·map·--144 o5·=·--·rational·map·--
145 ·····················ZZ145 ·····················ZZ
146 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])146 ·····source:·Proj(--------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
147 ··················10000019··0···1···2···3···4···5···6147 ··················10000019··0···1···2···3···4···5···6
148 ·····················ZZ148 ·····················ZZ
149 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])149 ·····target:·Proj(--------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);77 <td>··············<pre><code·class="language-macaulay2">i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);
  
78 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)</code></pre>78 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to·PP^8)</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·time·p·=·point·source·f81 <td>··············<pre><code·class="language-macaulay2">i2·:·time·p·=·point·source·f
82 ·--·used·0.159174s·(cpu);·0.117175s·(thread);·0s·(gc)82 ·--·used·0.175983s·(cpu);·0.122205s·(thread);·0s·(gc)
  
83 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+83 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+
84 ·············10········11···9·········11···8········11···7········11···6··84 ·············10········11···9·········11···8········11···7········11···6··
85 ·····------------------------------------------------------------------------85 ·····------------------------------------------------------------------------
86 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-86 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-
87 ···········11···5········11···4········11···3·········11···2········11···1··87 ···········11···5········11···4········11···3·········11···2········11···1··
88 ·····------------------------------------------------------------------------88 ·····------------------------------------------------------------------------
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96 ···························································33331··0···1196 ···························································33331··0···11
97 o2·:·Ideal·of·-------------------------------------------------------------------------------------------------------97 o2·:·Ideal·of·-------------------------------------------------------------------------------------------------------
98 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)98 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
99 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·11···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>99 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·11···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i3·:·time·p·==·f^*·f·p102 <td>··············<pre><code·class="language-macaulay2">i3·:·time·p·==·f^*·f·p
103 ·--·used·0.152976s·(cpu);·0.109782s·(thread);·0s·(gc)103 ·--·used·0.152868s·(cpu);·0.108276s·(thread);·0s·(gc)
  
104 o3·=·true</code></pre>104 o3·=·true</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ········</table>106 ········</table>
107 ······</div>107 ······</div>
108 ······<div>108 ······<div>
109 ········<h2>See·also</h2>109 ········<h2>See·also</h2>
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20 documentation)·,·see·_\x8p_\x8o_\x8i_\x8n_\x8t_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8).20 documentation)·,·see·_\x8p_\x8o_\x8i_\x8n_\x8t_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8).
21 Below·we·verify·the·birationality·of·a·rational·map.21 Below·we·verify·the·birationality·of·a·rational·map.
22 i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);22 i1·:·f·=·inverseMap·specialQuadraticTransformation(9,ZZ/33331);
  
23 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to23 o1·:·RationalMap·(cubic·rational·map·from·8-dimensional·subvariety·of·PP^11·to
24 PP^8)24 PP^8)
25 i2·:·time·p·=·point·source·f25 i2·:·time·p·=·point·source·f
26 ·--·used·0.159174s·(cpu);·0.117175s·(thread);·0s·(gc)26 ·--·used·0.175983s·(cpu);·0.122205s·(thread);·0s·(gc)
  
27 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+27 o2·=·ideal·(y···-·9235y··,·y··+·11075y··,·y··-·5847y··,·y··+·7396y··,·y··+
28 ·············10········11···9·········11···8········11···7········11···628 ·············10········11···9·········11···8········11···7········11···6
29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
30 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-30 ·····13530y··,·y··+·4359y··,·y··-·2924y··,·y··+·13040y··,·y··+·6904y··,·y··-
31 ···········11···5········11···4········11···3·········11···2········11···131 ···········11···5········11···4········11···3·········11···2········11···1
32 ·····------------------------------------------------------------------------32 ·····------------------------------------------------------------------------
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41 o2·:·Ideal·of·-----------------------------------------------------------------41 o2·:·Ideal·of·-----------------------------------------------------------------
42 --------------------------------------42 --------------------------------------
43 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y43 ··············(y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y·y··-·y·y··+·y·y··,·y
44 y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)44 y··-·y·y··+·y·y·,·y·y··-·y·y··+·y·y·)
45 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·1145 ················6·7····5·8····4·11···3·7····2·8····1·11···3·5····2·6····0·11
46 3·4····1·6····0·8···2·4····1·5····0·746 3·4····1·6····0·8···2·4····1·5····0·7
47 i3·:·time·p·==·f^*·f·p47 i3·:·time·p·==·f^*·f·p
48 ·--·used·0.152976s·(cpu);·0.109782s·(thread);·0s·(gc)48 ·--·used·0.152868s·(cpu);·0.108276s·(thread);·0s·(gc)
  
49 o3·=·true49 o3·=·true
50 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*50 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
51 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8K_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8P_\x8o_\x8i_\x8n_\x8t·--·Pick·a·random·K·rational·point·on·the·scheme·X51 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8K_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8P_\x8o_\x8i_\x8n_\x8t·--·Pick·a·random·K·rational·point·on·the·scheme·X
52 ······defined·by·I52 ······defined·by·I
53 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*53 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
54 ····*·point(PolynomialRing)54 ····*·point(PolynomialRing)
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90 ·····················0···4··············0···5·······1····0·2·····1·2····0·3·····2····1·3·····1·3····0·4·····2·3····1·4·····3····2·490 ·····················0···4··············0···5·······1····0·2·····1·2····0·3·····2····1·3·····1·3····0·4·····2·3····1·4·····3····2·4
  
91 o2·:·RingMap·GF·109561[t·..t·]·&lt;--·GF·109561[x·..x·]91 o2·:·RingMap·GF·109561[t·..t·]·&lt;--·GF·109561[x·..x·]
92 ························0···4·················0···5</code></pre>92 ························0···4·················0···5</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·projectiveDegrees(phi,Certify=>true)95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·projectiveDegrees(phi,Certify=>true)
96 ·--·used·0.292879s·(cpu);·0.0432214s·(thread);·0s·(gc)96 ·--·used·0.232918s·(cpu);·0.0344567s·(thread);·0s·(gc)
97 Certify:·output·certified!97 Certify:·output·certified!
  
98 o3·=·{1,·2,·4,·4,·2}98 o3·=·{1,·2,·4,·4,·2}
  
99 o3·:·List</code></pre>99 o3·:·List</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
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114 ·························0···5114 ·························0···5
115 o4·:·RingMap·------------------·&lt;--·GF·109561[t·..t·]115 o4·:·RingMap·------------------·&lt;--·GF·109561[t·..t·]
116 ·············x·x··-·x·x··+·x·x·················0···4116 ·············x·x··-·x·x··+·x·x·················0···4
117 ··············2·3····1·4····0·5</code></pre>117 ··············2·3····1·4····0·5</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees(psi,Certify=>true)120 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees(psi,Certify=>true)
121 ·--·used·0.230338s·(cpu);·0.0297309s·(thread);·0s·(gc)121 ·--·used·0.157863s·(cpu);·0.0409547s·(thread);·0s·(gc)
122 Certify:·output·certified!122 Certify:·output·certified!
  
123 o5·=·{2,·4,·4,·2,·1}123 o5·=·{2,·4,·4,·2,·1}
  
124 o5·:·List</code></pre>124 o5·:·List</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ········</table>126 ········</table>
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137 ···············ZZ·················ZZ137 ···············ZZ·················ZZ
138 o6·:·RingMap·------[x·..x·]·&lt;--·------[x·..x·]138 o6·:·RingMap·------[x·..x·]·&lt;--·------[x·..x·]
139 ·············300007··0···6······300007··0···6</code></pre>139 ·············300007··0···6······300007··0···6</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i7·:·time·projectiveDegrees·phi142 <td>··············<pre><code·class="language-macaulay2">i7·:·time·projectiveDegrees·phi
143 ·--·used·0.00292301s·(cpu);·3.7126e-05s·(thread);·0s·(gc)143 ·--·used·0.000623389s·(cpu);·4.3461e-05s·(thread);·0s·(gc)
  
144 o7·=·{1,·2,·4,·8,·8,·4,·1}144 o7·=·{1,·2,·4,·8,·8,·4,·1}
  
145 o7·:·List</code></pre>145 o7·:·List</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i8·:·time·projectiveDegrees(phi,NumDegrees=>1)148 <td>··············<pre><code·class="language-macaulay2">i8·:·time·projectiveDegrees(phi,NumDegrees=>1)
149 ·--·used·8.3025e-05s·(cpu);·1.4762e-05s·(thread);·0s·(gc)149 ·--·used·8.1052e-05s·(cpu);·1.8395e-05s·(thread);·0s·(gc)
  
150 o8·=·{4,·1}150 o8·=·{4,·1}
  
151 o8·:·List</code></pre>151 o8·:·List</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ········</table>153 ········</table>
154 ········<p>Another·way·to·use·this·method·is·by·passing·an·integer·<span·class="tt">i</span>·as·second·argument.·However,·this·is·equivalent·to·<span·class="tt">first·projectiveDegrees(phi,NumDegrees=>i)</span>·and·generally·it·is·not·faster.</p>154 ········<p>Another·way·to·use·this·method·is·by·passing·an·integer·<span·class="tt">i</span>·as·second·argument.·However,·this·is·equivalent·to·<span·class="tt">first·projectiveDegrees(phi,NumDegrees=>i)</span>·and·generally·it·is·not·faster.</p>
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53 t··+·t·t·,·-·t·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·a})53 t··+·t·t·,·-·t·t··+·t·t·,·-·t·t··+·t·t·,·-·t··+·t·t·,·a})
54 ·····················0···4··············0···5·······1····0·2·····1·2····0·354 ·····················0···4··············0···5·······1····0·2·····1·2····0·3
55 2····1·3·····1·3····0·4·····2·3····1·4·····3····2·455 2····1·3·····1·3····0·4·····2·3····1·4·····3····2·4
  
56 o2·:·RingMap·GF·109561[t·..t·]·<--·GF·109561[x·..x·]56 o2·:·RingMap·GF·109561[t·..t·]·<--·GF·109561[x·..x·]
57 ························0···4·················0···557 ························0···4·················0···5
58 i3·:·time·projectiveDegrees(phi,Certify=>true)58 i3·:·time·projectiveDegrees(phi,Certify=>true)
59 ·--·used·0.292879s·(cpu);·0.0432214s·(thread);·0s·(gc)59 ·--·used·0.232918s·(cpu);·0.0344567s·(thread);·0s·(gc)
60 Certify:·output·certified!60 Certify:·output·certified!
  
61 o3·=·{1,·2,·4,·4,·2}61 o3·=·{1,·2,·4,·4,·2}
  
62 o3·:·List62 o3·:·List
63 i4·:·psi=inverseMap(toMap(phi,Dominant=>infinity))63 i4·:·psi=inverseMap(toMap(phi,Dominant=>infinity))
  
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76 ··············GF·109561[x·..x·]76 ··············GF·109561[x·..x·]
77 ·························0···577 ·························0···5
78 o4·:·RingMap·------------------·<--·GF·109561[t·..t·]78 o4·:·RingMap·------------------·<--·GF·109561[t·..t·]
79 ·············x·x··-·x·x··+·x·x·················0···479 ·············x·x··-·x·x··+·x·x·················0···4
80 ··············2·3····1·4····0·580 ··············2·3····1·4····0·5
81 i5·:·time·projectiveDegrees(psi,Certify=>true)81 i5·:·time·projectiveDegrees(psi,Certify=>true)
82 ·--·used·0.230338s·(cpu);·0.0297309s·(thread);·0s·(gc)82 ·--·used·0.157863s·(cpu);·0.0409547s·(thread);·0s·(gc)
83 Certify:·output·certified!83 Certify:·output·certified!
  
84 o5·=·{2,·4,·4,·2,·1}84 o5·=·{2,·4,·4,·2,·1}
  
85 o5·:·List85 o5·:·List
86 i6·:·--·Cremona·transformation·of·P^6·defined·by·the·quadrics·through·a86 i6·:·--·Cremona·transformation·of·P^6·defined·by·the·quadrics·through·a
87 rational·octic·surface87 rational·octic·surface
Offset 120, 21 lines modifiedOffset 120, 21 lines modified
120 4·5·········5·········0·6·········1·6··········2·6··········3·6·········4·6120 4·5·········5·········0·6·········1·6··········2·6··········3·6·········4·6
121 5·6121 5·6
  
122 ···············ZZ·················ZZ122 ···············ZZ·················ZZ
123 o6·:·RingMap·------[x·..x·]·<--·------[x·..x·]123 o6·:·RingMap·------[x·..x·]·<--·------[x·..x·]
124 ·············300007··0···6······300007··0···6124 ·············300007··0···6······300007··0···6
125 i7·:·time·projectiveDegrees·phi125 i7·:·time·projectiveDegrees·phi
126 ·--·used·0.00292301s·(cpu);·3.7126e-05s·(thread);·0s·(gc)126 ·--·used·0.000623389s·(cpu);·4.3461e-05s·(thread);·0s·(gc)
  
127 o7·=·{1,·2,·4,·8,·8,·4,·1}127 o7·=·{1,·2,·4,·8,·8,·4,·1}
  
128 o7·:·List128 o7·:·List
129 i8·:·time·projectiveDegrees(phi,NumDegrees=>1)129 i8·:·time·projectiveDegrees(phi,NumDegrees=>1)
130 ·--·used·8.3025e-05s·(cpu);·1.4762e-05s·(thread);·0s·(gc)130 ·--·used·8.1052e-05s·(cpu);·1.8395e-05s·(thread);·0s·(gc)
  
131 o8·=·{4,·1}131 o8·=·{4,·1}
  
132 o8·:·List132 o8·:·List
133 Another·way·to·use·this·method·is·by·passing·an·integer·i·as·second·argument.133 Another·way·to·use·this·method·is·by·passing·an·integer·i·as·second·argument.
134 However,·this·is·equivalent·to·first·projectiveDegrees(phi,NumDegrees=>i)·and134 However,·this·is·equivalent·to·first·projectiveDegrees(phi,NumDegrees=>i)·and
135 generally·it·is·not·faster.135 generally·it·is·not·faster.
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90 ················ZZ90 ················ZZ
91 o2·:·Ideal·of·-----[x·..x·]91 o2·:·Ideal·of·-----[x·..x·]
92 ··············33331··0···6</code></pre>92 ··············33331··0···6</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·phi·=·rationalMap(V,3,2)95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·phi·=·rationalMap(V,3,2)
96 ·--·used·0.0835354s·(cpu);·0.0823515s·(thread);·0s·(gc)96 ·--·used·0.10052s·(cpu);·0.101724s·(thread);·0s·(gc)
  
97 o3·=·--·rational·map·--97 o3·=·--·rational·map·--
98 ····················ZZ98 ····················ZZ
99 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])99 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])
100 ··················33331··0···1···2···3···4···5···6100 ··················33331··0···1···2···3···4···5···6
101 ····················ZZ101 ····················ZZ
102 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,·y··])102 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,·y··])
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35 i1·:·ZZ/33331[x_0..x_6];·V·=·ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-35 i1·:·ZZ/33331[x_0..x_6];·V·=·ideal(x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-
36 x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);36 x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3,x_6);
  
37 ················ZZ37 ················ZZ
38 o2·:·Ideal·of·-----[x·..x·]38 o2·:·Ideal·of·-----[x·..x·]
39 ··············33331··0···639 ··············33331··0···6
40 i3·:·time·phi·=·rationalMap(V,3,2)40 i3·:·time·phi·=·rationalMap(V,3,2)
41 ·--·used·0.0835354s·(cpu);·0.0823515s·(thread);·0s·(gc)41 ·--·used·0.10052s·(cpu);·0.101724s·(thread);·0s·(gc)
  
42 o3·=·--·rational·map·--42 o3·=·--·rational·map·--
43 ····················ZZ43 ····················ZZ
44 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])44 ·····source:·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·,·x·])
45 ··················33331··0···1···2···3···4···5···645 ··················33331··0···1···2···3···4···5···6
46 ····················ZZ46 ····················ZZ
47 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,47 ·····target:·Proj(-----[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y··,·y··,·y··,
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104 o4·:·Ideal·of·X</code></pre>104 o4·:·Ideal·of·X</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i6·:·time·phi·=·rationalMap·D110 <td>··············<pre><code·class="language-macaulay2">i6·:·time·phi·=·rationalMap·D
111 ·--·used·0.0239816s·(cpu);·0.0242809s·(thread);·0s·(gc)111 ·--·used·0.0367502s·(cpu);·0.0370709s·(thread);·0s·(gc)
  
112 o6·=·--·rational·map·--112 o6·=·--·rational·map·--
113 ··································ZZ113 ··································ZZ
114 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by114 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by
115 ································65521··0···1···2···3···4···5115 ································65521··0···1···2···3···4···5
116 ·············{116 ·············{
117 ·················2··················2117 ·················2··················2
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210 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5210 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5
211 ·····················}211 ·····················}
  
212 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)</code></pre>212 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)</code></pre>
213 </td>··········</tr>213 </td>··········</tr>
214 ··········<tr>214 ··········<tr>
215 <td>··············<pre><code·class="language-macaulay2">i7·:·time·?·image(phi,&quot;F4&quot;)215 <td>··············<pre><code·class="language-macaulay2">i7·:·time·?·image(phi,&quot;F4&quot;)
216 ·--·used·0.674053s·(cpu);·0.496989s·(thread);·0s·(gc)216 ·--·used·1.0602s·(cpu);·0.502788s·(thread);·0s·(gc)
  
217 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153217 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153
218 ·····hypersurfaces·of·degree·2</code></pre>218 ·····hypersurfaces·of·degree·2</code></pre>
219 </td>··········</tr>219 </td>··········</tr>
220 ········</table>220 ········</table>
221 ········<p>See·also·the·package·<a·href="http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.16/share/doc/Macaulay2/Divisor/html/index.html">Divisor</a>,·which·provides·general·tools·for·working·with·divisors.</p>221 ········<p>See·also·the·package·<a·href="http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.16/share/doc/Macaulay2/Divisor/html/index.html">Divisor</a>,·which·provides·general·tools·for·working·with·divisors.</p>
222 ······</div>222 ······</div>
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41 o4·=·ideal(-·32646x··-·28377x··+·26433x··-·29566x··+·3783x··+·26696x·)41 o4·=·ideal(-·32646x··-·28377x··+·26433x··-·29566x··+·3783x··+·26696x·)
42 ···················0·········1·········2·········3········4·········542 ···················0·········1·········2·········3········4·········5
  
43 o4·:·Ideal·of·X43 o4·:·Ideal·of·X
44 i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};44 i5·:·D·=·new·Tally·from·{H·=>·2,C·=>·1};
45 i6·:·time·phi·=·rationalMap·D45 i6·:·time·phi·=·rationalMap·D
46 ·--·used·0.0239816s·(cpu);·0.0242809s·(thread);·0s·(gc)46 ·--·used·0.0367502s·(cpu);·0.0370709s·(thread);·0s·(gc)
  
47 o6·=·--·rational·map·--47 o6·=·--·rational·map·--
48 ··································ZZ48 ··································ZZ
49 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by49 ·····source:·subvariety·of·Proj(-----[x·,·x·,·x·,·x·,·x·,·x·])·defined·by
50 ································65521··0···1···2···3···4···550 ································65521··0···1···2···3···4···5
51 ·············{51 ·············{
52 ·················2··················252 ·················2··················2
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170 ··················································2·························2170 ··················································2·························2
171 ······················x·x·x··+·x·x·x··+·x·x·x··+·x·x··+·x·x·x··-·2x·x·x··+·x·x171 ······················x·x·x··+·x·x·x··+·x·x·x··+·x·x··+·x·x·x··-·2x·x·x··+·x·x
172 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5172 ·······················0·1·5····0·2·5····1·2·5····2·5····1·4·5·····2·4·5····4·5
173 ·····················}173 ·····················}
  
174 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)174 o6·:·RationalMap·(cubic·rational·map·from·surface·in·PP^5·to·PP^20)
175 i7·:·time·?·image(phi,"F4")175 i7·:·time·?·image(phi,"F4")
176 ·--·used·0.674053s·(cpu);·0.496989s·(thread);·0s·(gc)176 ·--·used·1.0602s·(cpu);·0.502788s·(thread);·0s·(gc)
  
177 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153177 o7·=·surface·of·degree·38·and·sectional·genus·20·in·PP^20·cut·out·by·153
178 ·····hypersurfaces·of·degree·2178 ·····hypersurfaces·of·degree·2
179 See·also·the·package·_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,·which·provides·general·tools·for·working·with179 See·also·the·package·_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,·which·provides·general·tools·for·working·with
180 divisors.180 divisors.
181 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*181 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
182 ····*·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·--·makes·a·rational·map182 ····*·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·--·makes·a·rational·map
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70 ······</div>70 ······</div>
71 ······<div>71 ······<div>
72 ········<h2>Description</h2>72 ········<h2>Description</h2>
73 ········<p>A·Cremona·transformation·is·said·to·be·special·if·the·base·locus·scheme·is·smooth·and·irreducible.·To·ensure·this·condition,·the·field·<span·class="tt">K</span>·must·be·large·enough·but·no·check·is·made.</p>73 ········<p>A·Cremona·transformation·is·said·to·be·special·if·the·base·locus·scheme·is·smooth·and·irreducible.·To·ensure·this·condition,·the·field·<span·class="tt">K</span>·must·be·large·enough·but·no·check·is·made.</p>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))
77 ·--·used·0.81958s·(cpu);·0.744604s·(thread);·0s·(gc)77 ·--·used·1.06038s·(cpu);·0.934817s·(thread);·0s·(gc)
  
78 o1·=·(rational·map·defined·by·forms·of·degree·3,78 o1·=·(rational·map·defined·by·forms·of·degree·3,
79 ······source·variety:·PP^3······················79 ······source·variety:·PP^3······················
80 ······target·variety:·PP^3······················80 ······target·variety:·PP^3······················
81 ······dominance:·true···························81 ······dominance:·true···························
82 ······birationality:·true·······················82 ······birationality:·true·······················
83 ······projective·degrees:·{1,·3,·3,·1}··········83 ······projective·degrees:·{1,·3,·3,·1}··········
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17 ············K,·according·to·the·classification·given·in·Table·1·of·_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l17 ············K,·according·to·the·classification·given·in·Table·1·of·_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l
18 ············_\x8c_\x8u_\x8b_\x8i_\x8c_\x8·_\x8C_\x8r_\x8e_\x8m_\x8o_\x8n_\x8a_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8P_\x86_\x8·_\x8a_\x8n_\x8d_\x8·_\x8P_\x87.18 ············_\x8c_\x8u_\x8b_\x8i_\x8c_\x8·_\x8C_\x8r_\x8e_\x8m_\x8o_\x8n_\x8a_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8P_\x86_\x8·_\x8a_\x8n_\x8d_\x8·_\x8P_\x87.
19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
20 A·Cremona·transformation·is·said·to·be·special·if·the·base·locus·scheme·is20 A·Cremona·transformation·is·said·to·be·special·if·the·base·locus·scheme·is
21 smooth·and·irreducible.·To·ensure·this·condition,·the·field·K·must·be·large21 smooth·and·irreducible.·To·ensure·this·condition,·the·field·K·must·be·large
22 enough·but·no·check·is·made.22 enough·but·no·check·is·made.
23 i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))23 i1·:·time·apply(1..12,i·->·describe·specialCremonaTransformation(i,ZZ/3331))
24 ·--·used·0.81958s·(cpu);·0.744604s·(thread);·0s·(gc)24 ·--·used·1.06038s·(cpu);·0.934817s·(thread);·0s·(gc)
  
25 o1·=·(rational·map·defined·by·forms·of·degree·3,25 o1·=·(rational·map·defined·by·forms·of·degree·3,
26 ······source·variety:·PP^326 ······source·variety:·PP^3
27 ······target·variety:·PP^327 ······target·variety:·PP^3
28 ······dominance:·true28 ······dominance:·true
29 ······birationality:·true29 ······birationality:·true
30 ······projective·degrees:·{1,·3,·3,·1}30 ······projective·degrees:·{1,·3,·3,·1}
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70 ······</div>70 ······</div>
71 ······<div>71 ······<div>
72 ········<h2>Description</h2>72 ········<h2>Description</h2>
73 ········<p>The·field·<span·class="tt">K</span>·is·required·to·be·large·enough.</p>73 ········<p>The·field·<span·class="tt">K</span>·is·required·to·be·large·enough.</p>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·specialCubicTransformation·976 <td>··············<pre><code·class="language-macaulay2">i1·:·time·specialCubicTransformation·9
77 ·--·used·0.0687208s·(cpu);·0.0696801s·(thread);·0s·(gc)77 ·--·used·0.101541s·(cpu);·0.10362s·(thread);·0s·(gc)
  
78 o1·=·--·rational·map·--78 o1·=·--·rational·map·--
79 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])79 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])
80 ······················0···1···2···3···4···5···680 ······················0···1···2···3···4···5···6
81 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])·defined·by81 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])·defined·by
82 ····································0···1···2···3···4···5···6···7···8···982 ····································0···1···2···3···4···5···6···7···8···9
83 ·············{83 ·············{
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136 ························0·1······0·1······1······0·2······0·1·2······1·2······0·2······1·2·····2·····0·3······0·1·3······1·3······0·2·3······1·2·3······2·3······0·3······1·3······2·3······3·····0·4·····0·1·4······1·4······0·2·4·····1·2·4·····2·4·····0·3·4······1·3·4······2·3·4······3·4·····0·4·····1·4·····2·4·····3·4····4······0·5······0·1·5······1·5·····0·2·5······1·2·5·····2·5······0·3·5······1·3·5······2·3·5······3·5······0·4·5·····1·4·5·····2·4·5·····3·4·5·····4·5·····0·5······1·5·····2·5······3·5····4·5·····5·····0·6·····0·1·6······1·6······0·2·6······1·2·6·····2·6······1·3·6······2·3·6······3·6····0·4·6·····1·4·6·····2·4·6·····3·4·6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6······1·6·····2·6······3·6·····4·6·····5·6136 ························0·1······0·1······1······0·2······0·1·2······1·2······0·2······1·2·····2·····0·3······0·1·3······1·3······0·2·3······1·2·3······2·3······0·3······1·3······2·3······3·····0·4·····0·1·4······1·4······0·2·4·····1·2·4·····2·4·····0·3·4······1·3·4······2·3·4······3·4·····0·4·····1·4·····2·4·····3·4····4······0·5······0·1·5······1·5·····0·2·5······1·2·5·····2·5······0·3·5······1·3·5······2·3·5······3·5······0·4·5·····1·4·5·····2·4·5·····3·4·5·····4·5·····0·5······1·5·····2·5······3·5····4·5·····5·····0·6·····0·1·6······1·6······0·2·6······1·2·6·····2·6······1·3·6······2·3·6······3·6····0·4·6·····1·4·6·····2·4·6·····3·4·6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6······1·6·····2·6······3·6·····4·6·····5·6
137 ·····················}137 ·····················}
  
138 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>138 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of·PP^9)</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·oo141 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·oo
142 ·--·used·0.0151069s·(cpu);·0.0154917s·(thread);·0s·(gc)142 ·--·used·0.0164238s·(cpu);·0.0179546s·(thread);·0s·(gc)
  
143 o2·=·rational·map·defined·by·forms·of·degree·3143 o2·=·rational·map·defined·by·forms·of·degree·3
144 ·····source·variety:·PP^6144 ·····source·variety:·PP^6
145 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^9145 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^9
146 ·····dominance:·true146 ·····dominance:·true
147 ·····birationality:·true147 ·····birationality:·true
148 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}148 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}
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16 ··········o·a·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p,·an·example·of·special·cubic·birational16 ··········o·a·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p,·an·example·of·special·cubic·birational
17 ············transformation·over·K,·according·to·the·classification·given·in17 ············transformation·over·K,·according·to·the·classification·given·in
18 ············Table·2·of·_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8·_\x8c_\x8u_\x8b_\x8i_\x8c_\x8·_\x8b_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e18 ············Table·2·of·_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8·_\x8c_\x8u_\x8b_\x8i_\x8c_\x8·_\x8b_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e
19 ············_\x8s_\x8p_\x8a_\x8c_\x8e_\x8s.19 ············_\x8s_\x8p_\x8a_\x8c_\x8e_\x8s.
20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
21 The·field·K·is·required·to·be·large·enough.21 The·field·K·is·required·to·be·large·enough.
22 i1·:·time·specialCubicTransformation·922 i1·:·time·specialCubicTransformation·9
23 ·--·used·0.0687208s·(cpu);·0.0696801s·(thread);·0s·(gc)23 ·--·used·0.101541s·(cpu);·0.10362s·(thread);·0s·(gc)
  
24 o1·=·--·rational·map·--24 o1·=·--·rational·map·--
25 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])25 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·])
26 ······················0···1···2···3···4···5···626 ······················0···1···2···3···4···5···6
27 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])27 ·····target:·subvariety·of·Proj(QQ[t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·,·t·])
28 defined·by28 defined·by
29 ····································0···1···2···3···4···5···6···7···8···929 ····································0···1···2···3···4···5···6···7···8···9
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324 6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6324 6·····4·6······0·5·6······1·5·6·····2·5·6······3·5·6······4·5·6·····5·6·····0·6
325 1·6·····2·6······3·6·····4·6·····5·6325 1·6·····2·6······3·6·····4·6·····5·6
326 ·····················}326 ·····················}
  
327 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of327 o1·:·RationalMap·(cubic·birational·map·from·PP^6·to·6-dimensional·subvariety·of
328 PP^9)328 PP^9)
329 i2·:·time·describe·oo329 i2·:·time·describe·oo
330 ·--·used·0.0151069s·(cpu);·0.0154917s·(thread);·0s·(gc)330 ·--·used·0.0164238s·(cpu);·0.0179546s·(thread);·0s·(gc)
  
331 o2·=·rational·map·defined·by·forms·of·degree·3331 o2·=·rational·map·defined·by·forms·of·degree·3
332 ·····source·variety:·PP^6332 ·····source·variety:·PP^6
333 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^9333 ·····target·variety:·complete·intersection·of·type·(2,2,2)·in·PP^9
334 ·····dominance:·true334 ·····dominance:·true
335 ·····birationality:·true335 ·····birationality:·true
336 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}336 ·····projective·degrees:·{1,·3,·9,·17,·21,·16,·8}
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70 ······</div>70 ······</div>
71 ······<div>71 ······<div>
72 ········<h2>Description</h2>72 ········<h2>Description</h2>
73 ········<p>The·field·<span·class="tt">K</span>·is·required·to·be·large·enough.</p>73 ········<p>The·field·<span·class="tt">K</span>·is·required·to·be·large·enough.</p>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·specialQuadraticTransformation·476 <td>··············<pre><code·class="language-macaulay2">i1·:·time·specialQuadraticTransformation·4
77 ·--·used·0.050462s·(cpu);·0.0541456s·(thread);·0s·(gc)77 ·--·used·0.0613429s·(cpu);·0.0584016s·(thread);·0s·(gc)
  
78 o1·=·--·rational·map·--78 o1·=·--·rational·map·--
79 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])79 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
80 ······················0···1···2···3···4···5···6···7···880 ······················0···1···2···3···4···5···6···7···8
81 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])·defined·by81 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])·defined·by
82 ····································0···1···2···3···4···5···6···7···8···982 ····································0···1···2···3···4···5···6···7···8···9
83 ·············{83 ·············{
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124 ·······················0·1····0·4····3·6····4·6····6····5·7124 ·······················0·1····0·4····3·6····4·6····6····5·7
125 ·····················}125 ·····················}
  
126 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>126 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)</code></pre>
127 </td>··········</tr>127 </td>··········</tr>
128 ··········<tr>128 ··········<tr>
129 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·oo129 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·oo
130 ·--·used·0.00522602s·(cpu);·0.00520613s·(thread);·0s·(gc)130 ·--·used·0.00841658s·(cpu);·0.00841796s·(thread);·0s·(gc)
  
131 o2·=·rational·map·defined·by·forms·of·degree·2131 o2·=·rational·map·defined·by·forms·of·degree·2
132 ·····source·variety:·PP^8132 ·····source·variety:·PP^8
133 ·····target·variety:·hypersurface·of·degree·3·in·PP^9133 ·····target·variety:·hypersurface·of·degree·3·in·PP^9
134 ·····dominance:·true134 ·····dominance:·true
135 ·····birationality:·true135 ·····birationality:·true
136 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}136 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}
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16 ··········o·a·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p,·an·example·of·special·quadratic·birational16 ··········o·a·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p,·an·example·of·special·quadratic·birational
17 ············transformation·over·K,·according·to·the·classification·given·in17 ············transformation·over·K,·according·to·the·classification·given·in
18 ············Table·1·of·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s_\x8·_\x8o_\x8f_\x8·_\x8s_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8·_\x8q_\x8u_\x8a_\x8d_\x8r_\x8a_\x8t_\x8i_\x8c_\x8·_\x8b_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s18 ············Table·1·of·_\x8E_\x8x_\x8a_\x8m_\x8p_\x8l_\x8e_\x8s_\x8·_\x8o_\x8f_\x8·_\x8s_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8·_\x8q_\x8u_\x8a_\x8d_\x8r_\x8a_\x8t_\x8i_\x8c_\x8·_\x8b_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8t_\x8r_\x8a_\x8n_\x8s_\x8f_\x8o_\x8r_\x8m_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s
19 ············_\x8i_\x8n_\x8t_\x8o_\x8·_\x8c_\x8o_\x8m_\x8p_\x8l_\x8e_\x8t_\x8e_\x8·_\x8i_\x8n_\x8t_\x8e_\x8r_\x8s_\x8e_\x8c_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8q_\x8u_\x8a_\x8d_\x8r_\x8i_\x8c_\x8s.19 ············_\x8i_\x8n_\x8t_\x8o_\x8·_\x8c_\x8o_\x8m_\x8p_\x8l_\x8e_\x8t_\x8e_\x8·_\x8i_\x8n_\x8t_\x8e_\x8r_\x8s_\x8e_\x8c_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8o_\x8f_\x8·_\x8q_\x8u_\x8a_\x8d_\x8r_\x8i_\x8c_\x8s.
20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
21 The·field·K·is·required·to·be·large·enough.21 The·field·K·is·required·to·be·large·enough.
22 i1·:·time·specialQuadraticTransformation·422 i1·:·time·specialQuadraticTransformation·4
23 ·--·used·0.050462s·(cpu);·0.0541456s·(thread);·0s·(gc)23 ·--·used·0.0613429s·(cpu);·0.0584016s·(thread);·0s·(gc)
  
24 o1·=·--·rational·map·--24 o1·=·--·rational·map·--
25 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])25 ·····source:·Proj(QQ[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
26 ······················0···1···2···3···4···5···6···7···826 ······················0···1···2···3···4···5···6···7···8
27 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])27 ·····target:·subvariety·of·Proj(QQ[y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·,·y·])
28 defined·by28 defined·by
29 ····································0···1···2···3···4···5···6···7···8···929 ····································0···1···2···3···4···5···6···7···8···9
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79 ···················································279 ···················································2
80 ······················x·x··-·x·x··+·x·x··-·x·x··-·x··-·x·x80 ······················x·x··-·x·x··+·x·x··-·x·x··-·x··-·x·x
81 ·······················0·1····0·4····3·6····4·6····6····5·781 ·······················0·1····0·4····3·6····4·6····6····5·7
82 ·····················}82 ·····················}
  
83 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)83 o1·:·RationalMap·(quadratic·birational·map·from·PP^8·to·hypersurface·in·PP^9)
84 i2·:·time·describe·oo84 i2·:·time·describe·oo
85 ·--·used·0.00522602s·(cpu);·0.00520613s·(thread);·0s·(gc)85 ·--·used·0.00841658s·(cpu);·0.00841796s·(thread);·0s·(gc)
  
86 o2·=·rational·map·defined·by·forms·of·degree·286 o2·=·rational·map·defined·by·forms·of·degree·2
87 ·····source·variety:·PP^887 ·····source·variety:·PP^8
88 ·····target·variety:·hypersurface·of·degree·3·in·PP^988 ·····target·variety:·hypersurface·of·degree·3·in·PP^9
89 ·····dominance:·true89 ·····dominance:·true
90 ·····birationality:·true90 ·····birationality:·true
91 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}91 ·····projective·degrees:·{1,·2,·4,·8,·16,·21,·17,·9,·3}
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82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·#str83 <td>··············<pre><code·class="language-macaulay2">i3·:·#str
  
84 o3·=·6927</code></pre>84 o3·=·6927</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·time·phi'·=·value·str;87 <td>··············<pre><code·class="language-macaulay2">i4·:·time·phi'·=·value·str;
88 ·--·used·0.0780309s·(cpu);·0.0363949s·(thread);·0s·(gc)88 ·--·used·0.0931018s·(cpu);·0.0443018s·(thread);·0s·(gc)
  
89 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)</code></pre>89 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i5·:·time·describe·phi'92 <td>··············<pre><code·class="language-macaulay2">i5·:·time·describe·phi'
93 ·--·used·0.00119947s·(cpu);·0.0043174s·(thread);·0s·(gc)93 ·--·used·0.00481386s·(cpu);·0.00571627s·(thread);·0s·(gc)
  
94 o5·=·rational·map·defined·by·forms·of·degree·394 o5·=·rational·map·defined·by·forms·of·degree·3
95 ·····source·variety:·PP^395 ·····source·variety:·PP^3
96 ·····target·variety:·smooth·quadric·hypersurface·in·PP^496 ·····target·variety:·smooth·quadric·hypersurface·in·PP^4
97 ·····dominance:·true97 ·····dominance:·true
98 ·····birationality:·true·(the·inverse·map·is·already·calculated)98 ·····birationality:·true·(the·inverse·map·is·already·calculated)
99 ·····projective·degrees:·{1,·3,·4,·2}99 ·····projective·degrees:·{1,·3,·4,·2}
100 ·····number·of·minimal·representatives:·1100 ·····number·of·minimal·representatives:·1
101 ·····dimension·base·locus:·1101 ·····dimension·base·locus:·1
102 ·····degree·base·locus:·5102 ·····degree·base·locus:·5
103 ·····coefficient·ring:·ZZ/33331</code></pre>103 ·····coefficient·ring:·ZZ/33331</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·inverse·phi'106 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·inverse·phi'
107 ·--·used·0.000593353s·(cpu);·0.00374519s·(thread);·0s·(gc)107 ·--·used·0.00607502s·(cpu);·0.00601148s·(thread);·0s·(gc)
  
108 o6·=·rational·map·defined·by·forms·of·degree·2108 o6·=·rational·map·defined·by·forms·of·degree·2
109 ·····source·variety:·smooth·quadric·hypersurface·in·PP^4109 ·····source·variety:·smooth·quadric·hypersurface·in·PP^4
110 ·····target·variety:·PP^3110 ·····target·variety:·PP^3
111 ·····dominance:·true111 ·····dominance:·true
112 ·····birationality:·true·(the·inverse·map·is·already·calculated)112 ·····birationality:·true·(the·inverse·map·is·already·calculated)
113 ·····projective·degrees:·{2,·4,·3,·1}113 ·····projective·degrees:·{2,·4,·3,·1}
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20 o1·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)20 o1·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)
21 i2·:·str·=·toExternalString·phi;21 i2·:·str·=·toExternalString·phi;
22 i3·:·#str22 i3·:·#str
  
23 o3·=·692723 o3·=·6927
24 i4·:·time·phi'·=·value·str;24 i4·:·time·phi'·=·value·str;
25 ·--·used·0.0780309s·(cpu);·0.0363949s·(thread);·0s·(gc)25 ·--·used·0.0931018s·(cpu);·0.0443018s·(thread);·0s·(gc)
  
26 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)26 o4·:·RationalMap·(cubic·birational·map·from·PP^3·to·hypersurface·in·PP^4)
27 i5·:·time·describe·phi'27 i5·:·time·describe·phi'
28 ·--·used·0.00119947s·(cpu);·0.0043174s·(thread);·0s·(gc)28 ·--·used·0.00481386s·(cpu);·0.00571627s·(thread);·0s·(gc)
  
29 o5·=·rational·map·defined·by·forms·of·degree·329 o5·=·rational·map·defined·by·forms·of·degree·3
30 ·····source·variety:·PP^330 ·····source·variety:·PP^3
31 ·····target·variety:·smooth·quadric·hypersurface·in·PP^431 ·····target·variety:·smooth·quadric·hypersurface·in·PP^4
32 ·····dominance:·true32 ·····dominance:·true
33 ·····birationality:·true·(the·inverse·map·is·already·calculated)33 ·····birationality:·true·(the·inverse·map·is·already·calculated)
34 ·····projective·degrees:·{1,·3,·4,·2}34 ·····projective·degrees:·{1,·3,·4,·2}
35 ·····number·of·minimal·representatives:·135 ·····number·of·minimal·representatives:·1
36 ·····dimension·base·locus:·136 ·····dimension·base·locus:·1
37 ·····degree·base·locus:·537 ·····degree·base·locus:·5
38 ·····coefficient·ring:·ZZ/3333138 ·····coefficient·ring:·ZZ/33331
39 i6·:·time·describe·inverse·phi'39 i6·:·time·describe·inverse·phi'
40 ·--·used·0.000593353s·(cpu);·0.00374519s·(thread);·0s·(gc)40 ·--·used·0.00607502s·(cpu);·0.00601148s·(thread);·0s·(gc)
  
41 o6·=·rational·map·defined·by·forms·of·degree·241 o6·=·rational·map·defined·by·forms·of·degree·2
42 ·····source·variety:·smooth·quadric·hypersurface·in·PP^442 ·····source·variety:·smooth·quadric·hypersurface·in·PP^4
43 ·····target·variety:·PP^343 ·····target·variety:·PP^3
44 ·····dominance:·true44 ·····dominance:·true
45 ·····birationality:·true·(the·inverse·map·is·already·calculated)45 ·····birationality:·true·(the·inverse·map·is·already·calculated)
46 ·····projective·degrees:·{2,·4,·3,·1}46 ·····projective·degrees:·{2,·4,·3,·1}
⊟
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48 ········<p>Below·is·an·example·using·the·methods·provided·by·this·package,·dealing·with·a·birational·transformation·$\Phi:\mathbb{P}^6·\dashrightarrow·\mathbb{G}(2,4)\subset\mathbb{P}^9$·of·bidegree·$(3,3)$.</p>48 ········<p>Below·is·an·example·using·the·methods·provided·by·this·package,·dealing·with·a·birational·transformation·$\Phi:\mathbb{P}^6·\dashrightarrow·\mathbb{G}(2,4)\subset\mathbb{P}^9$·of·bidegree·$(3,3)$.</p>
49 ········<table·class="examples">49 ········<table·class="examples">
50 ··········<tr>50 ··········<tr>
51 <td>··············<pre><code·class="language-macaulay2">i1·:·ZZ/300007[t_0..t_6];</code></pre>51 <td>··············<pre><code·class="language-macaulay2">i1·:·ZZ/300007[t_0..t_6];</code></pre>
52 </td>··········</tr>52 </td>··········</tr>
53 ··········<tr>53 ··········<tr>
54 <td>··············<pre><code·class="language-macaulay2">i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})54 <td>··············<pre><code·class="language-macaulay2">i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
55 ·--·used·0.00403256s·(cpu);·0.0041249s·(thread);·0s·(gc)55 ·--·used·0.00400105s·(cpu);·0.0034921s·(thread);·0s·(gc)
  
56 ············ZZ··············ZZ················3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····256 ············ZZ··············ZZ················3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····2
57 o2·=·map·(------[t·..t·],·------[x·..x·],·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})57 o2·=·map·(------[t·..t·],·------[x·..x·],·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})
58 ··········300007··0···6···300007··0···9·······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·658 ··········300007··0···6···300007··0···9·······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·6
  
59 ···············ZZ·················ZZ59 ···············ZZ·················ZZ
60 o2·:·RingMap·------[t·..t·]·&lt;--·------[x·..x·]60 o2·:·RingMap·------[t·..t·]·&lt;--·------[x·..x·]
61 ·············300007··0···6······300007··0···9</code></pre>61 ·············300007··0···6······300007··0···9</code></pre>
62 </td>··········</tr>62 </td>··········</tr>
63 ··········<tr>63 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·J·=·kernel(phi,2)64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·J·=·kernel(phi,2)
65 ·--·used·0.0398989s·(cpu);·0.0391807s·(thread);·0s·(gc)65 ·--·used·0.0383873s·(cpu);·0.0398905s·(thread);·0s·(gc)
  
66 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x·66 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x·
67 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·467 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4
68 ·····------------------------------------------------------------------------68 ·····------------------------------------------------------------------------
69 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)69 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
70 ········1·6····0·8···2·4····1·5····0·770 ········1·6····0·8···2·4····1·5····0·7
  
71 ················ZZ71 ················ZZ
72 o3·:·Ideal·of·------[x·..x·]72 o3·:·Ideal·of·------[x·..x·]
73 ··············300007··0···9</code></pre>73 ··············300007··0···9</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degreeMap·phi76 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degreeMap·phi
77 ·--·used·0.0909701s·(cpu);·0.0436168s·(thread);·0s·(gc)77 ·--·used·0.0857462s·(cpu);·0.0404878s·(thread);·0s·(gc)
  
78 o4·=·1</code></pre>78 o4·=·1</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees·phi81 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projectiveDegrees·phi
82 ·--·used·0.34797s·(cpu);·0.305061s·(thread);·0s·(gc)82 ·--·used·0.459454s·(cpu);·0.3758s·(thread);·0s·(gc)
  
83 o5·=·{1,·3,·9,·17,·21,·15,·5}83 o5·=·{1,·3,·9,·17,·21,·15,·5}
  
84 o5·:·List</code></pre>84 o5·:·List</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i6·:·time·projectiveDegrees(phi,NumDegrees=>0)87 <td>··············<pre><code·class="language-macaulay2">i6·:·time·projectiveDegrees(phi,NumDegrees=>0)
88 ·--·used·0.0521797s·(cpu);·0.0526681s·(thread);·0s·(gc)88 ·--·used·0.0645226s·(cpu);·0.0651927s·(thread);·0s·(gc)
  
89 o6·=·{5}89 o6·=·{5}
  
90 o6·:·List</code></pre>90 o6·:·List</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i7·:·time·phi·=·toMap(phi,Dominant=>J)93 <td>··············<pre><code·class="language-macaulay2">i7·:·time·phi·=·toMap(phi,Dominant=>J)
94 ·--·used·0.000225799s·(cpu);·0.00189703s·(thread);·0s·(gc)94 ·--·used·0.000414056s·(cpu);·0.00209904s·(thread);·0s·(gc)
  
95 ·······································································ZZ95 ·······································································ZZ
96 ·····································································------[x·..x·]96 ·····································································------[x·..x·]
97 ············ZZ·······················································300007··0···9··················································3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····297 ············ZZ·······················································300007··0···9··················································3················2····2················2········2······················2··················2····2·················2·······················3················2····2················2·································2···························2····2··································2········2······················2··················2························2·························2····2·················2·······················3················2····2
98 o7·=·map·(------[t·..t·],·----------------------------------------------------------------------------------------------------,·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})98 o7·=·map·(------[t·..t·],·----------------------------------------------------------------------------------------------------,·{-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t·t··+·t·t··+·t·t··-·t·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t·t··-·t·t··+·t·t·t·,·-·t·t··+·t·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t·t··+·t·t··+·t·t·t··-·t·t··-·t·t·t··+·t·t·t·,·-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t·})
99 ··········300007··0···6···(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·699 ··········300007··0···6···(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)······2·····1·2·3····0·3····1·4····0·2·4·····2·3····1·3····1·2·4····0·3·4····1·5····0·2·5·····2·3····2·4····1·3·4····0·4····1·2·5····0·3·5·····3·····2·3·4····1·4····2·5····1·3·5·····2·4····1·3·4····1·2·5····0·3·5····1·6····0·2·6·····2·3·4····1·4····2·5····0·4·5····1·2·6····0·3·6·····3·4····2·4····2·3·5····1·4·5····2·6····1·3·6·····2·4····2·3·5····1·4·5····0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4·····3·4·5····2·5····3·6····2·4·6
100 ····························6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7100 ····························6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
Offset 114, 15 lines modifiedOffset 114, 15 lines modified
114 ···············ZZ··························································300007··0···9114 ···············ZZ··························································300007··0···9
115 o7·:·RingMap·------[t·..t·]·&lt;--·----------------------------------------------------------------------------------------------------115 o7·:·RingMap·------[t·..t·]·&lt;--·----------------------------------------------------------------------------------------------------
116 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)116 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
117 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>117 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i8·:·time·psi·=·inverseMap·phi120 <td>··············<pre><code·class="language-macaulay2">i8·:·time·psi·=·inverseMap·phi
121 ·--·used·0.468032s·(cpu);·0.425579s·(thread);·0s·(gc)121 ·--·used·0.439703s·(cpu);·0.39935s·(thread);·0s·(gc)
  
122 ·······················································ZZ122 ·······················································ZZ
123 ·····················································------[x·..x·]123 ·····················································------[x·..x·]
124 ·····················································300007··0···9················································ZZ··············3················2···············2····2························2··························2·····2········2·······························2···································2···············2·············2·······················3·················································2·················2····2··································2····2·················2·················································3·························2······2····2······2··············································2124 ·····················································300007··0···9················································ZZ··············3················2···············2····2························2··························2·····2········2·······························2···································2···············2·············2·······················3·················································2·················2····2··································2····2·················2·················································3·························2······2····2······2··············································2
125 o8·=·map·(----------------------------------------------------------------------------------------------------,·------[t·..t·],·{x··-·2x·x·x··+·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x·x··-·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··-·2x·x·,·x·x··-·x·x··-·x·x·x··+·x·x·x··+·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x·x··-·x·x·x··+·x·x··-·x·x·x··+·x·x··-·x·x·x··-·x·x·x·,·x··-·x·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x·,·x·x··-·x·x·x··+·x·x··+·x·x··-·x·x·x··-·x·x·x··-·x·x·x·,·x·x··-·x·x··-·x·x·x··+·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x··-·2x·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··-·x·x·x··-·2x·x·})125 o8·=·map·(----------------------------------------------------------------------------------------------------,·------[t·..t·],·{x··-·2x·x·x··+·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x·x··-·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··-·2x·x·,·x·x··-·x·x··-·x·x·x··+·x·x·x··+·x·x·x··+·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x·x··-·x·x·x··+·x·x··-·x·x·x··+·x·x··-·x·x·x··-·x·x·x·,·x··-·x·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x·,·x·x··-·x·x·x··+·x·x··+·x·x··-·x·x·x··-·x·x·x··-·x·x·x·,·x·x··-·x·x··-·x·x·x··+·x·x··+·x·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··+·x·x·x·,·x··-·2x·x·x··-·x·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·x··-·2x·x·x··-·x·x·x··-·x·x·x··-·2x·x·})
126 ··········(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)··300007··0···6·····2·····1·2·3····0·3····1·2·5····0·5····1·6····0·2·6····0·4·6····1·7·····0·2·7····0·4·7·····0·9···2·3····1·3····1·2·6····0·3·6····0·5·6····1·8·····0·2·8····0·4·8····0·1·9···2·3····1·3·6····0·6····0·3·8····1·9····0·2·9····0·4·9···3····1·3·8····0·6·8····1·2·9·····0·3·9····0·5·9···3·6····2·3·8····0·8····2·9····1·3·9····0·6·9····0·7·9···3·6····3·8····2·6·8····1·8····2·3·9····2·5·9·····1·6·9····1·7·9····0·8·9···6·····3·6·8····5·6·8····2·8····4·8····3·9····5·9····2·6·9·····4·6·9····2·7·9····4·7·9·····0·9126 ··········(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)··300007··0···6·····2·····1·2·3····0·3····1·2·5····0·5····1·6····0·2·6····0·4·6····1·7·····0·2·7····0·4·7·····0·9···2·3····1·3····1·2·6····0·3·6····0·5·6····1·8·····0·2·8····0·4·8····0·1·9···2·3····1·3·6····0·6····0·3·8····1·9····0·2·9····0·4·9···3····1·3·8····0·6·8····1·2·9·····0·3·9····0·5·9···3·6····2·3·8····0·8····2·9····1·3·9····0·6·9····0·7·9···3·6····3·8····2·6·8····1·8····2·3·9····2·5·9·····1·6·9····1·7·9····0·8·9···6·····3·6·8····5·6·8····2·8····4·8····3·9····5·9····2·6·9·····4·6·9····2·7·9····4·7·9·····0·9
127 ············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7127 ············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
Offset 132, 38 lines modifiedOffset 132, 38 lines modified
132 ························································300007··0···9···················································ZZ132 ························································300007··0···9···················································ZZ
133 o8·:·RingMap·----------------------------------------------------------------------------------------------------·&lt;--·------[t·..t·]133 o8·:·RingMap·----------------------------------------------------------------------------------------------------·&lt;--·------[t·..t·]
134 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···6134 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···6
135 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>135 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7</code></pre>
136 </td>··········</tr>136 </td>··········</tr>
137 ··········<tr>137 ··········<tr>
138 <td>··············<pre><code·class="language-macaulay2">i9·:·time·isInverseMap(phi,psi)138 <td>··············<pre><code·class="language-macaulay2">i9·:·time·isInverseMap(phi,psi)
139 ·--·used·0.00801057s·(cpu);·0.00771425s·(thread);·0s·(gc)139 ·--·used·0.00783973s·(cpu);·0.0072009s·(thread);·0s·(gc)
  
140 o9·=·true</code></pre>140 o9·=·true</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i10·:·time·degreeMap·psi143 <td>··············<pre><code·class="language-macaulay2">i10·:·time·degreeMap·psi
144 ·--·used·0.201606s·(cpu);·0.15919s·(thread);·0s·(gc)144 ·--·used·0.234928s·(cpu);·0.182881s·(thread);·0s·(gc)
  
145 o10·=·1</code></pre>145 o10·=·1</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i11·:·time·projectiveDegrees·psi148 <td>··············<pre><code·class="language-macaulay2">i11·:·time·projectiveDegrees·psi
149 ·--·used·5.09383s·(cpu);·4.83436s·(thread);·0s·(gc)149 ·--·used·5.06976s·(cpu);·4.68993s·(thread);·0s·(gc)
  
150 o11·=·{5,·15,·21,·17,·9,·3,·1}150 o11·=·{5,·15,·21,·17,·9,·3,·1}
  
151 o11·:·List</code></pre>151 o11·:·List</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ········</table>153 ········</table>
154 ········<p>We·repeat·the·example·using·the·type·<a·title="the·class·of·all·rational·maps·between·absolutely·irreducible·projective·varieties·over·a·field"·href="___Rational__Map.html">RationalMap</a>·and·using·deterministic·methods.</p>154 ········<p>We·repeat·the·example·using·the·type·<a·title="the·class·of·all·rational·maps·between·absolutely·irreducible·projective·varieties·over·a·field"·href="___Rational__Map.html">RationalMap</a>·and·using·deterministic·methods.</p>
155 ········<table·class="examples">155 ········<table·class="examples">
156 ··········<tr>156 ··········<tr>
157 <td>··············<pre><code·class="language-macaulay2">i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})157 <td>··············<pre><code·class="language-macaulay2">i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
158 ·--·used·0.000721897s·(cpu);·0.00192268s·(thread);·0s·(gc)158 ·--·used·0.00180521s·(cpu);·0.00182693s·(thread);·0s·(gc)
  
159 o12·=·--·rational·map·--159 o12·=·--·rational·map·--
160 ·····················ZZ160 ·····················ZZ
161 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])161 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
162 ···················300007··0···1···2···3···4···5···6162 ···················300007··0···1···2···3···4···5···6
163 ·····················ZZ163 ·····················ZZ
164 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])164 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 210, 15 lines modifiedOffset 210, 15 lines modified
210 ··························4·····3·4·5····2·5····3·6····2·4·6210 ··························4·····3·4·5····2·5····3·6····2·4·6
211 ······················}211 ······················}
  
212 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)</code></pre>212 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)</code></pre>
213 </td>··········</tr>213 </td>··········</tr>
214 ··········<tr>214 ··········<tr>
215 <td>··············<pre><code·class="language-macaulay2">i13·:·time·phi·=·rationalMap(phi,Dominant=>2)215 <td>··············<pre><code·class="language-macaulay2">i13·:·time·phi·=·rationalMap(phi,Dominant=>2)
216 ·--·used·0.108803s·(cpu);·0.0695419s·(thread);·0s·(gc)216 ·--·used·0.10811s·(cpu);·0.068337s·(thread);·0s·(gc)
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25 map)·from·a·list·of·$m+1$·homogeneous·elements·of·the·same·degree·in·$K25 map)·from·a·list·of·$m+1$·homogeneous·elements·of·the·same·degree·in·$K
26 [x_0,...,x_n]/I$.26 [x_0,...,x_n]/I$.
27 Below·is·an·example·using·the·methods·provided·by·this·package,·dealing·with·a27 Below·is·an·example·using·the·methods·provided·by·this·package,·dealing·with·a
28 birational·transformation·$\Phi:\mathbb{P}^6·\dashrightarrow·\mathbb{G}28 birational·transformation·$\Phi:\mathbb{P}^6·\dashrightarrow·\mathbb{G}
29 (2,4)\subset\mathbb{P}^9$·of·bidegree·$(3,3)$.29 (2,4)\subset\mathbb{P}^9$·of·bidegree·$(3,3)$.
30 i1·:·ZZ/300007[t_0..t_6];30 i1·:·ZZ/300007[t_0..t_6];
31 i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})31 i2·:·time·phi·=·toMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
32 ·--·used·0.00403256s·(cpu);·0.0041249s·(thread);·0s·(gc)32 ·--·used·0.00400105s·(cpu);·0.0034921s·(thread);·0s·(gc)
  
33 ············ZZ··············ZZ················3················2····233 ············ZZ··············ZZ················3················2····2
34 2········2······················2··················2····2·················234 2········2······················2··················2····2·················2
35 3················2····2················2·································235 3················2····2················2·································2
36 2····2··································2········2······················236 2····2··································2········2······················2
37 2························2·························2····2·················237 2························2·························2····2·················2
38 3················2····238 3················2····2
Offset 52, 43 lines modifiedOffset 52, 43 lines modified
52 0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····452 0·5····1·3·6····0·4·6·····3·4····3·5····2·4·5····1·5····2·3·6····1·4·6·····4
53 3·4·5····2·5····3·6····2·4·653 3·4·5····2·5····3·6····2·4·6
  
54 ···············ZZ·················ZZ54 ···············ZZ·················ZZ
55 o2·:·RingMap·------[t·..t·]·<--·------[x·..x·]55 o2·:·RingMap·------[t·..t·]·<--·------[x·..x·]
56 ·············300007··0···6······300007··0···956 ·············300007··0···6······300007··0···9
57 i3·:·time·J·=·kernel(phi,2)57 i3·:·time·J·=·kernel(phi,2)
58 ·--·used·0.0398989s·(cpu);·0.0391807s·(thread);·0s·(gc)58 ·--·used·0.0383873s·(cpu);·0.0398905s·(thread);·0s·(gc)
  
59 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x59 o3·=·ideal·(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x
60 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·460 ·············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4
61 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
62 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)62 ·····-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
63 ········1·6····0·8···2·4····1·5····0·763 ········1·6····0·8···2·4····1·5····0·7
  
64 ················ZZ64 ················ZZ
65 o3·:·Ideal·of·------[x·..x·]65 o3·:·Ideal·of·------[x·..x·]
66 ··············300007··0···966 ··············300007··0···9
67 i4·:·time·degreeMap·phi67 i4·:·time·degreeMap·phi
68 ·--·used·0.0909701s·(cpu);·0.0436168s·(thread);·0s·(gc)68 ·--·used·0.0857462s·(cpu);·0.0404878s·(thread);·0s·(gc)
  
69 o4·=·169 o4·=·1
70 i5·:·time·projectiveDegrees·phi70 i5·:·time·projectiveDegrees·phi
71 ·--·used·0.34797s·(cpu);·0.305061s·(thread);·0s·(gc)71 ·--·used·0.459454s·(cpu);·0.3758s·(thread);·0s·(gc)
  
72 o5·=·{1,·3,·9,·17,·21,·15,·5}72 o5·=·{1,·3,·9,·17,·21,·15,·5}
  
73 o5·:·List73 o5·:·List
74 i6·:·time·projectiveDegrees(phi,NumDegrees=>0)74 i6·:·time·projectiveDegrees(phi,NumDegrees=>0)
75 ·--·used·0.0521797s·(cpu);·0.0526681s·(thread);·0s·(gc)75 ·--·used·0.0645226s·(cpu);·0.0651927s·(thread);·0s·(gc)
  
76 o6·=·{5}76 o6·=·{5}
  
77 o6·:·List77 o6·:·List
78 i7·:·time·phi·=·toMap(phi,Dominant=>J)78 i7·:·time·phi·=·toMap(phi,Dominant=>J)
79 ·--·used·0.000225799s·(cpu);·0.00189703s·(thread);·0s·(gc)79 ·--·used·0.000414056s·(cpu);·0.00209904s·(thread);·0s·(gc)
  
80 ·······································································ZZ80 ·······································································ZZ
81 ·····································································------[x81 ·····································································------[x
82 ..x·]82 ..x·]
83 ············ZZ·······················································300007··083 ············ZZ·······················································300007··0
84 9··················································3················2····284 9··················································3················2····2
85 2········2······················2··················2····2·················285 2········2······················2··················2····2·················2
Offset 123, 15 lines modifiedOffset 123, 15 lines modified
123 o7·:·RingMap·------[t·..t·]·<--·-----------------------------------------------123 o7·:·RingMap·------[t·..t·]·<--·-----------------------------------------------
124 -----------------------------------------------------124 -----------------------------------------------------
125 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-125 ·············300007··0···6······(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-
126 x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)126 x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)
127 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5127 ··································6·7····5·8····4·9···3·7····2·8····1·9···3·5
128 2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7128 2·6····0·9···3·4····1·6····0·8···2·4····1·5····0·7
129 i8·:·time·psi·=·inverseMap·phi129 i8·:·time·psi·=·inverseMap·phi
130 ·--·used·0.468032s·(cpu);·0.425579s·(thread);·0s·(gc)130 ·--·used·0.439703s·(cpu);·0.39935s·(thread);·0s·(gc)
  
131 ·······················································ZZ131 ·······················································ZZ
132 ·····················································------[x·..x·]132 ·····················································------[x·..x·]
133 ·····················································300007··0···9133 ·····················································300007··0···9
134 ZZ··············3················2···············2····2134 ZZ··············3················2···············2····2
135 2··························2·····2········2·······························2135 2··························2·····2········2·······························2
136 2···············2·············2·······················3136 2···············2·············2·······················3
Offset 164, 31 lines modifiedOffset 164, 31 lines modified
164 o8·:·RingMap·------------------------------------------------------------------164 o8·:·RingMap·------------------------------------------------------------------
165 ----------------------------------·<--·------[t·..t·]165 ----------------------------------·<--·------[t·..t·]
166 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x166 ·············(x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x··-·x·x··+·x·x·,·x·x
167 -·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···6167 -·x·x··+·x·x·,·x·x··-·x·x··+·x·x·)·····300007··0···6
168 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4168 ···············6·7····5·8····4·9···3·7····2·8····1·9···3·5····2·6····0·9···3·4
169 1·6····0·8···2·4····1·5····0·7169 1·6····0·8···2·4····1·5····0·7
170 i9·:·time·isInverseMap(phi,psi)170 i9·:·time·isInverseMap(phi,psi)
171 ·--·used·0.00801057s·(cpu);·0.00771425s·(thread);·0s·(gc)171 ·--·used·0.00783973s·(cpu);·0.0072009s·(thread);·0s·(gc)
  
172 o9·=·true172 o9·=·true
173 i10·:·time·degreeMap·psi173 i10·:·time·degreeMap·psi
174 ·--·used·0.201606s·(cpu);·0.15919s·(thread);·0s·(gc)174 ·--·used·0.234928s·(cpu);·0.182881s·(thread);·0s·(gc)
  
175 o10·=·1175 o10·=·1
176 i11·:·time·projectiveDegrees·psi176 i11·:·time·projectiveDegrees·psi
177 ·--·used·5.09383s·(cpu);·4.83436s·(thread);·0s·(gc)177 ·--·used·5.06976s·(cpu);·4.68993s·(thread);·0s·(gc)
  
178 o11·=·{5,·15,·21,·17,·9,·3,·1}178 o11·=·{5,·15,·21,·17,·9,·3,·1}
  
179 o11·:·List179 o11·:·List
180 We·repeat·the·example·using·the·type·_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·and·using·deterministic180 We·repeat·the·example·using·the·type·_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·and·using·deterministic
181 methods.181 methods.
182 i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})182 i12·:·time·phi·=·rationalMap·minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
183 ·--·used·0.000721897s·(cpu);·0.00192268s·(thread);·0s·(gc)183 ·--·used·0.00180521s·(cpu);·0.00182693s·(thread);·0s·(gc)
  
184 o12·=·--·rational·map·--184 o12·=·--·rational·map·--
185 ·····················ZZ185 ·····················ZZ
186 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])186 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
187 ···················300007··0···1···2···3···4···5···6187 ···················300007··0···1···2···3···4···5···6
188 ·····················ZZ188 ·····················ZZ
189 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])189 ······target:·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·])
Offset 233, 15 lines modifiedOffset 233, 15 lines modified
233 ··························3················2····2233 ··························3················2····2
234 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t234 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t
235 ··························4·····3·4·5····2·5····3·6····2·4·6235 ··························4·····3·4·5····2·5····3·6····2·4·6
236 ······················}236 ······················}
  
237 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)237 o12·:·RationalMap·(cubic·rational·map·from·PP^6·to·PP^9)
238 i13·:·time·phi·=·rationalMap(phi,Dominant=>2)238 i13·:·time·phi·=·rationalMap(phi,Dominant=>2)
239 ·--·used·0.108803s·(cpu);·0.0695419s·(thread);·0s·(gc)239 ·--·used·0.10811s·(cpu);·0.068337s·(thread);·0s·(gc)
  
240 o13·=·--·rational·map·--240 o13·=·--·rational·map·--
241 ·····················ZZ241 ·····················ZZ
242 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])242 ······source:·Proj(------[t·,·t·,·t·,·t·,·t·,·t·,·t·])
243 ···················300007··0···1···2···3···4···5···6243 ···················300007··0···1···2···3···4···5···6
244 ···································ZZ244 ···································ZZ
245 ······target:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x245 ······target:·subvariety·of·Proj(------[x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x·,·x
Offset 304, 15 lines modifiedOffset 304, 15 lines modified
304 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t304 ·······················-·t··+·2t·t·t··-·t·t··-·t·t··+·t·t·t
305 ··························4·····3·4·5····2·5····3·6····2·4·6305 ··························4·····3·4·5····2·5····3·6····2·4·6
306 ······················}306 ······················}
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11 cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p11 cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p
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./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump
    
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11 REdBbGdlYnJhTWFwIiwgImxpbmVudW0iID0+IDMxMDQsIElucHV0cyA9PiB7U1BBTntUVHsicGhp11 REdBbGdlYnJhTWFwIiwgImxpbmVudW0iID0+IDMxMDQsIElucHV0cyA9PiB7U1BBTntUVHsicGhp
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./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
    
Offset 155, 15 lines modifiedOffset 155, 15 lines modified
155 ··································2·····2·····2·······2·2·····2·2······2·2······2·2·····2·2········2·2·······2·2········2·······2·······2155 ··································2·····2·····2·······2·2·····2·2······2·2······2·2·····2·2········2·2·······2·2········2·······2·······2
156 ·······Differential·=>·{a,·b,·c,·a·T·,·b·T·,·c·T·,·a*b·c·T·,·b·c·T·,·-a·b·T·,·-a·c·T·,·b·c·T·T·,·-a·c·T·T·,·b·c·T·T·,·-a·T·T·,·c·T·T·,·b·T·T·}156 ·······Differential·=>·{a,·b,·c,·a·T·,·b·T·,·c·T·,·a*b·c·T·,·b·c·T·,·-a·b·T·,·-a·c·T·,·b·c·T·T·,·-a·c·T·T·,·b·c·T·T·,·-a·T·T·,·c·T·T·,·b·T·T·}
157 ····································1·····2·····3·········1·······4········6········5·······3·4········3·5·······2·4······1·7·····3·7·····2·7157 ····································1·····2·····3·········1·······4········6········5·······3·4········3·5·······2·4······1·7·····3·7·····2·7
  
158 o16·:·DGAlgebra158 o16·:·DGAlgebra
  
159 i17·:·HHg·=·HH·g159 i17·:·HHg·=·HH·g
160 ·--·used·0.574312s·(cpu);·0.0679513s·(thread);·0s·(gc)160 ·--·used·0.535111s·(cpu);·0.0683976s·(thread);·0s·(gc)
161 Finding·easy·relations···········:·161 Finding·easy·relations···········:·
162 ··························ZZ162 ··························ZZ
163 ·························---[a..c]163 ·························---[a..c]
164 ············ZZ···········101164 ············ZZ···········101
165 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})165 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
166 ···········101··1···2···········3···1·····1166 ···········101··1···2···········3···1·····1
167 ························(c,·b,·a·)167 ························(c,·b,·a·)
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1.06 KB
./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
    
Offset 30, 15 lines modifiedOffset 30, 15 lines modified
30 ······Underlying·algebra·=>·R[S·..S·]30 ······Underlying·algebra·=>·R[S·..S·]
31 ·······························1···431 ·······························1···4
32 ······Differential·=>·{a,·b,·c,·d}32 ······Differential·=>·{a,·b,·c,·d}
  
33 o4·:·DGAlgebra33 o4·:·DGAlgebra
  
34 i5·:·HB·=·HH·B34 i5·:·HB·=·HH·B
35 ·--·used·0.661141s·(cpu);·0.0994092s·(thread);·0s·(gc)35 ·--·used·0.540795s·(cpu);·0.101247s·(thread);·0s·(gc)
36 Finding·easy·relations···········:·36 Finding·easy·relations···········:·
37 o5·=·HB37 o5·=·HB
  
38 o5·:·PolynomialRing,·4·skew·commutative·variable(s)38 o5·:·PolynomialRing,·4·skew·commutative·variable(s)
  
39 i6·:·describe·HB39 i6·:·describe·HB
  
Offset 68, 15 lines modifiedOffset 68, 15 lines modified
68 ····································268 ····································2
69 ······Differential·=>·{a,·b,·c,·d,·a·T·}69 ······Differential·=>·{a,·b,·c,·d,·a·T·}
70 ······································170 ······································1
  
71 o9·:·DGAlgebra71 o9·:·DGAlgebra
  
72 i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)72 i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
73 ·--·used·0.998692s·(cpu);·0.135325s·(thread);·0s·(gc)73 ·--·used·0.876597s·(cpu);·0.148079s·(thread);·0s·(gc)
74 Finding·easy·relations···········:·74 Finding·easy·relations···········:·
75 ·······ZZ75 ·······ZZ
76 o10·=·---[X·..X·]76 o10·=·---[X·..X·]
77 ······101··1···377 ······101··1···3
  
78 o10·:·PolynomialRing,·3·skew·commutative·variable(s)78 o10·:·PolynomialRing,·3·skew·commutative·variable(s)
  
⊟
687 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
    
Offset 55, 15 lines modifiedOffset 55, 15 lines modified
55 ···············{2}·|·0·|55 ···············{2}·|·0·|
56 ···············{2}·|·0·|56 ···············{2}·|·0·|
57 ···············{2}·|·1·|57 ···············{2}·|·1·|
  
58 o6·:·ChainComplexMap58 o6·:·ChainComplexMap
  
59 i7·:·HHg·=·HH·g59 i7·:·HHg·=·HH·g
60 ·--·used·0.62249s·(cpu);·0.084406s·(thread);·0s·(gc)60 ·--·used·0.470934s·(cpu);·0.0622814s·(thread);·0s·(gc)
61 Finding·easy·relations···········:·61 Finding·easy·relations···········:·
62 ·························ZZ62 ·························ZZ
63 ························---[a..c]63 ························---[a..c]
64 ···········ZZ···········10164 ···········ZZ···········101
65 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})65 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
66 ··········101··1···2···········3···1·····166 ··········101··1···2···········3···1·····1
67 ·······················(c,·b,·a·)67 ·······················(c,·b,·a·)
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1.02 KB
./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
    
Offset 61, 15 lines modifiedOffset 61, 15 lines modified
61 ··················{9}·|·0·0·0·0·0·0·0·0·0·0·0·0·d·c·b·a·|················61 ··················{9}·|·0·0·0·0·0·0·0·0·0·0·0·0·d·c·b·a·|················
  
62 ·····4·:·cokernel·{12}·|·d·c·b·a·|·······································62 ·····4·:·cokernel·{12}·|·d·c·b·a·|·······································
  
63 o6·:·GradedModule63 o6·:·GradedModule
  
64 i7·:·HKR·=·HH·KR64 i7·:·HKR·=·HH·KR
65 ·--·used·0.702569s·(cpu);·0.121897s·(thread);·0s·(gc)65 ·--·used·0.681886s·(cpu);·0.0954678s·(thread);·0s·(gc)
66 Finding·easy·relations···········:·66 Finding·easy·relations···········:·
67 o7·=·HKR67 o7·=·HKR
  
68 o7·:·PolynomialRing,·4·skew·commutative·variable(s)68 o7·:·PolynomialRing,·4·skew·commutative·variable(s)
  
69 i8·:·ideal·HKR69 i8·:·ideal·HKR
  
Offset 80, 15 lines modifiedOffset 80, 15 lines modified
80 i9·:·R'·=·ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}80 i9·:·R'·=·ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-c^2*d^2}
  
81 o9·=·R'81 o9·=·R'
  
82 o9·:·QuotientRing82 o9·:·QuotientRing
  
83 i10·:·HKR'·=·HH·koszulComplexDGA·R'83 i10·:·HKR'·=·HH·koszulComplexDGA·R'
84 ·--·used·2.54767s·(cpu);·0.709695s·(thread);·0s·(gc)84 ·--·used·2.02187s·(cpu);·0.748472s·(thread);·0s·(gc)
85 Finding·easy·relations···········:·85 Finding·easy·relations···········:·
86 o10·=·HKR'86 o10·=·HKR'
  
87 o10·:·QuotientRing87 o10·:·QuotientRing
  
88 i11·:·numgens·HKR'88 i11·:·numgens·HKR'
  
⊟
509 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_cycles.out
    
Offset 18, 15 lines modifiedOffset 18, 15 lines modified
18 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))18 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
19 o3·=·{1,·4,·6,·4,·1}19 o3·=·{1,·4,·6,·4,·1}
  
20 o3·:·List20 o3·:·List
  
21 i4·:·HA·=·homologyAlgebra(A)21 i4·:·HA·=·homologyAlgebra(A)
22 ·--·used·0.883949s·(cpu);·0.115372s·(thread);·0s·(gc)22 ·--·used·1.53337s·(cpu);·0.275246s·(thread);·0s·(gc)
23 Finding·easy·relations···········:·23 Finding·easy·relations···········:·
24 o4·=·HA24 o4·=·HA
  
25 o4·:·PolynomialRing,·4·skew·commutative·variable(s)25 o4·:·PolynomialRing,·4·skew·commutative·variable(s)
  
26 i5·:·numgens·HA26 i5·:·numgens·HA
  
⊟
1.7 KB
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
    
Offset 18, 15 lines modifiedOffset 18, 15 lines modified
18 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))18 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
19 o3·=·{1,·4,·6,·4,·1}19 o3·=·{1,·4,·6,·4,·1}
  
20 o3·:·List20 o3·:·List
  
21 i4·:·HA·=·homologyAlgebra(A)21 i4·:·HA·=·homologyAlgebra(A)
22 ·--·used·0.69296s·(cpu);·0.0935568s·(thread);·0s·(gc)22 ·--·used·0.592566s·(cpu);·0.119349s·(thread);·0s·(gc)
23 Finding·easy·relations···········:·23 Finding·easy·relations···········:·
24 o4·=·HA24 o4·=·HA
  
25 o4·:·PolynomialRing,·4·skew·commutative·variable(s)25 o4·:·PolynomialRing,·4·skew·commutative·variable(s)
  
26 i5·:·R·=·ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}26 i5·:·R·=·ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
  
Offset 46, 15 lines modifiedOffset 46, 15 lines modified
46 i7·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))46 i7·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
47 o7·=·{1,·5,·10,·10,·4}47 o7·=·{1,·5,·10,·10,·4}
  
48 o7·:·List48 o7·:·List
  
49 i8·:·HA·=·homologyAlgebra(A)49 i8·:·HA·=·homologyAlgebra(A)
50 ·--·used·1.92263s·(cpu);·0.308938s·(thread);·0s·(gc)50 ·--·used·1.7032s·(cpu);·0.305095s·(thread);·0s·(gc)
51 Finding·easy·relations···········:·51 Finding·easy·relations···········:·
52 o8·=·HA52 o8·=·HA
  
53 o8·:·QuotientRing53 o8·:·QuotientRing
  
54 i9·:·numgens·HA54 i9·:·numgens·HA
  
Offset 114, 15 lines modifiedOffset 114, 15 lines modified
114 i15·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))114 i15·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
115 o15·=·{1,·7,·7,·1}115 o15·=·{1,·7,·7,·1}
  
116 o15·:·List116 o15·:·List
  
117 i16·:·HA·=·homologyAlgebra(A)117 i16·:·HA·=·homologyAlgebra(A)
118 ·--·used·1.05295s·(cpu);·0.14915s·(thread);·0s·(gc)118 ·--·used·0.794164s·(cpu);·0.133185s·(thread);·0s·(gc)
119 Finding·easy·relations···········:·119 Finding·easy·relations···········:·
120 o16·=·HA120 o16·=·HA
  
121 o16·:·QuotientRing121 o16·:·QuotientRing
  
122 i17·:·R·=·ZZ/101[a,b,c,d]122 i17·:·R·=·ZZ/101[a,b,c,d]
  
Offset 151, 14 lines modifiedOffset 151, 14 lines modified
151 ·······Underlying·algebra·=>·S[T·..T·]151 ·······Underlying·algebra·=>·S[T·..T·]
152 ································1···4152 ································1···4
153 ·······Differential·=>·{a,·b,·c,·d}153 ·······Differential·=>·{a,·b,·c,·d}
  
154 o20·:·DGAlgebra154 o20·:·DGAlgebra
  
155 i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)155 i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
156 ·--·used·0.176481s·(cpu);·0.037047s·(thread);·0s·(gc)156 ·--·used·0.52822s·(cpu);·0.0858266s·(thread);·0s·(gc)
157 Finding·easy·relations···········:·157 Finding·easy·relations···········:·
158 o21·=·HB158 o21·=·HB
  
159 o21·:·PolynomialRing,·4·skew·commutative·variable(s)159 o21·:·PolynomialRing,·4·skew·commutative·variable(s)
  
160 i22·:·160 i22·:·
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483 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
    
Offset 43, 15 lines modifiedOffset 43, 15 lines modified
43 o6·=·y·T43 o6·=·y·T
44 ········244 ········2
  
45 o6·:·R[T·..T·]45 o6·:·R[T·..T·]
46 ········1···346 ········1···3
  
47 i7·:·H·=·HH(KR)47 i7·:·H·=·HH(KR)
48 ·--·used·0.647953s·(cpu);·0.0882768s·(thread);·0s·(gc)48 ·--·used·0.531456s·(cpu);·0.0862601s·(thread);·0s·(gc)
49 Finding·easy·relations···········:·49 Finding·easy·relations···········:·
50 o7·=·H50 o7·=·H
  
51 o7·:·PolynomialRing,·3·skew·commutative·variable(s)51 o7·:·PolynomialRing,·3·skew·commutative·variable(s)
  
52 i8·:·homologyClass(KR,z1*z2)52 i8·:·homologyClass(KR,z1*z2)
  
⊟
549 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Module.out
    
Offset 34, 15 lines modifiedOffset 34, 15 lines modified
34 o5·=·R··<--·R··<--·R··<--·R··<--·R34 o5·=·R··<--·R··<--·R··<--·R··<--·R
35 ··································35 ··································
36 ·····0······1······2······3······436 ·····0······1······2······3······4
  
37 o5·:·ChainComplex37 o5·:·ChainComplex
  
38 i6·:·HKR·=·HH(KR)38 i6·:·HKR·=·HH(KR)
39 ·--·used·1.5631s·(cpu);·0.211641s·(thread);·0s·(gc)39 ·--·used·1.22896s·(cpu);·0.248658s·(thread);·0s·(gc)
40 Finding·easy·relations···········:·40 Finding·easy·relations···········:·
41 o6·=·HKR41 o6·=·HKR
  
42 o6·:·QuotientRing42 o6·:·QuotientRing
  
43 i7·:·degList·=·first·entries·vars·Q·/·degree·/·first43 i7·:·degList·=·first·entries·vars·Q·/·degree·/·first
  
⊟
586 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
    
Offset 68, 15 lines modifiedOffset 68, 15 lines modified
68 ···············268 ···············2
69 o9·=·(true,·x·y·T·T·T··-·x·x·y·T·T·T·)69 o9·=·(true,·x·y·T·T·T··-·x·x·y·T·T·T·)
70 ·············2·2·1·2·3····1·2·2·2·3·470 ·············2·2·1·2·3····1·2·2·2·3·4
  
71 o9·:·Sequence71 o9·:·Sequence
  
72 i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)72 i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)
73 ·--·used·2.87754s·(cpu);·0.732378s·(thread);·0s·(gc)73 ·--·used·2.46677s·(cpu);·0.739171s·(thread);·0s·(gc)
74 Finding·easy·relations···········:·74 Finding·easy·relations···········:·
75 ·············275 ·············2
76 o10·=·x·x·y·z·T·T·T·T76 o10·=·x·x·y·z·T·T·T·T
77 ·······1·2·2···2·3·4·577 ·······1·2·2···2·3·4·5
  
78 o10·:·R[T·..T·]78 o10·:·R[T·..T·]
79 ·········1···579 ·········1···5
⊟
655 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
    
Offset 27, 15 lines modifiedOffset 27, 15 lines modified
27 ·······························1···427 ·······························1···4
28 ······Differential·=>·{t·,·t·,·t·,·t·}28 ······Differential·=>·{t·,·t·,·t·,·t·}
29 ························1···2···3···429 ························1···2···3···4
  
30 o4·:·DGAlgebra30 o4·:·DGAlgebra
  
31 i5·:·H·=·HH(KR)31 i5·:·H·=·HH(KR)
32 ·--·used·2.10443s·(cpu);·0.351445s·(thread);·0s·(gc)32 ·--·used·1.95528s·(cpu);·0.384272s·(thread);·0s·(gc)
33 Finding·easy·relations···········:·33 Finding·easy·relations···········:·
34 o5·=·H34 o5·=·H
  
35 o5·:·QuotientRing35 o5·:·QuotientRing
  
36 i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);36 i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);
  
⊟
527 B
./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
    
Offset 11, 15 lines modifiedOffset 11, 15 lines modified
11 i3·:·S·=·R/ideal{a^3*b^3*c^3*d^3}11 i3·:·S·=·R/ideal{a^3*b^3*c^3*d^3}
  
12 o3·=·S12 o3·=·S
  
13 o3·:·QuotientRing13 o3·:·QuotientRing
  
14 i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)14 i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
15 ·--·used·1.45046s·(cpu);·0.466287s·(thread);·0s·(gc)15 ·--·used·2.56234s·(cpu);·0.709352s·(thread);·0s·(gc)
16 Finding·easy·relations···········:·16 Finding·easy·relations···········:·
17 o4·=·HB17 o4·=·HB
  
18 o4·:·QuotientRing18 o4·:·QuotientRing
  
19 i5·:·numgens·HB19 i5·:·numgens·HB
  
⊟
1.57 KB
./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
    
Offset 249, 15 lines modifiedOffset 249, 15 lines modified
249 ········</table>249 ········</table>
250 ········<div>250 ········<div>
251 ··········<p>One·can·also·obtain·the·map·on·homology·induced·by·a·DGAlgebra·map.</p>251 ··········<p>One·can·also·obtain·the·map·on·homology·induced·by·a·DGAlgebra·map.</p>
252 ········</div>252 ········</div>
253 ········<table·class="examples">253 ········<table·class="examples">
254 ··········<tr>254 ··········<tr>
255 <td>··············<pre><code·class="language-macaulay2">i17·:·HHg·=·HH·g255 <td>··············<pre><code·class="language-macaulay2">i17·:·HHg·=·HH·g
256 ·--·used·0.574312s·(cpu);·0.0679513s·(thread);·0s·(gc)256 ·--·used·0.535111s·(cpu);·0.0683976s·(thread);·0s·(gc)
257 Finding·easy·relations···········:·257 Finding·easy·relations···········:·
258 ··························ZZ258 ··························ZZ
259 ·························---[a..c]259 ·························---[a..c]
260 ············ZZ···········101260 ············ZZ···········101
261 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})261 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
262 ···········101··1···2···········3···1·····1262 ···········101··1···2···········3···1·····1
263 ························(c,·b,·a·)263 ························(c,·b,·a·)
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735 B
html2text {}
    
Offset 210, 15 lines modifiedOffset 210, 15 lines modified
210 a·c·T·,·b·c·T·T·,·-a·c·T·T·,·b·c·T·T·,·-a·T·T·,·c·T·T·,·b·T·T·}210 a·c·T·,·b·c·T·T·,·-a·c·T·T·,·b·c·T·T·,·-a·T·T·,·c·T·T·,·b·T·T·}
211 ····································1·····2·····3·········1·······4········6211 ····································1·····2·····3·········1·······4········6
212 5·······3·4········3·5·······2·4······1·7·····3·7·····2·7212 5·······3·4········3·5·······2·4······1·7·····3·7·····2·7
  
213 o16·:·DGAlgebra213 o16·:·DGAlgebra
214 One·can·also·obtain·the·map·on·homology·induced·by·a·DGAlgebra·map.214 One·can·also·obtain·the·map·on·homology·induced·by·a·DGAlgebra·map.
215 i17·:·HHg·=·HH·g215 i17·:·HHg·=·HH·g
216 ·--·used·0.574312s·(cpu);·0.0679513s·(thread);·0s·(gc)216 ·--·used·0.535111s·(cpu);·0.0683976s·(thread);·0s·(gc)
217 Finding·easy·relations···········:217 Finding·easy·relations···········:
218 ··························ZZ218 ··························ZZ
219 ·························---[a..c]219 ·························---[a..c]
220 ············ZZ···········101220 ············ZZ···········101
221 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})221 o17·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
222 ···········101··1···2···········3···1·····1222 ···········101··1···2···········3···1·····1
223 ························(c,·b,·a·)223 ························(c,·b,·a·)
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97 ········</table>97 ········</table>
98 ········<div>98 ········<div>
99 ··········<p>One·can·compute·the·homology·algebra·of·a·DGAlgebra·using·the·homology·(or·HH)·command.</p>99 ··········<p>One·can·compute·the·homology·algebra·of·a·DGAlgebra·using·the·homology·(or·HH)·command.</p>
100 ········</div>100 ········</div>
101 ········<table·class="examples">101 ········<table·class="examples">
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i5·:·HB·=·HH·B103 <td>··············<pre><code·class="language-macaulay2">i5·:·HB·=·HH·B
104 ·--·used·0.661141s·(cpu);·0.0994092s·(thread);·0s·(gc)104 ·--·used·0.540795s·(cpu);·0.101247s·(thread);·0s·(gc)
105 Finding·easy·relations···········:·105 Finding·easy·relations···········:·
106 o5·=·HB106 o5·=·HB
  
107 o5·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>107 o5·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i6·:·describe·HB110 <td>··············<pre><code·class="language-macaulay2">i6·:·describe·HB
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148 ······Differential·=>·{a,·b,·c,·d,·a·T·}148 ······Differential·=>·{a,·b,·c,·d,·a·T·}
149 ······································1149 ······································1
  
150 o9·:·DGAlgebra</code></pre>150 o9·:·DGAlgebra</code></pre>
151 </td>··········</tr>151 </td>··········</tr>
152 ··········<tr>152 ··········<tr>
153 <td>··············<pre><code·class="language-macaulay2">i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)153 <td>··············<pre><code·class="language-macaulay2">i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
154 ·--·used·0.998692s·(cpu);·0.135325s·(thread);·0s·(gc)154 ·--·used·0.876597s·(cpu);·0.148079s·(thread);·0s·(gc)
155 Finding·easy·relations···········:·155 Finding·easy·relations···········:·
156 ·······ZZ156 ·······ZZ
157 o10·=·---[X·..X·]157 o10·=·---[X·..X·]
158 ······101··1···3158 ······101··1···3
  
159 o10·:·PolynomialRing,·3·skew·commutative·variable(s)</code></pre>159 o10·:·PolynomialRing,·3·skew·commutative·variable(s)</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
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42 ·······························1···442 ·······························1···4
43 ······Differential·=>·{a,·b,·c,·d}43 ······Differential·=>·{a,·b,·c,·d}
  
44 o4·:·DGAlgebra44 o4·:·DGAlgebra
45 One·can·compute·the·homology·algebra·of·a·DGAlgebra·using·the·homology·(or·HH)45 One·can·compute·the·homology·algebra·of·a·DGAlgebra·using·the·homology·(or·HH)
46 command.46 command.
47 i5·:·HB·=·HH·B47 i5·:·HB·=·HH·B
48 ·--·used·0.661141s·(cpu);·0.0994092s·(thread);·0s·(gc)48 ·--·used·0.540795s·(cpu);·0.101247s·(thread);·0s·(gc)
49 Finding·easy·relations···········:49 Finding·easy·relations···········:
50 o5·=·HB50 o5·=·HB
  
51 o5·:·PolynomialRing,·4·skew·commutative·variable(s)51 o5·:·PolynomialRing,·4·skew·commutative·variable(s)
52 i6·:·describe·HB52 i6·:·describe·HB
  
53 ······ZZ53 ······ZZ
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86 ·······························1···586 ·······························1···5
87 ····································287 ····································2
88 ······Differential·=>·{a,·b,·c,·d,·a·T·}88 ······Differential·=>·{a,·b,·c,·d,·a·T·}
89 ······································189 ······································1
  
90 o9·:·DGAlgebra90 o9·:·DGAlgebra
91 i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)91 i10·:·homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
92 ·--·used·0.998692s·(cpu);·0.135325s·(thread);·0s·(gc)92 ·--·used·0.876597s·(cpu);·0.148079s·(thread);·0s·(gc)
93 Finding·easy·relations···········:93 Finding·easy·relations···········:
94 ·······ZZ94 ·······ZZ
95 o10·=·---[X·..X·]95 o10·=·---[X·..X·]
96 ······101··1···396 ······101··1···3
  
97 o10·:·PolynomialRing,·3·skew·commutative·variable(s)97 o10·:·PolynomialRing,·3·skew·commutative·variable(s)
98 i11·:·C·=·killCycles(B)98 i11·:·C·=·killCycles(B)
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132 ···············{2}·|·0·|132 ···············{2}·|·0·|
133 ···············{2}·|·1·|133 ···············{2}·|·1·|
  
134 o6·:·ChainComplexMap</code></pre>134 o6·:·ChainComplexMap</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i7·:·HHg·=·HH·g137 <td>··············<pre><code·class="language-macaulay2">i7·:·HHg·=·HH·g
138 ·--·used·0.62249s·(cpu);·0.084406s·(thread);·0s·(gc)138 ·--·used·0.470934s·(cpu);·0.0622814s·(thread);·0s·(gc)
139 Finding·easy·relations···········:·139 Finding·easy·relations···········:·
140 ·························ZZ140 ·························ZZ
141 ························---[a..c]141 ························---[a..c]
142 ···········ZZ···········101142 ···········ZZ···········101
143 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})143 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
144 ··········101··1···2···········3···1·····1144 ··········101··1···2···········3···1·····1
145 ·······················(c,·b,·a·)145 ·······················(c,·b,·a·)
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63 ·····2·:·R··<-------------·R··:·263 ·····2·:·R··<-------------·R··:·2
64 ···············{2}·|·0·|64 ···············{2}·|·0·|
65 ···············{2}·|·0·|65 ···············{2}·|·0·|
66 ···············{2}·|·1·|66 ···············{2}·|·1·|
  
67 o6·:·ChainComplexMap67 o6·:·ChainComplexMap
68 i7·:·HHg·=·HH·g68 i7·:·HHg·=·HH·g
69 ·--·used·0.62249s·(cpu);·0.084406s·(thread);·0s·(gc)69 ·--·used·0.470934s·(cpu);·0.0622814s·(thread);·0s·(gc)
70 Finding·easy·relations···········:70 Finding·easy·relations···········:
71 ·························ZZ71 ·························ZZ
72 ························---[a..c]72 ························---[a..c]
73 ···········ZZ···········10173 ···········ZZ···········101
74 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})74 o7·=·map·(---[X·..X·],·----------[X·],·{X·,·0,·0,·0})
75 ··········101··1···2···········3···1·····175 ··········101··1···2···········3···1·····1
76 ·······················(c,·b,·a·)76 ·······················(c,·b,·a·)
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130 ········</table>130 ········</table>
131 ········<div>131 ········<div>
132 ··········<p>Since·the·Koszul·complex·is·a·DG·algebra,·its·homology·is·itself·an·algebra.··One·can·obtain·this·algebra·using·the·command·homology,·homologyAlgebra,·or·HH·(all·commands·work).··This·algebra·structure·can·detect·whether·or·not·the·ring·is·a·complete·intersection·or·Gorenstein.</p>132 ··········<p>Since·the·Koszul·complex·is·a·DG·algebra,·its·homology·is·itself·an·algebra.··One·can·obtain·this·algebra·using·the·command·homology,·homologyAlgebra,·or·HH·(all·commands·work).··This·algebra·structure·can·detect·whether·or·not·the·ring·is·a·complete·intersection·or·Gorenstein.</p>
133 ········</div>133 ········</div>
134 ········<table·class="examples">134 ········<table·class="examples">
135 ··········<tr>135 ··········<tr>
136 <td>··············<pre><code·class="language-macaulay2">i7·:·HKR·=·HH·KR136 <td>··············<pre><code·class="language-macaulay2">i7·:·HKR·=·HH·KR
137 ·--·used·0.702569s·(cpu);·0.121897s·(thread);·0s·(gc)137 ·--·used·0.681886s·(cpu);·0.0954678s·(thread);·0s·(gc)
138 Finding·easy·relations···········:·138 Finding·easy·relations···········:·
139 o7·=·HKR139 o7·=·HKR
  
140 o7·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>140 o7·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i8·:·ideal·HKR143 <td>··············<pre><code·class="language-macaulay2">i8·:·ideal·HKR
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152 o9·=·R'152 o9·=·R'
  
153 o9·:·QuotientRing</code></pre>153 o9·:·QuotientRing</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i10·:·HKR'·=·HH·koszulComplexDGA·R'156 <td>··············<pre><code·class="language-macaulay2">i10·:·HKR'·=·HH·koszulComplexDGA·R'
157 ·--·used·2.54767s·(cpu);·0.709695s·(thread);·0s·(gc)157 ·--·used·2.02187s·(cpu);·0.748472s·(thread);·0s·(gc)
158 Finding·easy·relations···········:·158 Finding·easy·relations···········:·
159 o10·=·HKR'159 o10·=·HKR'
  
160 o10·:·QuotientRing</code></pre>160 o10·:·QuotientRing</code></pre>
161 </td>··········</tr>161 </td>··········</tr>
162 ··········<tr>162 ··········<tr>
163 <td>··············<pre><code·class="language-macaulay2">i11·:·numgens·HKR'163 <td>··············<pre><code·class="language-macaulay2">i11·:·numgens·HKR'
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75 o6·:·GradedModule75 o6·:·GradedModule
76 Since·the·Koszul·complex·is·a·DG·algebra,·its·homology·is·itself·an·algebra.76 Since·the·Koszul·complex·is·a·DG·algebra,·its·homology·is·itself·an·algebra.
77 One·can·obtain·this·algebra·using·the·command·homology,·homologyAlgebra,·or·HH77 One·can·obtain·this·algebra·using·the·command·homology,·homologyAlgebra,·or·HH
78 (all·commands·work).·This·algebra·structure·can·detect·whether·or·not·the·ring78 (all·commands·work).·This·algebra·structure·can·detect·whether·or·not·the·ring
79 is·a·complete·intersection·or·Gorenstein.79 is·a·complete·intersection·or·Gorenstein.
80 i7·:·HKR·=·HH·KR80 i7·:·HKR·=·HH·KR
81 ·--·used·0.702569s·(cpu);·0.121897s·(thread);·0s·(gc)81 ·--·used·0.681886s·(cpu);·0.0954678s·(thread);·0s·(gc)
82 Finding·easy·relations···········:82 Finding·easy·relations···········:
83 o7·=·HKR83 o7·=·HKR
  
84 o7·:·PolynomialRing,·4·skew·commutative·variable(s)84 o7·:·PolynomialRing,·4·skew·commutative·variable(s)
85 i8·:·ideal·HKR85 i8·:·ideal·HKR
  
86 o8·=·ideal·()86 o8·=·ideal·()
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92 i9·:·R'·=·ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-92 i9·:·R'·=·ZZ/101[a,b,c,d]/ideal{a^3,b^3,c^3,d^3,a*c,a*d,b*c,b*d,a^2*b^2-
93 c^2*d^2}93 c^2*d^2}
  
94 o9·=·R'94 o9·=·R'
  
95 o9·:·QuotientRing95 o9·:·QuotientRing
96 i10·:·HKR'·=·HH·koszulComplexDGA·R'96 i10·:·HKR'·=·HH·koszulComplexDGA·R'
97 ·--·used·2.54767s·(cpu);·0.709695s·(thread);·0s·(gc)97 ·--·used·2.02187s·(cpu);·0.748472s·(thread);·0s·(gc)
98 Finding·easy·relations···········:98 Finding·easy·relations···········:
99 o10·=·HKR'99 o10·=·HKR'
  
100 o10·:·QuotientRing100 o10·:·QuotientRing
101 i11·:·numgens·HKR'101 i11·:·numgens·HKR'
  
102 o11·=·34102 o11·=·34
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78 o3·=·{1,·4,·6,·4,·1}78 o3·=·{1,·4,·6,·4,·1}
  
79 o3·:·List</code></pre>79 o3·:·List</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i4·:·HA·=·homologyAlgebra(A)82 <td>··············<pre><code·class="language-macaulay2">i4·:·HA·=·homologyAlgebra(A)
83 ·--·used·0.883949s·(cpu);·0.115372s·(thread);·0s·(gc)83 ·--·used·1.53337s·(cpu);·0.275246s·(thread);·0s·(gc)
84 Finding·easy·relations···········:·84 Finding·easy·relations···········:·
85 o4·=·HA85 o4·=·HA
  
86 o4·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>86 o4·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·numgens·HA89 <td>··············<pre><code·class="language-macaulay2">i5·:·numgens·HA
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24 o2·:·DGAlgebra24 o2·:·DGAlgebra
25 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))25 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
26 o3·=·{1,·4,·6,·4,·1}26 o3·=·{1,·4,·6,·4,·1}
  
27 o3·:·List27 o3·:·List
28 i4·:·HA·=·homologyAlgebra(A)28 i4·:·HA·=·homologyAlgebra(A)
29 ·--·used·0.883949s·(cpu);·0.115372s·(thread);·0s·(gc)29 ·--·used·1.53337s·(cpu);·0.275246s·(thread);·0s·(gc)
30 Finding·easy·relations···········:30 Finding·easy·relations···········:
31 o4·=·HA31 o4·=·HA
  
32 o4·:·PolynomialRing,·4·skew·commutative·variable(s)32 o4·:·PolynomialRing,·4·skew·commutative·variable(s)
33 i5·:·numgens·HA33 i5·:·numgens·HA
  
34 o5·=·434 o5·=·4
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100 o3·=·{1,·4,·6,·4,·1}100 o3·=·{1,·4,·6,·4,·1}
  
101 o3·:·List</code></pre>101 o3·:·List</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i4·:·HA·=·homologyAlgebra(A)104 <td>··············<pre><code·class="language-macaulay2">i4·:·HA·=·homologyAlgebra(A)
105 ·--·used·0.69296s·(cpu);·0.0935568s·(thread);·0s·(gc)105 ·--·used·0.592566s·(cpu);·0.119349s·(thread);·0s·(gc)
106 Finding·easy·relations···········:·106 Finding·easy·relations···········:·
107 o4·=·HA107 o4·=·HA
  
108 o4·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>108 o4·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ········</table>110 ········</table>
111 ········<div>111 ········<div>
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137 o7·=·{1,·5,·10,·10,·4}137 o7·=·{1,·5,·10,·10,·4}
  
138 o7·:·List</code></pre>138 o7·:·List</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i8·:·HA·=·homologyAlgebra(A)141 <td>··············<pre><code·class="language-macaulay2">i8·:·HA·=·homologyAlgebra(A)
142 ·--·used·1.92263s·(cpu);·0.308938s·(thread);·0s·(gc)142 ·--·used·1.7032s·(cpu);·0.305095s·(thread);·0s·(gc)
143 Finding·easy·relations···········:·143 Finding·easy·relations···········:·
144 o8·=·HA144 o8·=·HA
  
145 o8·:·QuotientRing</code></pre>145 o8·:·QuotientRing</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i9·:·numgens·HA148 <td>··············<pre><code·class="language-macaulay2">i9·:·numgens·HA
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215 o15·=·{1,·7,·7,·1}215 o15·=·{1,·7,·7,·1}
  
216 o15·:·List</code></pre>216 o15·:·List</code></pre>
217 </td>··········</tr>217 </td>··········</tr>
218 ··········<tr>218 ··········<tr>
219 <td>··············<pre><code·class="language-macaulay2">i16·:·HA·=·homologyAlgebra(A)219 <td>··············<pre><code·class="language-macaulay2">i16·:·HA·=·homologyAlgebra(A)
220 ·--·used·1.05295s·(cpu);·0.14915s·(thread);·0s·(gc)220 ·--·used·0.794164s·(cpu);·0.133185s·(thread);·0s·(gc)
221 Finding·easy·relations···········:·221 Finding·easy·relations···········:·
222 o16·=·HA222 o16·=·HA
  
223 o16·:·QuotientRing</code></pre>223 o16·:·QuotientRing</code></pre>
224 </td>··········</tr>224 </td>··········</tr>
225 ········</table>225 ········</table>
226 ········<div>226 ········<div>
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265 ································1···4265 ································1···4
266 ·······Differential·=>·{a,·b,·c,·d}266 ·······Differential·=>·{a,·b,·c,·d}
  
267 o20·:·DGAlgebra</code></pre>267 o20·:·DGAlgebra</code></pre>
268 </td>··········</tr>268 </td>··········</tr>
269 ··········<tr>269 ··········<tr>
270 <td>··············<pre><code·class="language-macaulay2">i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)270 <td>··············<pre><code·class="language-macaulay2">i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
271 ·--·used·0.176481s·(cpu);·0.037047s·(thread);·0s·(gc)271 ·--·used·0.52822s·(cpu);·0.0858266s·(thread);·0s·(gc)
272 Finding·easy·relations···········:·272 Finding·easy·relations···········:·
273 o21·=·HB273 o21·=·HB
  
274 o21·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>274 o21·:·PolynomialRing,·4·skew·commutative·variable(s)</code></pre>
275 </td>··········</tr>275 </td>··········</tr>
276 ········</table>276 ········</table>
277 ······</div>277 ······</div>
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34 o2·:·DGAlgebra34 o2·:·DGAlgebra
35 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))35 i3·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
36 o3·=·{1,·4,·6,·4,·1}36 o3·=·{1,·4,·6,·4,·1}
  
37 o3·:·List37 o3·:·List
38 i4·:·HA·=·homologyAlgebra(A)38 i4·:·HA·=·homologyAlgebra(A)
39 ·--·used·0.69296s·(cpu);·0.0935568s·(thread);·0s·(gc)39 ·--·used·0.592566s·(cpu);·0.119349s·(thread);·0s·(gc)
40 Finding·easy·relations···········:40 Finding·easy·relations···········:
41 o4·=·HA41 o4·=·HA
  
42 o4·:·PolynomialRing,·4·skew·commutative·variable(s)42 o4·:·PolynomialRing,·4·skew·commutative·variable(s)
43 Note·that·HA·is·a·graded·commutative·polynomial·ring·(i.e.·an·exterior·algebra)43 Note·that·HA·is·a·graded·commutative·polynomial·ring·(i.e.·an·exterior·algebra)
44 since·R·is·a·complete·intersection.44 since·R·is·a·complete·intersection.
45 i5·:·R·=·ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}45 i5·:·R·=·ZZ/101[a,b,c,d]/ideal{a^4,b^4,c^4,d^4,a^3*b^3*c^3*d^3}
Offset 60, 15 lines modifiedOffset 60, 15 lines modified
60 o6·:·DGAlgebra60 o6·:·DGAlgebra
61 i7·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))61 i7·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
62 o7·=·{1,·5,·10,·10,·4}62 o7·=·{1,·5,·10,·10,·4}
  
63 o7·:·List63 o7·:·List
64 i8·:·HA·=·homologyAlgebra(A)64 i8·:·HA·=·homologyAlgebra(A)
65 ·--·used·1.92263s·(cpu);·0.308938s·(thread);·0s·(gc)65 ·--·used·1.7032s·(cpu);·0.305095s·(thread);·0s·(gc)
66 Finding·easy·relations···········:66 Finding·easy·relations···········:
67 o8·=·HA67 o8·=·HA
  
68 o8·:·QuotientRing68 o8·:·QuotientRing
69 i9·:·numgens·HA69 i9·:·numgens·HA
  
70 o9·=·1970 o9·=·19
Offset 121, 15 lines modifiedOffset 121, 15 lines modified
121 o14·:·DGAlgebra121 o14·:·DGAlgebra
122 i15·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))122 i15·:·apply(maxDegree·A·+·1,·i·->·numgens·prune·homology(i,A))
  
123 o15·=·{1,·7,·7,·1}123 o15·=·{1,·7,·7,·1}
  
124 o15·:·List124 o15·:·List
125 i16·:·HA·=·homologyAlgebra(A)125 i16·:·HA·=·homologyAlgebra(A)
126 ·--·used·1.05295s·(cpu);·0.14915s·(thread);·0s·(gc)126 ·--·used·0.794164s·(cpu);·0.133185s·(thread);·0s·(gc)
127 Finding·easy·relations···········:127 Finding·easy·relations···········:
128 o16·=·HA128 o16·=·HA
  
129 o16·:·QuotientRing129 o16·:·QuotientRing
130 One·can·check·that·HA·has·Poincare·duality·since·R·is·Gorenstein.130 One·can·check·that·HA·has·Poincare·duality·since·R·is·Gorenstein.
131 If·your·DGAlgebra·has·generators·in·even·degrees,·then·one·must·specify·the131 If·your·DGAlgebra·has·generators·in·even·degrees,·then·one·must·specify·the
132 options·GenDegreeLimit·and·RelDegreeLimit.132 options·GenDegreeLimit·and·RelDegreeLimit.
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156 o20·=·{Ring·=>·S······················}156 o20·=·{Ring·=>·S······················}
157 ·······Underlying·algebra·=>·S[T·..T·]157 ·······Underlying·algebra·=>·S[T·..T·]
158 ································1···4158 ································1···4
159 ·······Differential·=>·{a,·b,·c,·d}159 ·······Differential·=>·{a,·b,·c,·d}
  
160 o20·:·DGAlgebra160 o20·:·DGAlgebra
161 i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)161 i21·:·HB·=·homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
162 ·--·used·0.176481s·(cpu);·0.037047s·(thread);·0s·(gc)162 ·--·used·0.52822s·(cpu);·0.0858266s·(thread);·0s·(gc)
163 Finding·easy·relations···········:163 Finding·easy·relations···········:
164 o21·=·HB164 o21·=·HB
  
165 o21·:·PolynomialRing,·4·skew·commutative·variable(s)165 o21·:·PolynomialRing,·4·skew·commutative·variable(s)
166 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ho\x8om\x8mo\x8ol\x8lo\x8og\x8gy\x8yA\x8Al\x8lg\x8ge\x8eb\x8br\x8ra\x8a:\x8:·*\x8**\x8**\x8**\x8**\x8*166 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ho\x8om\x8mo\x8ol\x8lo\x8og\x8gy\x8yA\x8Al\x8lg\x8ge\x8eb\x8br\x8ra\x8a:\x8:·*\x8**\x8**\x8**\x8**\x8*
167 ····*·homologyAlgebra(DGAlgebra)167 ····*·homologyAlgebra(DGAlgebra)
168 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*168 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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123 ········2123 ········2
  
124 o6·:·R[T·..T·]124 o6·:·R[T·..T·]
125 ········1···3</code></pre>125 ········1···3</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i7·:·H·=·HH(KR)128 <td>··············<pre><code·class="language-macaulay2">i7·:·H·=·HH(KR)
129 ·--·used·0.647953s·(cpu);·0.0882768s·(thread);·0s·(gc)129 ·--·used·0.531456s·(cpu);·0.0862601s·(thread);·0s·(gc)
130 Finding·easy·relations···········:·130 Finding·easy·relations···········:·
131 o7·=·H131 o7·=·H
  
132 o7·:·PolynomialRing,·3·skew·commutative·variable(s)</code></pre>132 o7·:·PolynomialRing,·3·skew·commutative·variable(s)</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i8·:·homologyClass(KR,z1*z2)135 <td>··············<pre><code·class="language-macaulay2">i8·:·homologyClass(KR,z1*z2)
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54 ······254 ······2
55 o6·=·y·T55 o6·=·y·T
56 ········256 ········2
  
57 o6·:·R[T·..T·]57 o6·:·R[T·..T·]
58 ········1···358 ········1···3
59 i7·:·H·=·HH(KR)59 i7·:·H·=·HH(KR)
60 ·--·used·0.647953s·(cpu);·0.0882768s·(thread);·0s·(gc)60 ·--·used·0.531456s·(cpu);·0.0862601s·(thread);·0s·(gc)
61 Finding·easy·relations···········:61 Finding·easy·relations···········:
62 o7·=·H62 o7·=·H
  
63 o7·:·PolynomialRing,·3·skew·commutative·variable(s)63 o7·:·PolynomialRing,·3·skew·commutative·variable(s)
64 i8·:·homologyClass(KR,z1*z2)64 i8·:·homologyClass(KR,z1*z2)
  
65 o8·=·X·X65 o8·=·X·X
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119 ··································119 ··································
120 ·····0······1······2······3······4120 ·····0······1······2······3······4
  
121 o5·:·ChainComplex</code></pre>121 o5·:·ChainComplex</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i6·:·HKR·=·HH(KR)124 <td>··············<pre><code·class="language-macaulay2">i6·:·HKR·=·HH(KR)
125 ·--·used·1.5631s·(cpu);·0.211641s·(thread);·0s·(gc)125 ·--·used·1.22896s·(cpu);·0.248658s·(thread);·0s·(gc)
126 Finding·easy·relations···········:·126 Finding·easy·relations···········:·
127 o6·=·HKR127 o6·=·HKR
  
128 o6·:·QuotientRing</code></pre>128 o6·:·QuotientRing</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ········</table>130 ········</table>
131 ········<div>131 ········<div>
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55 ······1······4······6······4······155 ······1······4······6······4······1
56 o5·=·R··<--·R··<--·R··<--·R··<--·R56 o5·=·R··<--·R··<--·R··<--·R··<--·R
  
57 ·····0······1······2······3······457 ·····0······1······2······3······4
  
58 o5·:·ChainComplex58 o5·:·ChainComplex
59 i6·:·HKR·=·HH(KR)59 i6·:·HKR·=·HH(KR)
60 ·--·used·1.5631s·(cpu);·0.211641s·(thread);·0s·(gc)60 ·--·used·1.22896s·(cpu);·0.248658s·(thread);·0s·(gc)
61 Finding·easy·relations···········:61 Finding·easy·relations···········:
62 o6·=·HKR62 o6·=·HKR
  
63 o6·:·QuotientRing63 o6·:·QuotientRing
64 The·following·is·the·graded·canonical·module·of·R:64 The·following·is·the·graded·canonical·module·of·R:
65 i7·:·degList·=·first·entries·vars·Q·/·degree·/·first65 i7·:·degList·=·first·entries·vars·Q·/·degree·/·first
  
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176 ········</div>176 ········</div>
177 ········<div>177 ········<div>
178 ··········<p>Given·cycles·z1,z2,z3·such·that·z1*z2·and·z2*z3·are·boundaries,·the·Massey·triple·product·of·the·homology·classes·represented·by·z1,z2·and·z3·is·the·homology·class·of·lift12*z3·+·z1*lift23.··To·see·this,·we·compute·and·check:</p>178 ··········<p>Given·cycles·z1,z2,z3·such·that·z1*z2·and·z2*z3·are·boundaries,·the·Massey·triple·product·of·the·homology·classes·represented·by·z1,z2·and·z3·is·the·homology·class·of·lift12*z3·+·z1*lift23.··To·see·this,·we·compute·and·check:</p>
179 ········</div>179 ········</div>
180 ········<table·class="examples">180 ········<table·class="examples">
181 ··········<tr>181 ··········<tr>
182 <td>··············<pre><code·class="language-macaulay2">i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)182 <td>··············<pre><code·class="language-macaulay2">i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)
183 ·--·used·2.87754s·(cpu);·0.732378s·(thread);·0s·(gc)183 ·--·used·2.46677s·(cpu);·0.739171s·(thread);·0s·(gc)
184 Finding·easy·relations···········:·184 Finding·easy·relations···········:·
185 ·············2185 ·············2
186 o10·=·x·x·y·z·T·T·T·T186 o10·=·x·x·y·z·T·T·T·T
187 ·······1·2·2···2·3·4·5187 ·······1·2·2···2·3·4·5
  
188 o10·:·R[T·..T·]188 o10·:·R[T·..T·]
189 ·········1···5</code></pre>189 ·········1···5</code></pre>
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91 Note·that·the·first·return·value·of·_\x8g_\x8e_\x8t_\x8B_\x8o_\x8u_\x8n_\x8d_\x8a_\x8r_\x8y_\x8P_\x8r_\x8e_\x8i_\x8m_\x8a_\x8g_\x8e·indicates·that·the91 Note·that·the·first·return·value·of·_\x8g_\x8e_\x8t_\x8B_\x8o_\x8u_\x8n_\x8d_\x8a_\x8r_\x8y_\x8P_\x8r_\x8e_\x8i_\x8m_\x8a_\x8g_\x8e·indicates·that·the
92 inputs·are·indeed·boundaries,·and·the·second·value·is·the·lift·of·the·boundary92 inputs·are·indeed·boundaries,·and·the·second·value·is·the·lift·of·the·boundary
93 along·the·differential.93 along·the·differential.
94 Given·cycles·z1,z2,z3·such·that·z1*z2·and·z2*z3·are·boundaries,·the·Massey94 Given·cycles·z1,z2,z3·such·that·z1*z2·and·z2*z3·are·boundaries,·the·Massey
95 triple·product·of·the·homology·classes·represented·by·z1,z2·and·z3·is·the95 triple·product·of·the·homology·classes·represented·by·z1,z2·and·z3·is·the
96 homology·class·of·lift12*z3·+·z1*lift23.·To·see·this,·we·compute·and·check:96 homology·class·of·lift12*z3·+·z1*lift23.·To·see·this,·we·compute·and·check:
97 i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)97 i10·:·z123·=·masseyTripleProduct(KR,z1,z2,z3)
98 ·--·used·2.87754s·(cpu);·0.732378s·(thread);·0s·(gc)98 ·--·used·2.46677s·(cpu);·0.739171s·(thread);·0s·(gc)
99 Finding·easy·relations···········:99 Finding·easy·relations···········:
100 ·············2100 ·············2
101 o10·=·x·x·y·z·T·T·T·T101 o10·=·x·x·y·z·T·T·T·T
102 ·······1·2·2···2·3·4·5102 ·······1·2·2···2·3·4·5
  
103 o10·:·R[T·..T·]103 o10·:·R[T·..T·]
104 ·········1···5104 ·········1···5
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114 ······Differential·=>·{t·,·t·,·t·,·t·}114 ······Differential·=>·{t·,·t·,·t·,·t·}
115 ························1···2···3···4115 ························1···2···3···4
  
116 o4·:·DGAlgebra</code></pre>116 o4·:·DGAlgebra</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i5·:·H·=·HH(KR)119 <td>··············<pre><code·class="language-macaulay2">i5·:·H·=·HH(KR)
120 ·--·used·2.10443s·(cpu);·0.351445s·(thread);·0s·(gc)120 ·--·used·1.95528s·(cpu);·0.384272s·(thread);·0s·(gc)
121 Finding·easy·relations···········:·121 Finding·easy·relations···········:·
122 o5·=·H122 o5·=·H
  
123 o5·:·QuotientRing</code></pre>123 o5·:·QuotientRing</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);126 <td>··············<pre><code·class="language-macaulay2">i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);
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50 ······Underlying·algebra·=>·R[T·..T·]50 ······Underlying·algebra·=>·R[T·..T·]
51 ·······························1···451 ·······························1···4
52 ······Differential·=>·{t·,·t·,·t·,·t·}52 ······Differential·=>·{t·,·t·,·t·,·t·}
53 ························1···2···3···453 ························1···2···3···4
  
54 o4·:·DGAlgebra54 o4·:·DGAlgebra
55 i5·:·H·=·HH(KR)55 i5·:·H·=·HH(KR)
56 ·--·used·2.10443s·(cpu);·0.351445s·(thread);·0s·(gc)56 ·--·used·1.95528s·(cpu);·0.384272s·(thread);·0s·(gc)
57 Finding·easy·relations···········:57 Finding·easy·relations···········:
58 o5·=·H58 o5·=·H
  
59 o5·:·QuotientRing59 o5·:·QuotientRing
60 i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);60 i6·:·masseys·=·masseyTripleProduct(KR,1,1,1);
  
61 ··············5·······34361 ··············5·······343
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95 o3·=·S95 o3·=·S
  
96 o3·:·QuotientRing</code></pre>96 o3·:·QuotientRing</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)99 <td>··············<pre><code·class="language-macaulay2">i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
100 ·--·used·1.45046s·(cpu);·0.466287s·(thread);·0s·(gc)100 ·--·used·2.56234s·(cpu);·0.709352s·(thread);·0s·(gc)
101 Finding·easy·relations···········:·101 Finding·easy·relations···········:·
102 o4·=·HB102 o4·=·HB
  
103 o4·:·QuotientRing</code></pre>103 o4·:·QuotientRing</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·numgens·HB106 <td>··············<pre><code·class="language-macaulay2">i5·:·numgens·HB
⊟
419 B
html2text {}
    
Offset 28, 15 lines modifiedOffset 28, 15 lines modified
28 i2·:·M·=·coker·matrix·{{a^3*b^3*c^3*d^3}};28 i2·:·M·=·coker·matrix·{{a^3*b^3*c^3*d^3}};
29 i3·:·S·=·R/ideal{a^3*b^3*c^3*d^3}29 i3·:·S·=·R/ideal{a^3*b^3*c^3*d^3}
  
30 o3·=·S30 o3·=·S
  
31 o3·:·QuotientRing31 o3·:·QuotientRing
32 i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)32 i4·:·HB·=·torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
33 ·--·used·1.45046s·(cpu);·0.466287s·(thread);·0s·(gc)33 ·--·used·2.56234s·(cpu);·0.709352s·(thread);·0s·(gc)
34 Finding·easy·relations···········:34 Finding·easy·relations···········:
35 o4·=·HB35 o4·=·HB
  
36 o4·:·QuotientRing36 o4·:·QuotientRing
37 i5·:·numgens·HB37 i5·:·numgens·HB
  
38 o5·=·3538 o5·=·35
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775 B
./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/Depth/dump/rawdocumentation.dump
    
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8 U2VlZA==8 U2VlZA==
9 #:len=2199 #:len=219
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTZWVkIiwiU2VlZCIsIkRlcHRoIn0sIFByaW1hcnlU11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTZWVkIiwiU2VlZCIsIkRlcHRoIn0sIFByaW1hcnlU
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./usr/share/doc/Macaulay2/DeterminantalRepresentations/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=347 #:len=34
8 Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==8 Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==
9 #:len=3679 #:len=367
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp
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./usr/share/doc/Macaulay2/DiffAlg/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=107 #:len=10
8 bGluZWFyQ29tYg==8 bGluZWFyQ29tYg==
9 #:len=23909 #:len=2390
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp
11 b24gb2YgZWxlbWVudHMiLCAibGluZW51bSIgPT4gODY0LCBJbnB1dHMgPT4ge1NQQU57VFR7Ikwi11 b24gb2YgZWxlbWVudHMiLCAibGluZW51bSIgPT4gODY0LCBJbnB1dHMgPT4ge1NQQU57VFR7Ikwi
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./usr/share/doc/Macaulay2/Divisor/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=197 #:len=19
8 Zmxvb3IoUldlaWxEaXZpc29yKQ==8 Zmxvb3IoUldlaWxEaXZpc29yKQ==
9 #:len=2609 #:len=260
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Nywgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Nywgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl
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510 B
./usr/share/doc/Macaulay2/Divisor/example-output/___Basic__Divisor.out
    
Offset 16, 18 lines modifiedOffset 16, 18 lines modified
16 ···············cache·=>·CacheTable{...1...}16 ···············cache·=>·CacheTable{...1...}
17 ···············ring·=>·R17 ···············ring·=>·R
  
18 o3·:·HashTable18 o3·:·HashTable
  
19 i4·:·(2/3)*D19 i4·:·(2/3)*D
  
20 o4·=·2/3*Div(x)·+·2*Div(z)·+·4/3*Div(y)20 o4·=·2/3*Div(x)·+·4/3*Div(y)·+·2*Div(z)
  
21 o4·:·QWeilDivisor·on·R21 o4·:·QWeilDivisor·on·R
  
22 i5·:·0.6*D22 i5·:·0.6*D
  
23 o5·=·.6*Div(x)·+·1.8*Div(z)·+·1.2*Div(y)23 o5·=·.6*Div(x)·+·1.2*Div(y)·+·1.8*Div(z)
  
24 o5·:·RWeilDivisor·on·R24 o5·:·RWeilDivisor·on·R
  
25 i6·:·25 i6·:·
⊟
880 B
./usr/share/doc/Macaulay2/Divisor/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out
    
Offset 1, 32 lines modifiedOffset 1, 32 lines modified
1 --·-*-·M2-comint·-*-·hash:·110850518862001773291 --·-*-·M2-comint·-*-·hash:·11085051886200177329
  
2 i1·:·R·=·QQ[x,·y,·z];2 i1·:·R·=·QQ[x,·y,·z];
  
3 i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})3 i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})
  
4 o2·=·3*Div(y)·+·2*Div(z)·+·Div(x)4 o2·=·Div(x)·+·3*Div(y)·+·2*Div(z)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})6 i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})
  
7 o3·=·-2*Div(z)·+·3*Div(y)·+·-5*Div(x)7 o3·=·-5*Div(x)·+·-2*Div(z)·+·3*Div(y)
  
8 o3·:·WeilDivisor·on·R8 o3·:·WeilDivisor·on·R
  
9 i4·:·D1·+·D29 i4·:·D1·+·D2
  
10 o4·=·6*Div(y)·+·-4*Div(x)10 o4·=·-4*Div(x)·+·6*Div(y)
  
11 o4·:·WeilDivisor·on·R11 o4·:·WeilDivisor·on·R
  
12 i5·:·D1·-·D212 i5·:·D1·-·D2
  
13 o5·=·4*Div(z)·+·6*Div(x)13 o5·=·6*Div(x)·+·4*Div(z)
  
14 o5·:·WeilDivisor·on·R14 o5·:·WeilDivisor·on·R
  
15 i6·:·R·=·QQ[x,y];15 i6·:·R·=·QQ[x,y];
  
16 i7·:·D1·=·divisor({3,·1},·{ideal(x),·ideal(y)})16 i7·:·D1·=·divisor({3,·1},·{ideal(x),·ideal(y)})
  
⊟
1.09 KB
./usr/share/doc/Macaulay2/Divisor/example-output/___Number_sp_st_sp__Basic__Divisor.out
    
Offset 10, 27 lines modifiedOffset 10, 27 lines modified
  
10 o3·=·1/2*Div(x)·+·-5/3*Div(y)10 o3·=·1/2*Div(x)·+·-5/3*Div(y)
  
11 o3·:·QWeilDivisor·on·R11 o3·:·QWeilDivisor·on·R
  
12 i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},·CoefficientType=>RR)12 i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},·CoefficientType=>RR)
  
13 o4·=·1.5*Div(x)·+·0*Div(y)·+·-3.2*Div(-y^3+x^2)13 o4·=·-3.2*Div(-y^3+x^2)·+·1.5*Div(x)·+·0*Div(y)
  
14 o4·:·RWeilDivisor·on·R14 o4·:·RWeilDivisor·on·R
  
15 i5·:·8*D15 i5·:·8*D
  
16 o5·=·8*Div(y)·+·16*Div(x)·+·-8*Div(x+y)16 o5·=·-8*Div(x+y)·+·8*Div(y)·+·16*Div(x)
  
17 o5·:·WeilDivisor·on·R17 o5·:·WeilDivisor·on·R
  
18 i6·:·(-2/3)*D18 i6·:·(-2/3)*D
  
19 o6·=·-2/3*Div(y)·+·-4/3*Div(x)·+·2/3*Div(x+y)19 o6·=·2/3*Div(x+y)·+·-2/3*Div(y)·+·-4/3*Div(x)
  
20 o6·:·QWeilDivisor·on·R20 o6·:·QWeilDivisor·on·R
  
21 i7·:·0.0*D21 i7·:·0.0*D
  
22 o7·=·0,·the·zero·divisor22 o7·=·0,·the·zero·divisor
  
Offset 46, 18 lines modifiedOffset 46, 18 lines modified
  
46 o9·=·2.35667*Div(y)·+·-.707*Div(x)46 o9·=·2.35667*Div(y)·+·-.707*Div(x)
  
47 o9·:·RWeilDivisor·on·R47 o9·:·RWeilDivisor·on·R
  
48 i10·:·6*F48 i10·:·6*F
  
49 o10·=·9*Div(x)·+·-19.2*Div(-y^3+x^2)49 o10·=·-19.2*Div(-y^3+x^2)·+·9*Div(x)
  
50 o10·:·RWeilDivisor·on·R50 o10·:·RWeilDivisor·on·R
  
51 i11·:·(-3/2)*F51 i11·:·(-3/2)*F
  
52 o11·=·-2.25*Div(x)·+·4.8*Div(-y^3+x^2)52 o11·=·4.8*Div(-y^3+x^2)·+·-2.25*Div(x)
  
53 o11·:·RWeilDivisor·on·R53 o11·:·RWeilDivisor·on·R
  
54 i12·:·54 i12·:·
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457 B
./usr/share/doc/Macaulay2/Divisor/example-output/___O__O_sp__R__Weil__Divisor.out
    
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90 ······························290 ······························2
91 o15·:·R-module,·submodule·of·R91 o15·:·R-module,·submodule·of·R
  
92 i16·:·R·=·ZZ/11[x,y];92 i16·:·R·=·ZZ/11[x,y];
  
93 i17·:·D·=·divisor(x*y/(x+y))93 i17·:·D·=·divisor(x*y/(x+y))
  
94 o17·=·Div(x)·+·Div(y)·+·-Div(x+y)94 o17·=·-Div(x+y)·+·Div(x)·+·Div(y)
  
95 o17·:·WeilDivisor·on·R95 o17·:·WeilDivisor·on·R
  
96 i18·:·divisor(OO(D))96 i18·:·divisor(OO(D))
  
97 o18·=·0,·the·zero·divisor97 o18·=·0,·the·zero·divisor
  
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452 B
./usr/share/doc/Macaulay2/Divisor/example-output/_apply__To__Coefficients.out
    
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1 --·-*-·M2-comint·-*-·hash:·149379346520408128891 --·-*-·M2-comint·-*-·hash:·14937934652040812889
  
2 i1·:·R·=·QQ[x,·y,·z];2 i1·:·R·=·QQ[x,·y,·z];
  
3 i2·:·D·=·divisor(x*y^2/z)3 i2·:·D·=·divisor(x*y^2/z)
  
4 o2·=·Div(x)·+·-Div(z)·+·2*Div(y)4 o2·=·Div(x)·+·2*Div(y)·+·-Div(z)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·applyToCoefficients(D,·u->5*u)6 i3·:·applyToCoefficients(D,·u->5*u)
  
7 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)7 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)
  
⊟
855 B
./usr/share/doc/Macaulay2/Divisor/example-output/_ceiling_lp__R__Weil__Divisor_rp.out
    
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1 --·-*-·M2-comint·-*-·hash:·9921330779889496401 --·-*-·M2-comint·-*-·hash:·992133077988949640
  
2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);
  
3 i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)3 i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
4 o2·=·4/3*Div(y,·z)·+·1/2*Div(x,·z)4 o2·=·1/2*Div(x,·z)·+·4/3*Div(y,·z)
  
5 o2·:·QWeilDivisor·on·R5 o2·:·QWeilDivisor·on·R
  
6 i3·:·ceiling(·D·)6 i3·:·ceiling(·D·)
  
7 o3·=·2*Div(y,·z)·+·Div(x,·z)7 o3·=·Div(x,·z)·+·2*Div(y,·z)
  
8 o3·:·WeilDivisor·on·R8 o3·:·WeilDivisor·on·R
  
9 i4·:·floor(·D·)9 i4·:·floor(·D·)
  
10 o4·=·Div(y,·z)10 o4·=·Div(y,·z)
  
11 o4·:·WeilDivisor·on·R11 o4·:·WeilDivisor·on·R
  
12 i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)12 i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)
  
13 o5·=·-.7*Div(y,·z)·+·.3*Div(x,·z)13 o5·=·.3*Div(x,·z)·+·-.7*Div(y,·z)
  
14 o5·:·RWeilDivisor·on·R14 o5·:·RWeilDivisor·on·R
  
15 i6·:·ceiling(·E·)15 i6·:·ceiling(·E·)
  
16 o6·=·Div(x,·z)16 o6·=·Div(x,·z)
  
⊟
496 B
./usr/share/doc/Macaulay2/Divisor/example-output/_clean__Support.out
    
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1 --·-*-·M2-comint·-*-·hash:·28038184616617594591 --·-*-·M2-comint·-*-·hash:·2803818461661759459
  
2 i1·:·R·=·QQ[x,y,z];2 i1·:·R·=·QQ[x,y,z];
  
3 i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})3 i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})
  
4 o2·=·Div(x)·+·0*Div(y)·+·-2*Div(z)4 o2·=·0*Div(y)·+·-2*Div(z)·+·Div(x)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·cleanSupport(D)6 i3·:·cleanSupport(D)
  
7 o3·=·Div(x)·+·-2*Div(z)7 o3·=·-2*Div(z)·+·Div(x)
  
8 o3·:·WeilDivisor·on·R8 o3·:·WeilDivisor·on·R
  
9 i4·:·9 i4·:·
⊟
921 B
./usr/share/doc/Macaulay2/Divisor/example-output/_divisor.out
    
Offset 60, 29 lines modifiedOffset 60, 29 lines modified
  
60 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)60 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)
  
61 o14·:·WeilDivisor·on·A61 o14·:·WeilDivisor·on·A
  
62 i15·:·E·=·divisor(y2z)62 i15·:·E·=·divisor(y2z)
  
63 o15·=·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)·+·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)63 o15·=·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)·+·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)
  
64 o15·:·WeilDivisor·on·A64 o15·:·WeilDivisor·on·A
  
65 i16·:·R·=·ZZ/7[x,y];65 i16·:·R·=·ZZ/7[x,y];
  
66 i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)66 i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)
  
67 o17·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)67 o17·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
68 o17·:·QWeilDivisor·on·R68 o17·:·QWeilDivisor·on·R
  
69 i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)69 i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)
  
70 o18·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)70 o18·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
71 o18·:·QWeilDivisor·on·R71 o18·:·QWeilDivisor·on·R
  
72 i19·:·R·=·ZZ/11[x,y,u,v]/ideal(x*y-u*v);72 i19·:·R·=·ZZ/11[x,y,u,v]/ideal(x*y-u*v);
  
73 i20·:·D·=·divisor({1.1,·-3.14159},·{ideal(x,u),·ideal(x,·v)},·CoefficientType=>RR)73 i20·:·D·=·divisor({1.1,·-3.14159},·{ideal(x,u),·ideal(x,·v)},·CoefficientType=>RR)
  
⊟
1.62 KB
./usr/share/doc/Macaulay2/Divisor/example-output/_dualize.out
    
Offset 44, 51 lines modifiedOffset 44, 51 lines modified
44 i10·:·J·=·m^9;44 i10·:·J·=·m^9;
  
45 o10·:·Ideal·of·R45 o10·:·Ideal·of·R
  
46 i11·:·M·=·J*R^1;46 i11·:·M·=·J*R^1;
  
47 i12·:·time·dualize(J,·Strategy=>IdealStrategy);47 i12·:·time·dualize(J,·Strategy=>IdealStrategy);
48 ·--·used·0.0376684s·(cpu);·0.0407424s·(thread);·0s·(gc)48 ·--·used·0.0503945s·(cpu);·0.0496727s·(thread);·0s·(gc)
  
49 o12·:·Ideal·of·R49 o12·:·Ideal·of·R
  
50 i13·:·time·dualize(J,·Strategy=>ModuleStrategy);50 i13·:·time·dualize(J,·Strategy=>ModuleStrategy);
51 ·--·used·0.383657s·(cpu);·0.384438s·(thread);·0s·(gc)51 ·--·used·0.465893s·(cpu);·0.466334s·(thread);·0s·(gc)
  
52 o13·:·Ideal·of·R52 o13·:·Ideal·of·R
  
53 i14·:·time·dualize(M,·Strategy=>IdealStrategy);53 i14·:·time·dualize(M,·Strategy=>IdealStrategy);
54 ·--·used·0.462133s·(cpu);·0.43286s·(thread);·0s·(gc)54 ·--·used·0.661115s·(cpu);·0.613389s·(thread);·0s·(gc)
  
55 i15·:·time·dualize(M,·Strategy=>ModuleStrategy);55 i15·:·time·dualize(M,·Strategy=>ModuleStrategy);
56 ·--·used·0.00234692s·(cpu);·0.00241164s·(thread);·0s·(gc)56 ·--·used·0.00315628s·(cpu);·0.00331601s·(thread);·0s·(gc)
  
57 i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);57 i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);
58 ·--·used·9.1117e-05s·(cpu);·0.00231316s·(thread);·0s·(gc)58 ·--·used·6.1896e-05s·(cpu);·0.00256199s·(thread);·0s·(gc)
  
59 o16·:·Ideal·of·R59 o16·:·Ideal·of·R
  
60 i17·:·R·=·ZZ/7[x,y,u,v]/ideal(x*y-u*v);60 i17·:·R·=·ZZ/7[x,y,u,v]/ideal(x*y-u*v);
  
61 i18·:·I·=·ideal(x,u);61 i18·:·I·=·ideal(x,u);
  
62 o18·:·Ideal·of·R62 o18·:·Ideal·of·R
  
63 i19·:·J·=·I^15;63 i19·:·J·=·I^15;
  
64 o19·:·Ideal·of·R64 o19·:·Ideal·of·R
  
65 i20·:·time·dualize(J,·Strategy=>IdealStrategy);65 i20·:·time·dualize(J,·Strategy=>IdealStrategy);
66 ·--·used·0.124346s·(cpu);·0.0848201s·(thread);·0s·(gc)66 ·--·used·0.154716s·(cpu);·0.0985429s·(thread);·0s·(gc)
  
67 o20·:·Ideal·of·R67 o20·:·Ideal·of·R
  
68 i21·:·time·dualize(J,·Strategy=>ModuleStrategy);68 i21·:·time·dualize(J,·Strategy=>ModuleStrategy);
69 ·--·used·0.00452869s·(cpu);·0.00463597s·(thread);·0s·(gc)69 ·--·used·0.00398933s·(cpu);·0.00689616s·(thread);·0s·(gc)
  
70 o21·:·Ideal·of·R70 o21·:·Ideal·of·R
  
71 i22·:·R·=·QQ[x,y]/ideal(x*y);71 i22·:·R·=·QQ[x,y]/ideal(x*y);
  
72 i23·:·J·=·ideal(x,y);72 i23·:·J·=·ideal(x,y);
  
⊟
646 B
./usr/share/doc/Macaulay2/Divisor/example-output/_get__Prime__Divisors.out
    
Offset 1, 23 lines modifiedOffset 1, 23 lines modified
1 --·-*-·M2-comint·-*-·hash:·68160756605303982781 --·-*-·M2-comint·-*-·hash:·6816075660530398278
  
2 i1·:·R·=·QQ[x,·y,·z];2 i1·:·R·=·QQ[x,·y,·z];
  
3 i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})3 i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})
  
4 o2·=·2*Div(y)·+·0*Div(x^2+z)·+·-8*Div(x)4 o2·=·0*Div(x^2+z)·+·-8*Div(x)·+·2*Div(y)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·getPrimeDivisors(·D·)6 i3·:·getPrimeDivisors(·D·)
  
7 o3·=·{Div(y),·Div(x^2+z),·Div(x)}7 o3·=·{Div(x^2+z),·Div(x),·Div(y)}
  
8 o3·:·List8 o3·:·List
  
9 i4·:·getPrimeDivisors(·cleanSupport·D·)9 i4·:·getPrimeDivisors(·cleanSupport·D·)
  
10 o4·=·{Div(y),·Div(x)}10 o4·=·{Div(x),·Div(y)}
  
11 o4·:·List11 o4·:·List
  
12 i5·:·12 i5·:·
⊟
863 B
./usr/share/doc/Macaulay2/Divisor/example-output/_is__Cartier.out
    
Offset 12, 27 lines modifiedOffset 12, 27 lines modified
  
12 o3·=·false12 o3·=·false
  
13 i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);13 i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
  
14 i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})14 i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
15 o5·=·2*Div(y,·z)·+·Div(x,·z)15 o5·=·Div(x,·z)·+·2*Div(y,·z)
  
16 o5·:·WeilDivisor·on·R16 o5·:·WeilDivisor·on·R
  
17 i6·:·isCartier(·D·)17 i6·:·isCartier(·D·)
  
18 o6·=·false18 o6·=·false
  
19 i7·:·R·=·QQ[x,·y,·z];19 i7·:·R·=·QQ[x,·y,·z];
  
20 i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})20 i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})
  
21 o8·=·Div(x)·+·2*Div(y)21 o8·=·2*Div(y)·+·Div(x)
  
22 o8·:·WeilDivisor·on·R22 o8·:·WeilDivisor·on·R
  
23 i9·:·isCartier(·D·)23 i9·:·isCartier(·D·)
  
24 o9·=·true24 o9·=·true
  
Offset 48, 15 lines modifiedOffset 48, 15 lines modified
  
48 o12·=·true48 o12·=·true
  
49 i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);49 i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
  
50 i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})50 i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
51 o14·=·2*Div(y,·z)·+·Div(x,·z)51 o14·=·Div(x,·z)·+·2*Div(y,·z)
  
52 o14·:·WeilDivisor·on·R52 o14·:·WeilDivisor·on·R
  
53 i15·:·isCartier(D,·IsGraded·=>·true)53 i15·:·isCartier(D,·IsGraded·=>·true)
  
54 o15·=·true54 o15·=·true
  
⊟
1010 B
./usr/share/doc/Macaulay2/Divisor/example-output/_is__Linear__Equivalent.out
    
Offset 1, 38 lines modifiedOffset 1, 38 lines modified
1 --·-*-·M2-comint·-*-·hash:·60191193470828113961 --·-*-·M2-comint·-*-·hash:·6019119347082811396
  
2 i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);2 i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);
  
3 i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})3 i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
4 o2·=·3*Div(x,·z)·+·8*Div(y,·z)4 o2·=·8*Div(y,·z)·+·3*Div(x,·z)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})6 i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
7 o3·=·Div(x,·z)·+·8*Div(y,·z)7 o3·=·8*Div(y,·z)·+·Div(x,·z)
  
8 o3·:·WeilDivisor·on·R8 o3·:·WeilDivisor·on·R
  
9 i4·:·isLinearEquivalent(D1,·D2)9 i4·:·isLinearEquivalent(D1,·D2)
  
10 o4·=·true10 o4·=·true
  
11 i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);11 i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);
  
12 i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})12 i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
13 o6·=·8*Div(y,·z)·+·3*Div(x,·z)13 o6·=·3*Div(x,·z)·+·8*Div(y,·z)
  
14 o6·:·WeilDivisor·on·R14 o6·:·WeilDivisor·on·R
  
15 i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})15 i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
16 o7·=·8*Div(y,·z)·+·Div(x,·z)16 o7·=·Div(x,·z)·+·8*Div(y,·z)
  
17 o7·:·WeilDivisor·on·R17 o7·:·WeilDivisor·on·R
  
18 i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)18 i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)
  
19 o8·=·false19 o8·=·false
  
⊟
1.19 KB
./usr/share/doc/Macaulay2/Divisor/example-output/_is__Q__Linear__Equivalent.out
    
Offset 1, 20 lines modifiedOffset 1, 20 lines modified
1 --·-*-·M2-comint·-*-·hash:·139209593881088032161 --·-*-·M2-comint·-*-·hash:·13920959388108803216
  
2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
  
3 i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)3 i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
4 o2·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)4 o2·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)
  
5 o2·:·QWeilDivisor·on·R5 o2·:·QWeilDivisor·on·R
  
6 i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)6 i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)
  
7 o3·=·3/4*Div(y,·z)·+·5/2*Div(x,·z)7 o3·=·5/2*Div(x,·z)·+·3/4*Div(y,·z)
  
8 o3·:·QWeilDivisor·on·R8 o3·:·QWeilDivisor·on·R
  
9 i4·:·isQLinearEquivalent(10,·D,·E)9 i4·:·isQLinearEquivalent(10,·D,·E)
  
10 o4·=·true10 o4·=·true
  
Offset 36, 21 lines modifiedOffset 36, 21 lines modified
  
36 o9·=·true36 o9·=·true
  
37 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);37 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
  
38 i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)38 i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
39 o11·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)39 o11·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)
  
40 o11·:·QWeilDivisor·on·R40 o11·:·QWeilDivisor·on·R
  
41 i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)41 i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)
  
42 o12·=·-1/4*Div(x,·z)·+·3/2*Div(y,·z)42 o12·=·3/2*Div(y,·z)·+·-1/4*Div(x,·z)
  
43 o12·:·QWeilDivisor·on·R43 o12·:·QWeilDivisor·on·R
  
44 i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)44 i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)
  
45 o13·=·true45 o13·=·true
  
⊟
515 B
./usr/share/doc/Macaulay2/Divisor/example-output/_is__Reduced.out
    
Offset 1, 20 lines modifiedOffset 1, 20 lines modified
1 --·-*-·M2-comint·-*-·hash:·62633715804780901721 --·-*-·M2-comint·-*-·hash:·6263371580478090172
  
2 i1·:·R·=·QQ[x,·y,·z];2 i1·:·R·=·QQ[x,·y,·z];
  
3 i2·:·D1·=·divisor(x^2·*·y^3·*·z)3 i2·:·D1·=·divisor(x^2·*·y^3·*·z)
  
4 o2·=·3*Div(y)·+·Div(z)·+·2*Div(x)4 o2·=·2*Div(x)·+·3*Div(y)·+·Div(z)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·D2·=·divisor(x·*·y·*·z)6 i3·:·D2·=·divisor(x·*·y·*·z)
  
7 o3·=·Div(y)·+·Div(z)·+·Div(x)7 o3·=·Div(x)·+·Div(y)·+·Div(z)
  
8 o3·:·WeilDivisor·on·R8 o3·:·WeilDivisor·on·R
  
9 i4·:·isReduced(·D1·)9 i4·:·isReduced(·D1·)
  
10 o4·=·false10 o4·=·false
  
⊟
949 B
./usr/share/doc/Macaulay2/Divisor/example-output/_is__S__N__C.out
    
Offset 1, 14 lines modifiedOffset 1, 14 lines modified
1 --·-*-·M2-comint·-*-·hash:·23603715183041207181 --·-*-·M2-comint·-*-·hash:·2360371518304120718
  
2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);2 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
  
3 i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})3 i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
4 o2·=·-2*Div(y,·z)·+·Div(x,·z)4 o2·=·Div(x,·z)·+·-2*Div(y,·z)
  
5 o2·:·WeilDivisor·on·R5 o2·:·WeilDivisor·on·R
  
6 i3·:·isSNC(·D·)6 i3·:·isSNC(·D·)
  
7 o3·=·false7 o3·=·false
  
Offset 36, 15 lines modifiedOffset 36, 15 lines modified
  
36 o9·=·true36 o9·=·true
  
37 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);37 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
  
38 i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})38 i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
39 o11·=·Div(x,·z)·+·-2*Div(y,·z)39 o11·=·-2*Div(y,·z)·+·Div(x,·z)
  
40 o11·:·WeilDivisor·on·R40 o11·:·WeilDivisor·on·R
  
41 i12·:·isSNC(·D,·IsGraded·=>·true·)41 i12·:·isSNC(·D,·IsGraded·=>·true·)
  
42 o12·=·true42 o12·=·true
  
Offset 60, 15 lines modifiedOffset 60, 15 lines modified
  
60 o15·=·true60 o15·=·true
  
61 i16·:·R·=·QQ[x,y,z];61 i16·:·R·=·QQ[x,y,z];
  
62 i17·:·D·=·divisor(x*y*(x+y))62 i17·:·D·=·divisor(x*y*(x+y))
  
63 o17·=·Div(x)·+·Div(x+y)·+·Div(y)63 o17·=·Div(x+y)·+·Div(y)·+·Div(x)
  
64 o17·:·WeilDivisor·on·R64 o17·:·WeilDivisor·on·R
  
65 i18·:·isSNC(·D,·IsGraded·=>·true)65 i18·:·isSNC(·D,·IsGraded·=>·true)
  
66 o18·=·false66 o18·=·false
  
⊟
523 B
./usr/share/doc/Macaulay2/Divisor/example-output/_map__To__Projective__Space.out
    
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16 o3·:·RingMap·R·<--·QQ[YY·..YY·]16 o3·:·RingMap·R·<--·QQ[YY·..YY·]
17 ························1····217 ························1····2
  
18 i4·:·R·=·ZZ/7[x,y,z];18 i4·:·R·=·ZZ/7[x,y,z];
  
19 i5·:·D·=·divisor(x*y)19 i5·:·D·=·divisor(x*y)
  
20 o5·=·Div(y)·+·Div(x)20 o5·=·Div(x)·+·Div(y)
  
21 o5·:·WeilDivisor·on·R21 o5·:·WeilDivisor·on·R
  
22 i6·:·mapToProjectiveSpace(D,·Variable=>"Z")22 i6·:·mapToProjectiveSpace(D,·Variable=>"Z")
  
23 ·············ZZ············2·············2········223 ·············ZZ············2·············2········2
24 o6·=·map·(R,·--[Z·..Z·],·{x·,·x*y,·x*z,·y·,·y*z,·z·})24 o6·=·map·(R,·--[Z·..Z·],·{x·,·x*y,·x*z,·y·,·y*z,·z·})
⊟
944 B
./usr/share/doc/Macaulay2/Divisor/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out
    
Offset 6, 21 lines modifiedOffset 6, 21 lines modified
  
6 i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});6 i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});
  
7 o3·:·RingMap·T·<--·R7 o3·:·RingMap·T·<--·R
  
8 i4·:·D·=·divisor(y*z)8 i4·:·D·=·divisor(y*z)
  
9 o4·=·3*Div(z,·y,·x)·+·3*Div(w,·z,·y)9 o4·=·3*Div(w,·z,·y)·+·3*Div(z,·y,·x)
  
10 o4·:·WeilDivisor·on·R10 o4·:·WeilDivisor·on·R
  
11 i5·:·pullback(f,·D,·Strategy=>Primes)11 i5·:·pullback(f,·D,·Strategy=>Primes)
  
12 o5·=·3*Div(a)·+·3*Div(b)12 o5·=·3*Div(b)·+·3*Div(a)
  
13 o5·:·WeilDivisor·on·T13 o5·:·WeilDivisor·on·T
  
14 i6·:·pullback(f,·D,·Strategy=>Sheaves)14 i6·:·pullback(f,·D,·Strategy=>Sheaves)
  
15 o6·=·3*Div(b)·+·3*Div(a)15 o6·=·3*Div(b)·+·3*Div(a)
  
Offset 36, 18 lines modifiedOffset 36, 18 lines modified
  
36 i10·:·D·=·divisor(x*y*(x+y));36 i10·:·D·=·divisor(x*y*(x+y));
  
37 o10·:·WeilDivisor·on·R37 o10·:·WeilDivisor·on·R
  
38 i11·:·D1·=·pullback(f,·D)38 i11·:·D1·=·pullback(f,·D)
  
39 o11·=·Div(a+1)·+·Div(a)·+·3*Div(b)39 o11·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
40 o11·:·WeilDivisor·on·S40 o11·:·WeilDivisor·on·S
  
41 i12·:·f^*·D41 i12·:·f^*·D
  
42 o12·=·Div(a+1)·+·Div(a)·+·3*Div(b)42 o12·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
43 o12·:·WeilDivisor·on·S43 o12·:·WeilDivisor·on·S
  
44 i13·:·44 i13·:·
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3.48 KB
./usr/share/doc/Macaulay2/Divisor/example-output/_reflexify.out
    
Offset 103, 104 lines modifiedOffset 103, 104 lines modified
103 o21·:·Ideal·of·R103 o21·:·Ideal·of·R
  
104 i22·:·J·=·I^21;104 i22·:·J·=·I^21;
  
105 o22·:·Ideal·of·R105 o22·:·Ideal·of·R
  
106 i23·:·time·reflexify(J);106 i23·:·time·reflexify(J);
107 ·--·used·0.179189s·(cpu);·0.147909s·(thread);·0s·(gc)107 ·--·used·0.203596s·(cpu);·0.160719s·(thread);·0s·(gc)
  
108 o23·:·Ideal·of·R108 o23·:·Ideal·of·R
  
109 i24·:·time·reflexify(J*R^1);109 i24·:·time·reflexify(J*R^1);
110 ·--·used·0.309212s·(cpu);·0.273311s·(thread);·0s·(gc)110 ·--·used·0.372987s·(cpu);·0.327706s·(thread);·0s·(gc)
  
111 i25·:·R·=·ZZ/13[x,y,z]/ideal(x^3·+·y^3-z^11*x*y);111 i25·:·R·=·ZZ/13[x,y,z]/ideal(x^3·+·y^3-z^11*x*y);
  
112 i26·:·I·=·ideal(x-4*y,·z);112 i26·:·I·=·ideal(x-4*y,·z);
  
113 o26·:·Ideal·of·R113 o26·:·Ideal·of·R
  
114 i27·:·J·=·I^20;114 i27·:·J·=·I^20;
  
115 o27·:·Ideal·of·R115 o27·:·Ideal·of·R
  
116 i28·:·M·=·J*R^1;116 i28·:·M·=·J*R^1;
  
117 i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)117 i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)
118 ·--·used·0.196918s·(cpu);·0.123476s·(thread);·0s·(gc)118 ·--·used·0.205717s·(cpu);·0.113598s·(thread);·0s·(gc)
  
119 ··············2············2·····9·······9···11119 ··············2············2·····9·······9···11
120 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)120 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
121 o29·:·Ideal·of·R121 o29·:·Ideal·of·R
  
122 i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)122 i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)
123 ·--·used·4.72641s·(cpu);·3.73712s·(thread);·0s·(gc)123 ·--·used·5.42104s·(cpu);·4.32059s·(thread);·0s·(gc)
  
124 ··············2············2·····9·······9···11124 ··············2············2·····9·······9···11
125 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)125 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
126 o30·:·Ideal·of·R126 o30·:·Ideal·of·R
  
127 i31·:·J1·==·J2127 i31·:·J1·==·J2
  
128 o31·=·true128 o31·=·true
  
129 i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);129 i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
130 ·--·used·4.94131s·(cpu);·3.99555s·(thread);·0s·(gc)130 ·--·used·5.42293s·(cpu);·4.27129s·(thread);·0s·(gc)
  
131 i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);131 i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
132 ·--·used·0.428155s·(cpu);·0.313441s·(thread);·0s·(gc)132 ·--·used·0.450809s·(cpu);·0.330221s·(thread);·0s·(gc)
  
133 i34·:·R·=·QQ[x,y,u,v]/ideal(x*y-u*v);133 i34·:·R·=·QQ[x,y,u,v]/ideal(x*y-u*v);
  
134 i35·:·I·=·ideal(x,u);134 i35·:·I·=·ideal(x,u);
  
135 o35·:·Ideal·of·R135 o35·:·Ideal·of·R
  
136 i36·:·J·=·I^20;136 i36·:·J·=·I^20;
  
137 o36·:·Ideal·of·R137 o36·:·Ideal·of·R
  
138 i37·:·M·=·I^20*R^1;138 i37·:·M·=·I^20*R^1;
  
139 i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)139 i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)
140 ·--·used·0.79094s·(cpu);·0.303707s·(thread);·0s·(gc)140 ·--·used·0.924912s·(cpu);·0.306257s·(thread);·0s·(gc)
  
141 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·141 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·
142 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,142 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
143 ······-----------------------------------------------------------------------143 ······-----------------------------------------------------------------------
144 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·144 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·
145 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,145 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
146 ······-----------------------------------------------------------------------146 ······-----------------------------------------------------------------------
147 ·······19····20147 ·······19····20
148 ······x··u,·x··)148 ······x··u,·x··)
  
149 o38·:·Ideal·of·R149 o38·:·Ideal·of·R
  
150 i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)150 i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)
151 ·--·used·0.0145054s·(cpu);·0.0149865s·(thread);·0s·(gc)151 ·--·used·0.0133773s·(cpu);·0.0129823s·(thread);·0s·(gc)
  
152 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·152 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·
153 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,153 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
154 ······-----------------------------------------------------------------------154 ······-----------------------------------------------------------------------
155 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·155 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·
156 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,156 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
157 ······-----------------------------------------------------------------------157 ······-----------------------------------------------------------------------
158 ·······19····20158 ·······19····20
159 ······x··u,·x··)159 ······x··u,·x··)
  
160 o39·:·Ideal·of·R160 o39·:·Ideal·of·R
  
161 i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);161 i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
162 ·--·used·0.194438s·(cpu);·0.0762503s·(thread);·0s·(gc)162 ·--·used·0.223368s·(cpu);·0.0770611s·(thread);·0s·(gc)
  
163 i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);163 i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
164 ·--·used·0.00399938s·(cpu);·0.0057327s·(thread);·0s·(gc)164 ·--·used·0.00763461s·(cpu);·0.00806738s·(thread);·0s·(gc)
  
165 i42·:·R·=·QQ[x,y]/ideal(x*y);165 i42·:·R·=·QQ[x,y]/ideal(x*y);
  
166 i43·:·I·=·ideal(x,y);166 i43·:·I·=·ideal(x,y);
  
167 o43·:·Ideal·of·R167 o43·:·Ideal·of·R
  
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./usr/share/doc/Macaulay2/Divisor/example-output/_reflexive__Power.out
    
Offset 23, 44 lines modifiedOffset 23, 44 lines modified
23 i5·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);23 i5·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);
  
24 i6·:·I·=·ideal(x-z,y-2*z);24 i6·:·I·=·ideal(x-z,y-2*z);
  
25 o6·:·Ideal·of·R25 o6·:·Ideal·of·R
  
26 i7·:·time·J20a·=·reflexivePower(20,·I);26 i7·:·time·J20a·=·reflexivePower(20,·I);
27 ·--·used·0.0731817s·(cpu);·0.0389498s·(thread);·0s·(gc)27 ·--·used·0.0913967s·(cpu);·0.0405707s·(thread);·0s·(gc)
  
28 o7·:·Ideal·of·R28 o7·:·Ideal·of·R
  
29 i8·:·I20·=·I^20;29 i8·:·I20·=·I^20;
  
30 o8·:·Ideal·of·R30 o8·:·Ideal·of·R
  
31 i9·:·time·J20b·=·reflexify(I20);31 i9·:·time·J20b·=·reflexify(I20);
32 ·--·used·0.098693s·(cpu);·0.0988768s·(thread);·0s·(gc)32 ·--·used·0.125986s·(cpu);·0.125611s·(thread);·0s·(gc)
  
33 o9·:·Ideal·of·R33 o9·:·Ideal·of·R
  
34 i10·:·J20a·==·J20b34 i10·:·J20a·==·J20b
  
35 o10·=·true35 o10·=·true
  
36 i11·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);36 i11·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);
  
37 i12·:·I·=·ideal(x-z,y-2*z);37 i12·:·I·=·ideal(x-z,y-2*z);
  
38 o12·:·Ideal·of·R38 o12·:·Ideal·of·R
  
39 i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);39 i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);
40 ·--·used·0.0232514s·(cpu);·0.0251434s·(thread);·0s·(gc)40 ·--·used·0.0346998s·(cpu);·0.0330071s·(thread);·0s·(gc)
  
41 o13·:·Ideal·of·R41 o13·:·Ideal·of·R
  
42 i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);42 i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);
43 ·--·used·0.0986717s·(cpu);·0.0788927s·(thread);·0s·(gc)43 ·--·used·0.137004s·(cpu);·0.097576s·(thread);·0s·(gc)
  
44 o14·:·Ideal·of·R44 o14·:·Ideal·of·R
  
45 i15·:·J1·==·J245 i15·:·J1·==·J2
  
46 o15·=·true46 o15·=·true
  
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1.68 KB
./usr/share/doc/Macaulay2/Divisor/html/___Basic__Divisor.html
    
Offset 80, 22 lines modifiedOffset 80, 22 lines modified
80 ···············ring·=>·R80 ···············ring·=>·R
  
81 o3·:·HashTable</code></pre>81 o3·:·HashTable</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·(2/3)*D84 <td>··············<pre><code·class="language-macaulay2">i4·:·(2/3)*D
  
85 o4·=·2/3*Div(x)·+·2*Div(z)·+·4/3*Div(y)85 o4·=·2/3*Div(x)·+·4/3*Div(y)·+·2*Div(z)
  
86 o4·:·QWeilDivisor·on·R</code></pre>86 o4·:·QWeilDivisor·on·R</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·0.6*D89 <td>··············<pre><code·class="language-macaulay2">i5·:·0.6*D
  
90 o5·=·.6*Div(x)·+·1.8*Div(z)·+·1.2*Div(y)90 o5·=·.6*Div(x)·+·1.2*Div(y)·+·1.8*Div(z)
  
91 o5·:·RWeilDivisor·on·R</code></pre>91 o5·:·RWeilDivisor·on·R</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ········</table>93 ········</table>
94 ······</div>94 ······</div>
95 ······<div·class="waystouse">95 ······<div·class="waystouse">
96 ········<h2>Types·of·<span·class="tt">BasicDivisor</span>:</h2>96 ········<h2>Types·of·<span·class="tt">BasicDivisor</span>:</h2>
⊟
910 B
html2text {}
    
Offset 29, 20 lines modifiedOffset 29, 20 lines modified
29 ···············{z}·=>·{3}29 ···············{z}·=>·{3}
30 ···············cache·=>·CacheTable{...1...}30 ···············cache·=>·CacheTable{...1...}
31 ···············ring·=>·R31 ···············ring·=>·R
  
32 o3·:·HashTable32 o3·:·HashTable
33 i4·:·(2/3)*D33 i4·:·(2/3)*D
  
34 o4·=·2/3*Div(x)·+·2*Div(z)·+·4/3*Div(y)34 o4·=·2/3*Div(x)·+·4/3*Div(y)·+·2*Div(z)
  
35 o4·:·QWeilDivisor·on·R35 o4·:·QWeilDivisor·on·R
36 i5·:·0.6*D36 i5·:·0.6*D
  
37 o5·=·.6*Div(x)·+·1.8*Div(z)·+·1.2*Div(y)37 o5·=·.6*Div(x)·+·1.2*Div(y)·+·1.8*Div(z)
  
38 o5·:·RWeilDivisor·on·R38 o5·:·RWeilDivisor·on·R
39 *\x8**\x8**\x8**\x8**\x8*·T\x8Ty\x8yp\x8pe\x8es\x8s·o\x8of\x8f·B\x8Ba\x8as\x8si\x8ic\x8cD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*39 *\x8**\x8**\x8**\x8**\x8*·T\x8Ty\x8yp\x8pe\x8es\x8s·o\x8of\x8f·B\x8Ba\x8as\x8si\x8ic\x8cD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
40 ····*·RWeilDivisor40 ····*·RWeilDivisor
41 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8an\x8n·o\x8ob\x8bj\x8je\x8ec\x8ct\x8t·o\x8of\x8f·c\x8cl\x8la\x8as\x8ss\x8s·B\x8Ba\x8as\x8si\x8ic\x8cD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8an\x8n·o\x8ob\x8bj\x8je\x8ec\x8ct\x8t·o\x8of\x8f·c\x8cl\x8la\x8as\x8ss\x8s·B\x8Ba\x8as\x8si\x8ic\x8cD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
42 ····*·applyToCoefficients(BasicDivisor,Function)·--·see·_\x8a_\x8p_\x8p_\x8l_\x8y_\x8T_\x8o_\x8C_\x8o_\x8e_\x8f_\x8f_\x8i_\x8c_\x8i_\x8e_\x8n_\x8t_\x8s·-42 ····*·applyToCoefficients(BasicDivisor,Function)·--·see·_\x8a_\x8p_\x8p_\x8l_\x8y_\x8T_\x8o_\x8C_\x8o_\x8e_\x8f_\x8f_\x8i_\x8c_\x8i_\x8e_\x8n_\x8t_\x8s·-
43 ······-·apply·a·function·to·the·coefficients·of·a·divisor43 ······-·apply·a·function·to·the·coefficients·of·a·divisor
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2.32 KB
./usr/share/doc/Macaulay2/Divisor/html/___Basic__Divisor_sp_pl_sp__Basic__Divisor.html
    
Offset 79, 36 lines modifiedOffset 79, 36 lines modified
79 ········<table·class="examples">79 ········<table·class="examples">
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>81 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})84 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})
  
85 o2·=·3*Div(y)·+·2*Div(z)·+·Div(x)85 o2·=·Div(x)·+·3*Div(y)·+·2*Div(z)
  
86 o2·:·WeilDivisor·on·R</code></pre>86 o2·:·WeilDivisor·on·R</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})89 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})
  
90 o3·=·-2*Div(z)·+·3*Div(y)·+·-5*Div(x)90 o3·=·-5*Div(x)·+·-2*Div(z)·+·3*Div(y)
  
91 o3·:·WeilDivisor·on·R</code></pre>91 o3·:·WeilDivisor·on·R</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·D1·+·D294 <td>··············<pre><code·class="language-macaulay2">i4·:·D1·+·D2
  
95 o4·=·6*Div(y)·+·-4*Div(x)95 o4·=·-4*Div(x)·+·6*Div(y)
  
96 o4·:·WeilDivisor·on·R</code></pre>96 o4·:·WeilDivisor·on·R</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·D1·-·D299 <td>··············<pre><code·class="language-macaulay2">i5·:·D1·-·D2
  
100 o5·=·4*Div(z)·+·6*Div(x)100 o5·=·6*Div(x)·+·4*Div(z)
  
101 o5·:·WeilDivisor·on·R</code></pre>101 o5·:·WeilDivisor·on·R</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ········</table>103 ········</table>
104 ········<div>104 ········<div>
105 ··········<p>We·can·also·add·or·subtract·divisors·with·different·coefficients.</p>105 ··········<p>We·can·also·add·or·subtract·divisors·with·different·coefficients.</p>
106 ········</div>106 ········</div>
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17 ····*·Outputs:17 ····*·Outputs:
18 ··········o·an·instance·of·the·type·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,18 ··········o·an·instance·of·the·type·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,
19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
20 We·can·add·or·subtract·two·divisors:20 We·can·add·or·subtract·two·divisors:
21 i1·:·R·=·QQ[x,·y,·z];21 i1·:·R·=·QQ[x,·y,·z];
22 i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})22 i2·:·D1·=·divisor({1,·3,·2},·{ideal(x),·ideal(y),·ideal(z)})
  
23 o2·=·3*Div(y)·+·2*Div(z)·+·Div(x)23 o2·=·Div(x)·+·3*Div(y)·+·2*Div(z)
  
24 o2·:·WeilDivisor·on·R24 o2·:·WeilDivisor·on·R
25 i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})25 i3·:·D2·=·divisor({-2,·3,·-5},·{ideal(z),·ideal(y),·ideal(x)})
  
26 o3·=·-2*Div(z)·+·3*Div(y)·+·-5*Div(x)26 o3·=·-5*Div(x)·+·-2*Div(z)·+·3*Div(y)
  
27 o3·:·WeilDivisor·on·R27 o3·:·WeilDivisor·on·R
28 i4·:·D1·+·D228 i4·:·D1·+·D2
  
29 o4·=·6*Div(y)·+·-4*Div(x)29 o4·=·-4*Div(x)·+·6*Div(y)
  
30 o4·:·WeilDivisor·on·R30 o4·:·WeilDivisor·on·R
31 i5·:·D1·-·D231 i5·:·D1·-·D2
  
32 o5·=·4*Div(z)·+·6*Div(x)32 o5·=·6*Div(x)·+·4*Div(z)
  
33 o5·:·WeilDivisor·on·R33 o5·:·WeilDivisor·on·R
34 We·can·also·add·or·subtract·divisors·with·different·coefficients.34 We·can·also·add·or·subtract·divisors·with·different·coefficients.
35 i6·:·R·=·QQ[x,y];35 i6·:·R·=·QQ[x,y];
36 i7·:·D1·=·divisor({3,·1},·{ideal(x),·ideal(y)})36 i7·:·D1·=·divisor({3,·1},·{ideal(x),·ideal(y)})
  
37 o7·=·3*Div(x)·+·Div(y)37 o7·=·3*Div(x)·+·Div(y)
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89 o3·=·1/2*Div(x)·+·-5/3*Div(y)89 o3·=·1/2*Div(x)·+·-5/3*Div(y)
  
90 o3·:·QWeilDivisor·on·R</code></pre>90 o3·:·QWeilDivisor·on·R</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},·CoefficientType=>RR)93 <td>··············<pre><code·class="language-macaulay2">i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},·CoefficientType=>RR)
  
94 o4·=·1.5*Div(x)·+·0*Div(y)·+·-3.2*Div(-y^3+x^2)94 o4·=·-3.2*Div(-y^3+x^2)·+·1.5*Div(x)·+·0*Div(y)
  
95 o4·:·RWeilDivisor·on·R</code></pre>95 o4·:·RWeilDivisor·on·R</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i5·:·8*D98 <td>··············<pre><code·class="language-macaulay2">i5·:·8*D
  
99 o5·=·8*Div(y)·+·16*Div(x)·+·-8*Div(x+y)99 o5·=·-8*Div(x+y)·+·8*Div(y)·+·16*Div(x)
  
100 o5·:·WeilDivisor·on·R</code></pre>100 o5·:·WeilDivisor·on·R</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i6·:·(-2/3)*D103 <td>··············<pre><code·class="language-macaulay2">i6·:·(-2/3)*D
  
104 o6·=·-2/3*Div(y)·+·-4/3*Div(x)·+·2/3*Div(x+y)104 o6·=·2/3*Div(x+y)·+·-2/3*Div(y)·+·-4/3*Div(x)
  
105 o6·:·QWeilDivisor·on·R</code></pre>105 o6·:·QWeilDivisor·on·R</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i7·:·0.0*D108 <td>··············<pre><code·class="language-macaulay2">i7·:·0.0*D
  
109 o7·=·0,·the·zero·divisor109 o7·=·0,·the·zero·divisor
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131 o9·=·2.35667*Div(y)·+·-.707*Div(x)131 o9·=·2.35667*Div(y)·+·-.707*Div(x)
  
132 o9·:·RWeilDivisor·on·R</code></pre>132 o9·:·RWeilDivisor·on·R</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i10·:·6*F135 <td>··············<pre><code·class="language-macaulay2">i10·:·6*F
  
136 o10·=·9*Div(x)·+·-19.2*Div(-y^3+x^2)136 o10·=·-19.2*Div(-y^3+x^2)·+·9*Div(x)
  
137 o10·:·RWeilDivisor·on·R</code></pre>137 o10·:·RWeilDivisor·on·R</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i11·:·(-3/2)*F140 <td>··············<pre><code·class="language-macaulay2">i11·:·(-3/2)*F
  
141 o11·=·-2.25*Div(x)·+·4.8*Div(-y^3+x^2)141 o11·=·4.8*Div(-y^3+x^2)·+·-2.25*Div(x)
  
142 o11·:·RWeilDivisor·on·R</code></pre>142 o11·:·RWeilDivisor·on·R</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ········</table>144 ········</table>
145 ······</div>145 ······</div>
146 ······<div·class="waystouse">146 ······<div·class="waystouse">
147 ········<h2>Ways·to·use·this·method:</h2>147 ········<h2>Ways·to·use·this·method:</h2>
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23 o3·=·1/2*Div(x)·+·-5/3*Div(y)23 o3·=·1/2*Div(x)·+·-5/3*Div(y)
  
24 o3·:·QWeilDivisor·on·R24 o3·:·QWeilDivisor·on·R
25 i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},25 i4·:·F·=·divisor({1.5,·0,·-3.2},·{ideal(x),·ideal(y),·ideal(x^2-y^3)},
26 CoefficientType=>RR)26 CoefficientType=>RR)
  
27 o4·=·1.5*Div(x)·+·0*Div(y)·+·-3.2*Div(-y^3+x^2)27 o4·=·-3.2*Div(-y^3+x^2)·+·1.5*Div(x)·+·0*Div(y)
  
28 o4·:·RWeilDivisor·on·R28 o4·:·RWeilDivisor·on·R
29 i5·:·8*D29 i5·:·8*D
  
30 o5·=·8*Div(y)·+·16*Div(x)·+·-8*Div(x+y)30 o5·=·-8*Div(x+y)·+·8*Div(y)·+·16*Div(x)
  
31 o5·:·WeilDivisor·on·R31 o5·:·WeilDivisor·on·R
32 i6·:·(-2/3)*D32 i6·:·(-2/3)*D
  
33 o6·=·-2/3*Div(y)·+·-4/3*Div(x)·+·2/3*Div(x+y)33 o6·=·2/3*Div(x+y)·+·-2/3*Div(y)·+·-4/3*Div(x)
  
34 o6·:·QWeilDivisor·on·R34 o6·:·QWeilDivisor·on·R
35 i7·:·0.0*D35 i7·:·0.0*D
  
36 o7·=·0,·the·zero·divisor36 o7·=·0,·the·zero·divisor
  
37 o7·:·RWeilDivisor·on·R37 o7·:·RWeilDivisor·on·R
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53 i9·:·(-1.414)*E53 i9·:·(-1.414)*E
  
54 o9·=·2.35667*Div(y)·+·-.707*Div(x)54 o9·=·2.35667*Div(y)·+·-.707*Div(x)
  
55 o9·:·RWeilDivisor·on·R55 o9·:·RWeilDivisor·on·R
56 i10·:·6*F56 i10·:·6*F
  
57 o10·=·9*Div(x)·+·-19.2*Div(-y^3+x^2)57 o10·=·-19.2*Div(-y^3+x^2)·+·9*Div(x)
  
58 o10·:·RWeilDivisor·on·R58 o10·:·RWeilDivisor·on·R
59 i11·:·(-3/2)*F59 i11·:·(-3/2)*F
  
60 o11·=·-2.25*Div(x)·+·4.8*Div(-y^3+x^2)60 o11·=·4.8*Div(-y^3+x^2)·+·-2.25*Div(x)
  
61 o11·:·RWeilDivisor·on·R61 o11·:·RWeilDivisor·on·R
62 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*62 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
63 ····*·_\x8N_\x8u_\x8m_\x8b_\x8e_\x8r_\x8·_\x8*_\x8·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r·--·multiply·a·divisor·by·a·number63 ····*·_\x8N_\x8u_\x8m_\x8b_\x8e_\x8r_\x8·_\x8*_\x8·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r·--·multiply·a·divisor·by·a·number
64 ····*·QQ·*·RWeilDivisor64 ····*·QQ·*·RWeilDivisor
65 ····*·QQ·*·WeilDivisor65 ····*·QQ·*·WeilDivisor
66 ····*·RR·*·QWeilDivisor66 ····*·RR·*·QWeilDivisor
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195 ········<table·class="examples">195 ········<table·class="examples">
196 ··········<tr>196 ··········<tr>
197 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·ZZ/11[x,y];</code></pre>197 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·ZZ/11[x,y];</code></pre>
198 </td>··········</tr>198 </td>··········</tr>
199 ··········<tr>199 ··········<tr>
200 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor(x*y/(x+y))200 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor(x*y/(x+y))
  
201 o17·=·Div(x)·+·Div(y)·+·-Div(x+y)201 o17·=·-Div(x+y)·+·Div(x)·+·Div(y)
  
202 o17·:·WeilDivisor·on·R</code></pre>202 o17·:·WeilDivisor·on·R</code></pre>
203 </td>··········</tr>203 </td>··········</tr>
204 ··········<tr>204 ··········<tr>
205 <td>··············<pre><code·class="language-macaulay2">i18·:·divisor(OO(D))205 <td>··············<pre><code·class="language-macaulay2">i18·:·divisor(OO(D))
  
206 o18·=·0,·the·zero·divisor206 o18·=·0,·the·zero·divisor
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101 Note·that·you·can·call·the·divisor·constructor·on·the·module·you·construct,·but101 Note·that·you·can·call·the·divisor·constructor·on·the·module·you·construct,·but
102 it·will·only·produce·a·divisor·up·to·linear·equivalence·(which·can·mean102 it·will·only·produce·a·divisor·up·to·linear·equivalence·(which·can·mean
103 different·things·depending·on·whether·or·not·you·are·keeping·track·of·the103 different·things·depending·on·whether·or·not·you·are·keeping·track·of·the
104 grading).104 grading).
105 i16·:·R·=·ZZ/11[x,y];105 i16·:·R·=·ZZ/11[x,y];
106 i17·:·D·=·divisor(x*y/(x+y))106 i17·:·D·=·divisor(x*y/(x+y))
  
107 o17·=·Div(x)·+·Div(y)·+·-Div(x+y)107 o17·=·-Div(x+y)·+·Div(x)·+·Div(y)
  
108 o17·:·WeilDivisor·on·R108 o17·:·WeilDivisor·on·R
109 i18·:·divisor(OO(D))109 i18·:·divisor(OO(D))
  
110 o18·=·0,·the·zero·divisor110 o18·=·0,·the·zero·divisor
  
111 o18·:·WeilDivisor·on·R111 o18·:·WeilDivisor·on·R
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83 ········<table·class="examples">83 ········<table·class="examples">
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>85 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor(x*y^2/z)88 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor(x*y^2/z)
  
89 o2·=·Div(x)·+·-Div(z)·+·2*Div(y)89 o2·=·Div(x)·+·2*Div(y)·+·-Div(z)
  
90 o2·:·WeilDivisor·on·R</code></pre>90 o2·:·WeilDivisor·on·R</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i3·:·applyToCoefficients(D,·u->5*u)93 <td>··············<pre><code·class="language-macaulay2">i3·:·applyToCoefficients(D,·u->5*u)
  
94 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)94 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)
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26 the·output·D·is·the·same·as·the·class·of·the·input·D1·(WeilDivisor,26 the·output·D·is·the·same·as·the·class·of·the·input·D1·(WeilDivisor,
27 QWeilDivisor,·RWeilDivisor,·BasicDivisor).·If·Safe·is·set·to·true·(the·default27 QWeilDivisor,·RWeilDivisor,·BasicDivisor).·If·Safe·is·set·to·true·(the·default
28 is·false),·then·the·function·will·check·to·make·sure·the·output·is·a·valid28 is·false),·then·the·function·will·check·to·make·sure·the·output·is·a·valid
29 divisor.29 divisor.
30 i1·:·R·=·QQ[x,·y,·z];30 i1·:·R·=·QQ[x,·y,·z];
31 i2·:·D·=·divisor(x*y^2/z)31 i2·:·D·=·divisor(x*y^2/z)
  
32 o2·=·Div(x)·+·-Div(z)·+·2*Div(y)32 o2·=·Div(x)·+·2*Div(y)·+·-Div(z)
  
33 o2·:·WeilDivisor·on·R33 o2·:·WeilDivisor·on·R
34 i3·:·applyToCoefficients(D,·u->5*u)34 i3·:·applyToCoefficients(D,·u->5*u)
  
35 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)35 o3·=·5*Div(x)·+·10*Div(y)·+·-5*Div(z)
  
36 o3·:·WeilDivisor·on·R36 o3·:·WeilDivisor·on·R
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76 ········<table·class="examples">76 ········<table·class="examples">
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);</code></pre>78 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)81 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
82 o2·=·4/3*Div(y,·z)·+·1/2*Div(x,·z)82 o2·=·1/2*Div(x,·z)·+·4/3*Div(y,·z)
  
83 o2·:·QWeilDivisor·on·R</code></pre>83 o2·:·QWeilDivisor·on·R</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i3·:·ceiling(·D·)86 <td>··············<pre><code·class="language-macaulay2">i3·:·ceiling(·D·)
  
87 o3·=·2*Div(y,·z)·+·Div(x,·z)87 o3·=·Div(x,·z)·+·2*Div(y,·z)
  
88 o3·:·WeilDivisor·on·R</code></pre>88 o3·:·WeilDivisor·on·R</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i4·:·floor(·D·)91 <td>··············<pre><code·class="language-macaulay2">i4·:·floor(·D·)
  
92 o4·=·Div(y,·z)92 o4·=·Div(y,·z)
  
93 o4·:·WeilDivisor·on·R</code></pre>93 o4·:·WeilDivisor·on·R</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)96 <td>··············<pre><code·class="language-macaulay2">i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)
  
97 o5·=·-.7*Div(y,·z)·+·.3*Div(x,·z)97 o5·=·.3*Div(x,·z)·+·-.7*Div(y,·z)
  
98 o5·:·RWeilDivisor·on·R</code></pre>98 o5·:·RWeilDivisor·on·R</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i6·:·ceiling(·E·)101 <td>··············<pre><code·class="language-macaulay2">i6·:·ceiling(·E·)
  
102 o6·=·Div(x,·z)102 o6·=·Div(x,·z)
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16 ··········o·an·instance·of·the·type·_\x8W_\x8e_\x8i_\x8l_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,16 ··········o·an·instance·of·the·type·_\x8W_\x8e_\x8i_\x8l_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 Start·with·a·rational·or·real·Weil·divisor.·We·form·a·new·divisor·whose18 Start·with·a·rational·or·real·Weil·divisor.·We·form·a·new·divisor·whose
19 coefficients·are·obtained·by·applying·the·ceiling·or·floor·function·to·them.19 coefficients·are·obtained·by·applying·the·ceiling·or·floor·function·to·them.
20 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);20 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*y·-·z^2);
21 i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)21 i2·:·D·=·divisor({1/2,·4/3},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
22 o2·=·4/3*Div(y,·z)·+·1/2*Div(x,·z)22 o2·=·1/2*Div(x,·z)·+·4/3*Div(y,·z)
  
23 o2·:·QWeilDivisor·on·R23 o2·:·QWeilDivisor·on·R
24 i3·:·ceiling(·D·)24 i3·:·ceiling(·D·)
  
25 o3·=·2*Div(y,·z)·+·Div(x,·z)25 o3·=·Div(x,·z)·+·2*Div(y,·z)
  
26 o3·:·WeilDivisor·on·R26 o3·:·WeilDivisor·on·R
27 i4·:·floor(·D·)27 i4·:·floor(·D·)
  
28 o4·=·Div(y,·z)28 o4·=·Div(y,·z)
  
29 o4·:·WeilDivisor·on·R29 o4·:·WeilDivisor·on·R
30 i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)30 i5·:·E·=·divisor({0.3,·-0.7},·{ideal(x,·z),·ideal(y,z)},·CoefficientType·=>·RR)
  
31 o5·=·-.7*Div(y,·z)·+·.3*Div(x,·z)31 o5·=·.3*Div(x,·z)·+·-.7*Div(y,·z)
  
32 o5·:·RWeilDivisor·on·R32 o5·:·RWeilDivisor·on·R
33 i6·:·ceiling(·E·)33 i6·:·ceiling(·E·)
  
34 o6·=·Div(x,·z)34 o6·=·Div(x,·z)
  
35 o6·:·WeilDivisor·on·R35 o6·:·WeilDivisor·on·R
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73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,y,z];</code></pre>75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,y,z];</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})78 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})
  
79 o2·=·Div(x)·+·0*Div(y)·+·-2*Div(z)79 o2·=·0*Div(y)·+·-2*Div(z)·+·Div(x)
  
80 o2·:·WeilDivisor·on·R</code></pre>80 o2·:·WeilDivisor·on·R</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·cleanSupport(D)83 <td>··············<pre><code·class="language-macaulay2">i3·:·cleanSupport(D)
  
84 o3·=·Div(x)·+·-2*Div(z)84 o3·=·-2*Div(z)·+·Div(x)
  
85 o3·:·WeilDivisor·on·R</code></pre>85 o3·:·WeilDivisor·on·R</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ········</table>87 ········</table>
88 ······</div>88 ······</div>
89 ······<div·class="waystouse">89 ······<div·class="waystouse">
90 ········<h2>Ways·to·use·<span·class="tt">cleanSupport</span>:</h2>90 ········<h2>Ways·to·use·<span·class="tt">cleanSupport</span>:</h2>
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14 ··········o·an·instance·of·the·type·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,14 ··········o·an·instance·of·the·type·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r,
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 This·function·returns·a·divisor·where·all·entries·with·coefficient·zero·are16 This·function·returns·a·divisor·where·all·entries·with·coefficient·zero·are
17 removed.17 removed.
18 i1·:·R·=·QQ[x,y,z];18 i1·:·R·=·QQ[x,y,z];
19 i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})19 i2·:·D·=·divisor({1,0,-2},·{ideal(x),·ideal(y),·ideal(z)})
  
20 o2·=·Div(x)·+·0*Div(y)·+·-2*Div(z)20 o2·=·0*Div(y)·+·-2*Div(z)·+·Div(x)
  
21 o2·:·WeilDivisor·on·R21 o2·:·WeilDivisor·on·R
22 i3·:·cleanSupport(D)22 i3·:·cleanSupport(D)
  
23 o3·=·Div(x)·+·-2*Div(z)23 o3·=·-2*Div(z)·+·Div(x)
  
24 o3·:·WeilDivisor·on·R24 o3·:·WeilDivisor·on·R
25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8cl\x8le\x8ea\x8an\x8nS\x8Su\x8up\x8pp\x8po\x8or\x8rt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8cl\x8le\x8ea\x8an\x8nS\x8Su\x8up\x8pp\x8po\x8or\x8rt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
26 ····*·cleanSupport(BasicDivisor)26 ····*·cleanSupport(BasicDivisor)
27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
28 The·object·_\x8c_\x8l_\x8e_\x8a_\x8n_\x8S_\x8u_\x8p_\x8p_\x8o_\x8r_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.28 The·object·_\x8c_\x8l_\x8e_\x8a_\x8n_\x8S_\x8u_\x8p_\x8p_\x8o_\x8r_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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187 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)187 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)
  
188 o14·:·WeilDivisor·on·A</code></pre>188 o14·:·WeilDivisor·on·A</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ··········<tr>190 ··········<tr>
191 <td>··············<pre><code·class="language-macaulay2">i15·:·E·=·divisor(y2z)191 <td>··············<pre><code·class="language-macaulay2">i15·:·E·=·divisor(y2z)
  
192 o15·=·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)·+·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)192 o15·=·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)·+·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)
  
193 o15·:·WeilDivisor·on·A</code></pre>193 o15·:·WeilDivisor·on·A</code></pre>
194 </td>··········</tr>194 </td>··········</tr>
195 ········</table>195 ········</table>
196 ········<div>196 ········<div>
197 ··········<p>We·can·construct·a·Q-divisor·as·well.··Here·are·two·ways·to·do·it·(we·work·in·$A^2$·this·time).</p>197 ··········<p>We·can·construct·a·Q-divisor·as·well.··Here·are·two·ways·to·do·it·(we·work·in·$A^2$·this·time).</p>
198 ········</div>198 ········</div>
199 ········<table·class="examples">199 ········<table·class="examples">
200 ··········<tr>200 ··········<tr>
201 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·ZZ/7[x,y];</code></pre>201 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·ZZ/7[x,y];</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ··········<tr>203 ··········<tr>
204 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)204 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)
  
205 o17·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)205 o17·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
206 o17·:·QWeilDivisor·on·R</code></pre>206 o17·:·QWeilDivisor·on·R</code></pre>
207 </td>··········</tr>207 </td>··········</tr>
208 ··········<tr>208 ··········<tr>
209 <td>··············<pre><code·class="language-macaulay2">i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)209 <td>··············<pre><code·class="language-macaulay2">i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)
  
210 o18·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)210 o18·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
211 o18·:·QWeilDivisor·on·R</code></pre>211 o18·:·QWeilDivisor·on·R</code></pre>
212 </td>··········</tr>212 </td>··········</tr>
213 ········</table>213 ········</table>
214 ········<div>214 ········<div>
215 ··········<p>Or·an·R-divisor.··This·time·we·work·in·the·cone·over·$P^1·\times·P^1$.</p>215 ··········<p>Or·an·R-divisor.··This·time·we·work·in·the·cone·over·$P^1·\times·P^1$.</p>
216 ········</div>216 ········</div>
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96 i14·:·D·=·divisor(x3)96 i14·:·D·=·divisor(x3)
  
97 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)97 o14·=·3*Div(xz2,·xyz,·xy2,·x2z,·x2y,·x3)
  
98 o14·:·WeilDivisor·on·A98 o14·:·WeilDivisor·on·A
99 i15·:·E·=·divisor(y2z)99 i15·:·E·=·divisor(y2z)
  
100 o15·=·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)·+·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)100 o15·=·Div(z3,·yz2,·y2z,·xz2,·xyz,·x2z)·+·2*Div(yz2,·y2z,·y3,·xyz,·xy2,·x2y)
  
101 o15·:·WeilDivisor·on·A101 o15·:·WeilDivisor·on·A
102 We·can·construct·a·Q-divisor·as·well.·Here·are·two·ways·to·do·it·(we·work·in102 We·can·construct·a·Q-divisor·as·well.·Here·are·two·ways·to·do·it·(we·work·in
103 $A^2$·this·time).103 $A^2$·this·time).
104 i16·:·R·=·ZZ/7[x,y];104 i16·:·R·=·ZZ/7[x,y];
105 i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)105 i17·:·D·=·divisor({-1/2,·2/1},·{ideal(y^2-x^3),·ideal(x)},·CoefficientType=>QQ)
  
106 o17·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)106 o17·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
107 o17·:·QWeilDivisor·on·R107 o17·:·QWeilDivisor·on·R
108 i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)108 i18·:·D·=·(-1/2)*divisor(y^2-x^3)·+·(2/1)*divisor(x)
  
109 o18·=·-1/2*Div(-x^3+y^2)·+·2*Div(x)109 o18·=·2*Div(x)·+·-1/2*Div(-x^3+y^2)
  
110 o18·:·QWeilDivisor·on·R110 o18·:·QWeilDivisor·on·R
111 Or·an·R-divisor.·This·time·we·work·in·the·cone·over·$P^1·\times·P^1$.111 Or·an·R-divisor.·This·time·we·work·in·the·cone·over·$P^1·\times·P^1$.
112 i19·:·R·=·ZZ/11[x,y,u,v]/ideal(x*y-u*v);112 i19·:·R·=·ZZ/11[x,y,u,v]/ideal(x*y-u*v);
113 i20·:·D·=·divisor({1.1,·-3.14159},·{ideal(x,u),·ideal(x,·v)},113 i20·:·D·=·divisor({1.1,·-3.14159},·{ideal(x,u),·ideal(x,·v)},
114 CoefficientType=>RR)114 CoefficientType=>RR)
  
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145 o10·:·Ideal·of·R</code></pre>145 o10·:·Ideal·of·R</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i11·:·M·=·J*R^1;</code></pre>148 <td>··············<pre><code·class="language-macaulay2">i11·:·M·=·J*R^1;</code></pre>
149 </td>··········</tr>149 </td>··········</tr>
150 ··········<tr>150 ··········<tr>
151 <td>··············<pre><code·class="language-macaulay2">i12·:·time·dualize(J,·Strategy=>IdealStrategy);151 <td>··············<pre><code·class="language-macaulay2">i12·:·time·dualize(J,·Strategy=>IdealStrategy);
152 ·--·used·0.0376684s·(cpu);·0.0407424s·(thread);·0s·(gc)152 ·--·used·0.0503945s·(cpu);·0.0496727s·(thread);·0s·(gc)
  
153 o12·:·Ideal·of·R</code></pre>153 o12·:·Ideal·of·R</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i13·:·time·dualize(J,·Strategy=>ModuleStrategy);156 <td>··············<pre><code·class="language-macaulay2">i13·:·time·dualize(J,·Strategy=>ModuleStrategy);
157 ·--·used·0.383657s·(cpu);·0.384438s·(thread);·0s·(gc)157 ·--·used·0.465893s·(cpu);·0.466334s·(thread);·0s·(gc)
  
158 o13·:·Ideal·of·R</code></pre>158 o13·:·Ideal·of·R</code></pre>
159 </td>··········</tr>159 </td>··········</tr>
160 ··········<tr>160 ··········<tr>
161 <td>··············<pre><code·class="language-macaulay2">i14·:·time·dualize(M,·Strategy=>IdealStrategy);161 <td>··············<pre><code·class="language-macaulay2">i14·:·time·dualize(M,·Strategy=>IdealStrategy);
162 ·--·used·0.462133s·(cpu);·0.43286s·(thread);·0s·(gc)</code></pre>162 ·--·used·0.661115s·(cpu);·0.613389s·(thread);·0s·(gc)</code></pre>
163 </td>··········</tr>163 </td>··········</tr>
164 ··········<tr>164 ··········<tr>
165 <td>··············<pre><code·class="language-macaulay2">i15·:·time·dualize(M,·Strategy=>ModuleStrategy);165 <td>··············<pre><code·class="language-macaulay2">i15·:·time·dualize(M,·Strategy=>ModuleStrategy);
166 ·--·used·0.00234692s·(cpu);·0.00241164s·(thread);·0s·(gc)</code></pre>166 ·--·used·0.00315628s·(cpu);·0.00331601s·(thread);·0s·(gc)</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);169 <td>··············<pre><code·class="language-macaulay2">i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);
170 ·--·used·9.1117e-05s·(cpu);·0.00231316s·(thread);·0s·(gc)170 ·--·used·6.1896e-05s·(cpu);·0.00256199s·(thread);·0s·(gc)
  
171 o16·:·Ideal·of·R</code></pre>171 o16·:·Ideal·of·R</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ········</table>173 ········</table>
174 ········<div>174 ········<div>
175 ··········<p>For·monomial·ideals·in·toric·rings,·frequently·ModuleStrategy·appears·faster.</p>175 ··········<p>For·monomial·ideals·in·toric·rings,·frequently·ModuleStrategy·appears·faster.</p>
176 ········</div>176 ········</div>
Offset 189, 21 lines modifiedOffset 189, 21 lines modified
189 ··········<tr>189 ··········<tr>
190 <td>··············<pre><code·class="language-macaulay2">i19·:·J·=·I^15;190 <td>··············<pre><code·class="language-macaulay2">i19·:·J·=·I^15;
  
191 o19·:·Ideal·of·R</code></pre>191 o19·:·Ideal·of·R</code></pre>
192 </td>··········</tr>192 </td>··········</tr>
193 ··········<tr>193 ··········<tr>
194 <td>··············<pre><code·class="language-macaulay2">i20·:·time·dualize(J,·Strategy=>IdealStrategy);194 <td>··············<pre><code·class="language-macaulay2">i20·:·time·dualize(J,·Strategy=>IdealStrategy);
195 ·--·used·0.124346s·(cpu);·0.0848201s·(thread);·0s·(gc)195 ·--·used·0.154716s·(cpu);·0.0985429s·(thread);·0s·(gc)
  
196 o20·:·Ideal·of·R</code></pre>196 o20·:·Ideal·of·R</code></pre>
197 </td>··········</tr>197 </td>··········</tr>
198 ··········<tr>198 ··········<tr>
199 <td>··············<pre><code·class="language-macaulay2">i21·:·time·dualize(J,·Strategy=>ModuleStrategy);199 <td>··············<pre><code·class="language-macaulay2">i21·:·time·dualize(J,·Strategy=>ModuleStrategy);
200 ·--·used·0.00452869s·(cpu);·0.00463597s·(thread);·0s·(gc)200 ·--·used·0.00398933s·(cpu);·0.00689616s·(thread);·0s·(gc)
  
201 o21·:·Ideal·of·R</code></pre>201 o21·:·Ideal·of·R</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ········</table>203 ········</table>
204 ········<div>204 ········<div>
205 ··········<p><span·class="tt">KnownDomain</span>·is·an·option·for·<span·class="tt">dualize</span>.··If·it·is·<span·class="tt">false</span>·(default·is·<span·class="tt">true</span>),·then·the·computer·will·first·check·whether·the·ring·is·a·domain,·if·it·is·not·then·it·will·revert·to·<span·class="tt">ModuleStrategy</span>.··If·<span·class="tt">KnownDomain</span>·is·set·to·<span·class="tt">true</span>·for·a·non-domain,·then·the·function·can·return·an·incorrect·answer.</p>205 ··········<p><span·class="tt">KnownDomain</span>·is·an·option·for·<span·class="tt">dualize</span>.··If·it·is·<span·class="tt">false</span>·(default·is·<span·class="tt">true</span>),·then·the·computer·will·first·check·whether·the·ring·is·a·domain,·if·it·is·not·then·it·will·revert·to·<span·class="tt">ModuleStrategy</span>.··If·<span·class="tt">KnownDomain</span>·is·set·to·<span·class="tt">true</span>·for·a·non-domain,·then·the·function·can·return·an·incorrect·answer.</p>
206 ········</div>206 ········</div>
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61 o9·:·Ideal·of·R61 o9·:·Ideal·of·R
62 i10·:·J·=·m^9;62 i10·:·J·=·m^9;
  
63 o10·:·Ideal·of·R63 o10·:·Ideal·of·R
64 i11·:·M·=·J*R^1;64 i11·:·M·=·J*R^1;
65 i12·:·time·dualize(J,·Strategy=>IdealStrategy);65 i12·:·time·dualize(J,·Strategy=>IdealStrategy);
66 ·--·used·0.0376684s·(cpu);·0.0407424s·(thread);·0s·(gc)66 ·--·used·0.0503945s·(cpu);·0.0496727s·(thread);·0s·(gc)
  
67 o12·:·Ideal·of·R67 o12·:·Ideal·of·R
68 i13·:·time·dualize(J,·Strategy=>ModuleStrategy);68 i13·:·time·dualize(J,·Strategy=>ModuleStrategy);
69 ·--·used·0.383657s·(cpu);·0.384438s·(thread);·0s·(gc)69 ·--·used·0.465893s·(cpu);·0.466334s·(thread);·0s·(gc)
  
70 o13·:·Ideal·of·R70 o13·:·Ideal·of·R
71 i14·:·time·dualize(M,·Strategy=>IdealStrategy);71 i14·:·time·dualize(M,·Strategy=>IdealStrategy);
72 ·--·used·0.462133s·(cpu);·0.43286s·(thread);·0s·(gc)72 ·--·used·0.661115s·(cpu);·0.613389s·(thread);·0s·(gc)
73 i15·:·time·dualize(M,·Strategy=>ModuleStrategy);73 i15·:·time·dualize(M,·Strategy=>ModuleStrategy);
74 ·--·used·0.00234692s·(cpu);·0.00241164s·(thread);·0s·(gc)74 ·--·used·0.00315628s·(cpu);·0.00331601s·(thread);·0s·(gc)
75 i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);75 i16·:·time·embedAsIdeal·dualize(M,·Strategy=>ModuleStrategy);
76 ·--·used·9.1117e-05s·(cpu);·0.00231316s·(thread);·0s·(gc)76 ·--·used·6.1896e-05s·(cpu);·0.00256199s·(thread);·0s·(gc)
  
77 o16·:·Ideal·of·R77 o16·:·Ideal·of·R
78 For·monomial·ideals·in·toric·rings,·frequently·ModuleStrategy·appears·faster.78 For·monomial·ideals·in·toric·rings,·frequently·ModuleStrategy·appears·faster.
79 i17·:·R·=·ZZ/7[x,y,u,v]/ideal(x*y-u*v);79 i17·:·R·=·ZZ/7[x,y,u,v]/ideal(x*y-u*v);
80 i18·:·I·=·ideal(x,u);80 i18·:·I·=·ideal(x,u);
  
81 o18·:·Ideal·of·R81 o18·:·Ideal·of·R
82 i19·:·J·=·I^15;82 i19·:·J·=·I^15;
  
83 o19·:·Ideal·of·R83 o19·:·Ideal·of·R
84 i20·:·time·dualize(J,·Strategy=>IdealStrategy);84 i20·:·time·dualize(J,·Strategy=>IdealStrategy);
85 ·--·used·0.124346s·(cpu);·0.0848201s·(thread);·0s·(gc)85 ·--·used·0.154716s·(cpu);·0.0985429s·(thread);·0s·(gc)
  
86 o20·:·Ideal·of·R86 o20·:·Ideal·of·R
87 i21·:·time·dualize(J,·Strategy=>ModuleStrategy);87 i21·:·time·dualize(J,·Strategy=>ModuleStrategy);
88 ·--·used·0.00452869s·(cpu);·0.00463597s·(thread);·0s·(gc)88 ·--·used·0.00398933s·(cpu);·0.00689616s·(thread);·0s·(gc)
  
89 o21·:·Ideal·of·R89 o21·:·Ideal·of·R
90 KnownDomain·is·an·option·for·dualize.·If·it·is·false·(default·is·true),·then90 KnownDomain·is·an·option·for·dualize.·If·it·is·false·(default·is·true),·then
91 the·computer·will·first·check·whether·the·ring·is·a·domain,·if·it·is·not·then91 the·computer·will·first·check·whether·the·ring·is·a·domain,·if·it·is·not·then
92 it·will·revert·to·ModuleStrategy.·If·KnownDomain·is·set·to·true·for·a·non-92 it·will·revert·to·ModuleStrategy.·If·KnownDomain·is·set·to·true·for·a·non-
93 domain,·then·the·function·can·return·an·incorrect·answer.93 domain,·then·the·function·can·return·an·incorrect·answer.
94 i22·:·R·=·QQ[x,y]/ideal(x*y);94 i22·:·R·=·QQ[x,y]/ideal(x*y);
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Offset 73, 29 lines modifiedOffset 73, 29 lines modified
73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})78 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})
  
79 o2·=·2*Div(y)·+·0*Div(x^2+z)·+·-8*Div(x)79 o2·=·0*Div(x^2+z)·+·-8*Div(x)·+·2*Div(y)
  
80 o2·:·WeilDivisor·on·R</code></pre>80 o2·:·WeilDivisor·on·R</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·getPrimeDivisors(·D·)83 <td>··············<pre><code·class="language-macaulay2">i3·:·getPrimeDivisors(·D·)
  
84 o3·=·{Div(y),·Div(x^2+z),·Div(x)}84 o3·=·{Div(x^2+z),·Div(x),·Div(y)}
  
85 o3·:·List</code></pre>85 o3·:·List</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·getPrimeDivisors(·cleanSupport·D·)88 <td>··············<pre><code·class="language-macaulay2">i4·:·getPrimeDivisors(·cleanSupport·D·)
  
89 o4·=·{Div(y),·Div(x)}89 o4·=·{Div(x),·Div(y)}
  
90 o4·:·List</code></pre>90 o4·:·List</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ········</table>92 ········</table>
93 ······</div>93 ······</div>
94 ······<div·class="waystouse">94 ······<div·class="waystouse">
95 ········<h2>Ways·to·use·<span·class="tt">getPrimeDivisors</span>:</h2>95 ········<h2>Ways·to·use·<span·class="tt">getPrimeDivisors</span>:</h2>
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15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 This·function·returns·the·list·of·prime·divisors·of·a·given·divisor.·The·prime16 This·function·returns·the·list·of·prime·divisors·of·a·given·divisor.·The·prime
17 divisors·are·all·of·the·class·WeilDivisor.·If·you·do·not·call·cleanSupport,·you17 divisors·are·all·of·the·class·WeilDivisor.·If·you·do·not·call·cleanSupport,·you
18 may·obtain·divisors·with·zero·coefficients.18 may·obtain·divisors·with·zero·coefficients.
19 i1·:·R·=·QQ[x,·y,·z];19 i1·:·R·=·QQ[x,·y,·z];
20 i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})20 i2·:·D·=·divisor({-8,·2,·0},·{ideal(x),·ideal(y),·ideal(x^2+z)})
  
21 o2·=·2*Div(y)·+·0*Div(x^2+z)·+·-8*Div(x)21 o2·=·0*Div(x^2+z)·+·-8*Div(x)·+·2*Div(y)
  
22 o2·:·WeilDivisor·on·R22 o2·:·WeilDivisor·on·R
23 i3·:·getPrimeDivisors(·D·)23 i3·:·getPrimeDivisors(·D·)
  
24 o3·=·{Div(y),·Div(x^2+z),·Div(x)}24 o3·=·{Div(x^2+z),·Div(x),·Div(y)}
  
25 o3·:·List25 o3·:·List
26 i4·:·getPrimeDivisors(·cleanSupport·D·)26 i4·:·getPrimeDivisors(·cleanSupport·D·)
  
27 o4·=·{Div(y),·Div(x)}27 o4·=·{Div(x),·Div(y)}
  
28 o4·:·List28 o4·:·List
29 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8tP\x8Pr\x8ri\x8im\x8me\x8eD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8rs\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8tP\x8Pr\x8ri\x8im\x8me\x8eD\x8Di\x8iv\x8vi\x8is\x8so\x8or\x8rs\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
30 ····*·getPrimeDivisors(BasicDivisor)30 ····*·getPrimeDivisors(BasicDivisor)
31 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*31 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
32 The·object·_\x8g_\x8e_\x8t_\x8P_\x8r_\x8i_\x8m_\x8e_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.32 The·object·_\x8g_\x8e_\x8t_\x8P_\x8r_\x8i_\x8m_\x8e_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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Offset 99, 15 lines modifiedOffset 99, 15 lines modified
99 ········<table·class="examples">99 ········<table·class="examples">
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>101 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})104 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
105 o5·=·2*Div(y,·z)·+·Div(x,·z)105 o5·=·Div(x,·z)·+·2*Div(y,·z)
  
106 o5·:·WeilDivisor·on·R</code></pre>106 o5·:·WeilDivisor·on·R</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i6·:·isCartier(·D·)109 <td>··············<pre><code·class="language-macaulay2">i6·:·isCartier(·D·)
  
110 o6·=·false</code></pre>110 o6·=·false</code></pre>
Offset 119, 15 lines modifiedOffset 119, 15 lines modified
119 ········<table·class="examples">119 ········<table·class="examples">
120 ··········<tr>120 ··········<tr>
121 <td>··············<pre><code·class="language-macaulay2">i7·:·R·=·QQ[x,·y,·z];</code></pre>121 <td>··············<pre><code·class="language-macaulay2">i7·:·R·=·QQ[x,·y,·z];</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})124 <td>··············<pre><code·class="language-macaulay2">i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})
  
125 o8·=·Div(x)·+·2*Div(y)125 o8·=·2*Div(y)·+·Div(x)
  
126 o8·:·WeilDivisor·on·R</code></pre>126 o8·:·WeilDivisor·on·R</code></pre>
127 </td>··········</tr>127 </td>··········</tr>
128 ··········<tr>128 ··········<tr>
129 <td>··············<pre><code·class="language-macaulay2">i9·:·isCartier(·D·)129 <td>··············<pre><code·class="language-macaulay2">i9·:·isCartier(·D·)
  
130 o9·=·true</code></pre>130 o9·=·true</code></pre>
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156 ········<table·class="examples">156 ········<table·class="examples">
157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>158 <td>··············<pre><code·class="language-macaulay2">i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>
159 </td>··········</tr>159 </td>··········</tr>
160 ··········<tr>160 ··········<tr>
161 <td>··············<pre><code·class="language-macaulay2">i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})161 <td>··············<pre><code·class="language-macaulay2">i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
162 o14·=·2*Div(y,·z)·+·Div(x,·z)162 o14·=·Div(x,·z)·+·2*Div(y,·z)
  
163 o14·:·WeilDivisor·on·R</code></pre>163 o14·:·WeilDivisor·on·R</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ··········<tr>165 ··········<tr>
166 <td>··············<pre><code·class="language-macaulay2">i15·:·isCartier(D,·IsGraded·=>·true)166 <td>··············<pre><code·class="language-macaulay2">i15·:·isCartier(D,·IsGraded·=>·true)
  
167 o15·=·true</code></pre>167 o15·=·true</code></pre>
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26 i3·:·isCartier(·D·)26 i3·:·isCartier(·D·)
  
27 o3·=·false27 o3·=·false
28 Neither·is·this·divisor.28 Neither·is·this·divisor.
29 i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);29 i4·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
30 i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})30 i5·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
31 o5·=·2*Div(y,·z)·+·Div(x,·z)31 o5·=·Div(x,·z)·+·2*Div(y,·z)
  
32 o5·:·WeilDivisor·on·R32 o5·:·WeilDivisor·on·R
33 i6·:·isCartier(·D·)33 i6·:·isCartier(·D·)
  
34 o6·=·false34 o6·=·false
35 Of·course·the·next·divisor·is·Cartier.35 Of·course·the·next·divisor·is·Cartier.
36 i7·:·R·=·QQ[x,·y,·z];36 i7·:·R·=·QQ[x,·y,·z];
37 i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})37 i8·:·D·=·divisor({1,·2},·{ideal(x),·ideal(y)})
  
38 o8·=·Div(x)·+·2*Div(y)38 o8·=·2*Div(y)·+·Div(x)
  
39 o8·:·WeilDivisor·on·R39 o8·:·WeilDivisor·on·R
40 i9·:·isCartier(·D·)40 i9·:·isCartier(·D·)
  
41 o9·=·true41 o9·=·true
42 If·the·option·IsGraded·is·set·to·true·(it·is·false·by·default),·this·will·check42 If·the·option·IsGraded·is·set·to·true·(it·is·false·by·default),·this·will·check
43 as·if·D·is·a·divisor·on·the·$Proj$·of·the·ambient·graded·ring.43 as·if·D·is·a·divisor·on·the·$Proj$·of·the·ambient·graded·ring.
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 o11·:·WeilDivisor·on·R56 o11·:·WeilDivisor·on·R
57 i12·:·isCartier(D,·IsGraded·=>·true)57 i12·:·isCartier(D,·IsGraded·=>·true)
  
58 o12·=·true58 o12·=·true
59 i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);59 i13·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
60 i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})60 i14·:·D·=·divisor({1,·2},·{ideal(x,·z),·ideal(y,·z)})
  
61 o14·=·2*Div(y,·z)·+·Div(x,·z)61 o14·=·Div(x,·z)·+·2*Div(y,·z)
  
62 o14·:·WeilDivisor·on·R62 o14·:·WeilDivisor·on·R
63 i15·:·isCartier(D,·IsGraded·=>·true)63 i15·:·isCartier(D,·IsGraded·=>·true)
  
64 o15·=·true64 o15·=·true
65 The·output·value·of·this·function·is·stored·in·the·divisor's·cache·with·the65 The·output·value·of·this·function·is·stored·in·the·divisor's·cache·with·the
66 value·of·the·last·IsGraded·option.·If·you·change·the·IsGraded·option,·the·value66 value·of·the·last·IsGraded·option.·If·you·change·the·IsGraded·option,·the·value
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81 ········<table·class="examples">81 ········<table·class="examples">
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);</code></pre>83 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})86 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
87 o2·=·3*Div(x,·z)·+·8*Div(y,·z)87 o2·=·8*Div(y,·z)·+·3*Div(x,·z)
  
88 o2·:·WeilDivisor·on·R</code></pre>88 o2·:·WeilDivisor·on·R</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})91 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
92 o3·=·Div(x,·z)·+·8*Div(y,·z)92 o3·=·8*Div(y,·z)·+·Div(x,·z)
  
93 o3·:·WeilDivisor·on·R</code></pre>93 o3·:·WeilDivisor·on·R</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i4·:·isLinearEquivalent(D1,·D2)96 <td>··············<pre><code·class="language-macaulay2">i4·:·isLinearEquivalent(D1,·D2)
  
97 o4·=·true</code></pre>97 o4·=·true</code></pre>
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108 ········<table·class="examples">108 ········<table·class="examples">
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);</code></pre>110 <td>··············<pre><code·class="language-macaulay2">i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})113 <td>··············<pre><code·class="language-macaulay2">i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
114 o6·=·8*Div(y,·z)·+·3*Div(x,·z)114 o6·=·3*Div(x,·z)·+·8*Div(y,·z)
  
115 o6·:·WeilDivisor·on·R</code></pre>115 o6·:·WeilDivisor·on·R</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})118 <td>··············<pre><code·class="language-macaulay2">i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
119 o7·=·8*Div(y,·z)·+·Div(x,·z)119 o7·=·Div(x,·z)·+·8*Div(y,·z)
  
120 o7·:·WeilDivisor·on·R</code></pre>120 o7·:·WeilDivisor·on·R</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)123 <td>··············<pre><code·class="language-macaulay2">i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)
  
124 o8·=·false</code></pre>124 o8·=·false</code></pre>
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18 ··········o·flag,·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,18 ··········o·flag,·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,
19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
20 Given·two·Weil·divisors,·this·method·checks·whether·they·are·linearly20 Given·two·Weil·divisors,·this·method·checks·whether·they·are·linearly
21 equivalent.21 equivalent.
22 i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);22 i1·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);
23 i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})23 i2·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
24 o2·=·3*Div(x,·z)·+·8*Div(y,·z)24 o2·=·8*Div(y,·z)·+·3*Div(x,·z)
  
25 o2·:·WeilDivisor·on·R25 o2·:·WeilDivisor·on·R
26 i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})26 i3·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
27 o3·=·Div(x,·z)·+·8*Div(y,·z)27 o3·=·8*Div(y,·z)·+·Div(x,·z)
  
28 o3·:·WeilDivisor·on·R28 o3·:·WeilDivisor·on·R
29 i4·:·isLinearEquivalent(D1,·D2)29 i4·:·isLinearEquivalent(D1,·D2)
  
30 o4·=·true30 o4·=·true
31 If·IsGraded·is·set·to·true·(by·default·it·is·false),·then·it·treats·the31 If·IsGraded·is·set·to·true·(by·default·it·is·false),·then·it·treats·the
32 divisors·as·divisors·on·the·$Proj$·of·their·ambient·ring.32 divisors·as·divisors·on·the·$Proj$·of·their·ambient·ring.
33 i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);33 i5·:·R·=·QQ[x,·y,·z]/·ideal(x·*·y·-·z^2);
34 i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})34 i6·:·D1·=·divisor({3,·8},·{ideal(x,·z),·ideal(y,·z)})
  
35 o6·=·8*Div(y,·z)·+·3*Div(x,·z)35 o6·=·3*Div(x,·z)·+·8*Div(y,·z)
  
36 o6·:·WeilDivisor·on·R36 o6·:·WeilDivisor·on·R
37 i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})37 i7·:·D2·=·divisor({8,·1},·{ideal(y,·z),·ideal(x,·z)})
  
38 o7·=·8*Div(y,·z)·+·Div(x,·z)38 o7·=·Div(x,·z)·+·8*Div(y,·z)
  
39 o7·:·WeilDivisor·on·R39 o7·:·WeilDivisor·on·R
40 i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)40 i8·:·isLinearEquivalent(D1,·D2,·IsGraded·=>·true)
  
41 o8·=·false41 o8·=·false
42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
43 ····*·_\x8O_\x8O_\x8·_\x8R_\x8W_\x8e_\x8i_\x8l_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r43 ····*·_\x8O_\x8O_\x8·_\x8R_\x8W_\x8e_\x8i_\x8l_\x8D_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r
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83 ········<table·class="examples">83 ········<table·class="examples">
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>85 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)88 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
89 o2·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)89 o2·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)
  
90 o2·:·QWeilDivisor·on·R</code></pre>90 o2·:·QWeilDivisor·on·R</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)93 <td>··············<pre><code·class="language-macaulay2">i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)
  
94 o3·=·3/4*Div(y,·z)·+·5/2*Div(x,·z)94 o3·=·5/2*Div(x,·z)·+·3/4*Div(y,·z)
  
95 o3·:·QWeilDivisor·on·R</code></pre>95 o3·:·QWeilDivisor·on·R</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i4·:·isQLinearEquivalent(10,·D,·E)98 <td>··············<pre><code·class="language-macaulay2">i4·:·isQLinearEquivalent(10,·D,·E)
  
99 o4·=·true</code></pre>99 o4·=·true</code></pre>
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138 ········<table·class="examples">138 ········<table·class="examples">
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>140 <td>··············<pre><code·class="language-macaulay2">i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)143 <td>··············<pre><code·class="language-macaulay2">i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
144 o11·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)144 o11·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)
  
145 o11·:·QWeilDivisor·on·R</code></pre>145 o11·:·QWeilDivisor·on·R</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)148 <td>··············<pre><code·class="language-macaulay2">i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)
  
149 o12·=·-1/4*Div(x,·z)·+·3/2*Div(y,·z)149 o12·=·3/2*Div(y,·z)·+·-1/4*Div(x,·z)
  
150 o12·:·QWeilDivisor·on·R</code></pre>150 o12·:·QWeilDivisor·on·R</code></pre>
151 </td>··········</tr>151 </td>··········</tr>
152 ··········<tr>152 ··········<tr>
153 <td>··············<pre><code·class="language-macaulay2">i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)153 <td>··············<pre><code·class="language-macaulay2">i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)
  
154 o13·=·true</code></pre>154 o13·=·true</code></pre>
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20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
21 Given·two·rational·divisors,·this·method·returns·true·if·they·linearly21 Given·two·rational·divisors,·this·method·returns·true·if·they·linearly
22 equivalent·after·clearing·denominators·or·if·some·further·multiple·up·to·n22 equivalent·after·clearing·denominators·or·if·some·further·multiple·up·to·n
23 makes·them·linearly·equivalent.·Otherwise·it·returns·false.23 makes·them·linearly·equivalent.·Otherwise·it·returns·false.
24 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);24 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
25 i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)25 i2·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>·QQ)
  
26 o2·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)26 o2·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)
  
27 o2·:·QWeilDivisor·on·R27 o2·:·QWeilDivisor·on·R
28 i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)28 i3·:·E·=·divisor({3/4,·5/2},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>·QQ)
  
29 o3·=·3/4*Div(y,·z)·+·5/2*Div(x,·z)29 o3·=·5/2*Div(x,·z)·+·3/4*Div(y,·z)
  
30 o3·:·QWeilDivisor·on·R30 o3·:·QWeilDivisor·on·R
31 i4·:·isQLinearEquivalent(10,·D,·E)31 i4·:·isQLinearEquivalent(10,·D,·E)
  
32 o4·=·true32 o4·=·true
33 In·the·above·ring,·every·pair·of·divisors·is·Q-linearly·equivalent·because·the33 In·the·above·ring,·every·pair·of·divisors·is·Q-linearly·equivalent·because·the
34 Weil·divisor·class·group·is·isomorphic·to·Z/2.·However,·if·we·don't·set·n·high34 Weil·divisor·class·group·is·isomorphic·to·Z/2.·However,·if·we·don't·set·n·high
Offset 53, 21 lines modifiedOffset 53, 21 lines modified
53 o9·=·true53 o9·=·true
54 If·IsGraded=>true·(the·default·is·false),·then·it·treats·the·divisors·as·if54 If·IsGraded=>true·(the·default·is·false),·then·it·treats·the·divisors·as·if
55 they·are·divisors·on·the·$Proj$·of·their·ambient·ring.55 they·are·divisors·on·the·$Proj$·of·their·ambient·ring.
56 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);56 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2);
57 i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>57 i11·:·D·=·divisor({1/2,·3/4},·{ideal(x,·z),·ideal(y,·z)},·CoefficientType·=>
58 QQ)58 QQ)
  
59 o11·=·1/2*Div(x,·z)·+·3/4*Div(y,·z)59 o11·=·3/4*Div(y,·z)·+·1/2*Div(x,·z)
  
60 o11·:·QWeilDivisor·on·R60 o11·:·QWeilDivisor·on·R
61 i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>61 i12·:·E·=·divisor({3/2,·-1/4},·{ideal(y,·z),·ideal(x,·z)},·CoefficientType·=>
62 QQ)62 QQ)
  
63 o12·=·-1/4*Div(x,·z)·+·3/2*Div(y,·z)63 o12·=·3/2*Div(y,·z)·+·-1/4*Div(x,·z)
  
64 o12·:·QWeilDivisor·on·R64 o12·:·QWeilDivisor·on·R
65 i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)65 i13·:·isQLinearEquivalent(10,·D,·E,·IsGraded·=>·true)
  
66 o13·=·true66 o13·=·true
67 i14·:·isQLinearEquivalent(10,·3*D,·E,·IsGraded·=>·true)67 i14·:·isQLinearEquivalent(10,·3*D,·E,·IsGraded·=>·true)
  
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73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>75 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z];</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor(x^2·*·y^3·*·z)78 <td>··············<pre><code·class="language-macaulay2">i2·:·D1·=·divisor(x^2·*·y^3·*·z)
  
79 o2·=·3*Div(y)·+·Div(z)·+·2*Div(x)79 o2·=·2*Div(x)·+·3*Div(y)·+·Div(z)
  
80 o2·:·WeilDivisor·on·R</code></pre>80 o2·:·WeilDivisor·on·R</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor(x·*·y·*·z)83 <td>··············<pre><code·class="language-macaulay2">i3·:·D2·=·divisor(x·*·y·*·z)
  
84 o3·=·Div(y)·+·Div(z)·+·Div(x)84 o3·=·Div(x)·+·Div(y)·+·Div(z)
  
85 o3·:·WeilDivisor·on·R</code></pre>85 o3·:·WeilDivisor·on·R</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·isReduced(·D1·)88 <td>··············<pre><code·class="language-macaulay2">i4·:·isReduced(·D1·)
  
89 o4·=·false</code></pre>89 o4·=·false</code></pre>
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13 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,13 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 This·function·returns·true·if·the·divisor·is·reduced·(all·coefficients·equal·to15 This·function·returns·true·if·the·divisor·is·reduced·(all·coefficients·equal·to
16 1),·otherwise·it·returns·false.16 1),·otherwise·it·returns·false.
17 i1·:·R·=·QQ[x,·y,·z];17 i1·:·R·=·QQ[x,·y,·z];
18 i2·:·D1·=·divisor(x^2·*·y^3·*·z)18 i2·:·D1·=·divisor(x^2·*·y^3·*·z)
  
19 o2·=·3*Div(y)·+·Div(z)·+·2*Div(x)19 o2·=·2*Div(x)·+·3*Div(y)·+·Div(z)
  
20 o2·:·WeilDivisor·on·R20 o2·:·WeilDivisor·on·R
21 i3·:·D2·=·divisor(x·*·y·*·z)21 i3·:·D2·=·divisor(x·*·y·*·z)
  
22 o3·=·Div(y)·+·Div(z)·+·Div(x)22 o3·=·Div(x)·+·Div(y)·+·Div(z)
  
23 o3·:·WeilDivisor·on·R23 o3·:·WeilDivisor·on·R
24 i4·:·isReduced(·D1·)24 i4·:·isReduced(·D1·)
  
25 o4·=·false25 o4·=·false
26 i5·:·isReduced(·D2·)26 i5·:·isReduced(·D2·)
  
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79 ········<table·class="examples">79 ········<table·class="examples">
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>81 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})84 <td>··············<pre><code·class="language-macaulay2">i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
85 o2·=·-2*Div(y,·z)·+·Div(x,·z)85 o2·=·Div(x,·z)·+·-2*Div(y,·z)
  
86 o2·:·WeilDivisor·on·R</code></pre>86 o2·:·WeilDivisor·on·R</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·isSNC(·D·)89 <td>··············<pre><code·class="language-macaulay2">i3·:·isSNC(·D·)
  
90 o3·=·false</code></pre>90 o3·=·false</code></pre>
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133 ········<table·class="examples">133 ········<table·class="examples">
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>135 <td>··············<pre><code·class="language-macaulay2">i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);</code></pre>
136 </td>··········</tr>136 </td>··········</tr>
137 ··········<tr>137 ··········<tr>
138 <td>··············<pre><code·class="language-macaulay2">i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})138 <td>··············<pre><code·class="language-macaulay2">i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
139 o11·=·Div(x,·z)·+·-2*Div(y,·z)139 o11·=·-2*Div(y,·z)·+·Div(x,·z)
  
140 o11·:·WeilDivisor·on·R</code></pre>140 o11·:·WeilDivisor·on·R</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i12·:·isSNC(·D,·IsGraded·=>·true·)143 <td>··············<pre><code·class="language-macaulay2">i12·:·isSNC(·D,·IsGraded·=>·true·)
  
144 o12·=·true</code></pre>144 o12·=·true</code></pre>
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167 ········<table·class="examples">167 ········<table·class="examples">
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·QQ[x,y,z];</code></pre>169 <td>··············<pre><code·class="language-macaulay2">i16·:·R·=·QQ[x,y,z];</code></pre>
170 </td>··········</tr>170 </td>··········</tr>
171 ··········<tr>171 ··········<tr>
172 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor(x*y*(x+y))172 <td>··············<pre><code·class="language-macaulay2">i17·:·D·=·divisor(x*y*(x+y))
  
173 o17·=·Div(x)·+·Div(x+y)·+·Div(y)173 o17·=·Div(x+y)·+·Div(y)·+·Div(x)
  
174 o17·:·WeilDivisor·on·R</code></pre>174 o17·:·WeilDivisor·on·R</code></pre>
175 </td>··········</tr>175 </td>··········</tr>
176 ··········<tr>176 ··········<tr>
177 <td>··············<pre><code·class="language-macaulay2">i18·:·isSNC(·D,·IsGraded·=>·true)177 <td>··············<pre><code·class="language-macaulay2">i18·:·isSNC(·D,·IsGraded·=>·true)
  
178 o18·=·false</code></pre>178 o18·=·false</code></pre>
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16 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,16 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 This·function·returns·true·if·the·divisor·is·simple·normal·crossings,·this18 This·function·returns·true·if·the·divisor·is·simple·normal·crossings,·this
19 includes·checking·that·the·ambient·ring·is·regular.19 includes·checking·that·the·ambient·ring·is·regular.
20 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);20 i1·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
21 i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})21 i2·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
22 o2·=·-2*Div(y,·z)·+·Div(x,·z)22 o2·=·Div(x,·z)·+·-2*Div(y,·z)
  
23 o2·:·WeilDivisor·on·R23 o2·:·WeilDivisor·on·R
24 i3·:·isSNC(·D·)24 i3·:·isSNC(·D·)
  
25 o3·=·false25 o3·=·false
26 i4·:·R·=·QQ[x,·y];26 i4·:·R·=·QQ[x,·y];
27 i5·:·D·=·divisor(x*y*(x+y))27 i5·:·D·=·divisor(x*y*(x+y))
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46 o9·=·true46 o9·=·true
47 If·IsGraded·is·set·to·true·(default·false),·then·the·divisor·is·treated·as·if47 If·IsGraded·is·set·to·true·(default·false),·then·the·divisor·is·treated·as·if
48 it·is·on·the·$Proj$·of·the·ambient·ring.·In·particular,·non-SNC·behavior·at·the48 it·is·on·the·$Proj$·of·the·ambient·ring.·In·particular,·non-SNC·behavior·at·the
49 origin·is·ignored.49 origin·is·ignored.
50 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);50 i10·:·R·=·QQ[x,·y,·z]·/·ideal(x·*·y·-·z^2·);
51 i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})51 i11·:·D·=·divisor({1,·-2},·{ideal(x,·z),·ideal(y,·z)})
  
52 o11·=·Div(x,·z)·+·-2*Div(y,·z)52 o11·=·-2*Div(y,·z)·+·Div(x,·z)
  
53 o11·:·WeilDivisor·on·R53 o11·:·WeilDivisor·on·R
54 i12·:·isSNC(·D,·IsGraded·=>·true·)54 i12·:·isSNC(·D,·IsGraded·=>·true·)
  
55 o12·=·true55 o12·=·true
56 i13·:·R·=·QQ[x,·y];56 i13·:·R·=·QQ[x,·y];
57 i14·:·D·=·divisor(x*y*(x+y))57 i14·:·D·=·divisor(x*y*(x+y))
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64 o14·:·WeilDivisor·on·R64 o14·:·WeilDivisor·on·R
65 i15·:·isSNC(·D,·IsGraded·=>·true·)65 i15·:·isSNC(·D,·IsGraded·=>·true·)
  
66 o15·=·true66 o15·=·true
67 i16·:·R·=·QQ[x,y,z];67 i16·:·R·=·QQ[x,y,z];
68 i17·:·D·=·divisor(x*y*(x+y))68 i17·:·D·=·divisor(x*y*(x+y))
  
69 o17·=·Div(x)·+·Div(x+y)·+·Div(y)69 o17·=·Div(x+y)·+·Div(y)·+·Div(x)
  
70 o17·:·WeilDivisor·on·R70 o17·:·WeilDivisor·on·R
71 i18·:·isSNC(·D,·IsGraded·=>·true)71 i18·:·isSNC(·D,·IsGraded·=>·true)
  
72 o18·=·false72 o18·=·false
73 The·output·value·of·this·function·is·stored·in·the·divisor's·cache·with·the73 The·output·value·of·this·function·is·stored·in·the·divisor's·cache·with·the
74 value·of·the·last·IsGraded·option.·If·you·change·the·IsGraded·option,·the·value74 value·of·the·last·IsGraded·option.·If·you·change·the·IsGraded·option,·the·value
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107 ········<table·class="examples">107 ········<table·class="examples">
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·ZZ/7[x,y,z];</code></pre>109 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·ZZ/7[x,y,z];</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·divisor(x*y)112 <td>··············<pre><code·class="language-macaulay2">i5·:·D·=·divisor(x*y)
  
113 o5·=·Div(y)·+·Div(x)113 o5·=·Div(x)·+·Div(y)
  
114 o5·:·WeilDivisor·on·R</code></pre>114 o5·:·WeilDivisor·on·R</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ··········<tr>116 ··········<tr>
117 <td>··············<pre><code·class="language-macaulay2">i6·:·mapToProjectiveSpace(D,·Variable=>&quot;Z&quot;)117 <td>··············<pre><code·class="language-macaulay2">i6·:·mapToProjectiveSpace(D,·Variable=>&quot;Z&quot;)
  
118 ·············ZZ············2·············2········2118 ·············ZZ············2·············2········2
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37 o3·:·RingMap·R·<--·QQ[YY·..YY·]37 o3·:·RingMap·R·<--·QQ[YY·..YY·]
38 ························1····238 ························1····2
39 The·user·may·also·specify·the·variable·name·of·the·new·projective·space.39 The·user·may·also·specify·the·variable·name·of·the·new·projective·space.
40 i4·:·R·=·ZZ/7[x,y,z];40 i4·:·R·=·ZZ/7[x,y,z];
41 i5·:·D·=·divisor(x*y)41 i5·:·D·=·divisor(x*y)
  
42 o5·=·Div(y)·+·Div(x)42 o5·=·Div(x)·+·Div(y)
  
43 o5·:·WeilDivisor·on·R43 o5·:·WeilDivisor·on·R
44 i6·:·mapToProjectiveSpace(D,·Variable=>"Z")44 i6·:·mapToProjectiveSpace(D,·Variable=>"Z")
  
45 ·············ZZ············2·············2········245 ·············ZZ············2·············2········2
46 o6·=·map·(R,·--[Z·..Z·],·{x·,·x*y,·x*z,·y·,·y*z,·z·})46 o6·=·map·(R,·--[Z·..Z·],·{x·,·x*y,·x*z,·y·,·y*z,·z·})
47 ··············7··1···647 ··············7··1···6
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91 <td>··············<pre><code·class="language-macaulay2">i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});91 <td>··············<pre><code·class="language-macaulay2">i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});
  
92 o3·:·RingMap·T·&lt;--·R</code></pre>92 o3·:·RingMap·T·&lt;--·R</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i4·:·D·=·divisor(y*z)95 <td>··············<pre><code·class="language-macaulay2">i4·:·D·=·divisor(y*z)
  
96 o4·=·3*Div(z,·y,·x)·+·3*Div(w,·z,·y)96 o4·=·3*Div(w,·z,·y)·+·3*Div(z,·y,·x)
  
97 o4·:·WeilDivisor·on·R</code></pre>97 o4·:·WeilDivisor·on·R</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i5·:·pullback(f,·D,·Strategy=>Primes)100 <td>··············<pre><code·class="language-macaulay2">i5·:·pullback(f,·D,·Strategy=>Primes)
  
101 o5·=·3*Div(a)·+·3*Div(b)101 o5·=·3*Div(b)·+·3*Div(a)
  
102 o5·:·WeilDivisor·on·T</code></pre>102 o5·:·WeilDivisor·on·T</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i6·:·pullback(f,·D,·Strategy=>Sheaves)105 <td>··············<pre><code·class="language-macaulay2">i6·:·pullback(f,·D,·Strategy=>Sheaves)
  
106 o6·=·3*Div(b)·+·3*Div(a)106 o6·=·3*Div(b)·+·3*Div(a)
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133 <td>··············<pre><code·class="language-macaulay2">i10·:·D·=·divisor(x*y*(x+y));133 <td>··············<pre><code·class="language-macaulay2">i10·:·D·=·divisor(x*y*(x+y));
  
134 o10·:·WeilDivisor·on·R</code></pre>134 o10·:·WeilDivisor·on·R</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i11·:·D1·=·pullback(f,·D)137 <td>··············<pre><code·class="language-macaulay2">i11·:·D1·=·pullback(f,·D)
  
138 o11·=·Div(a+1)·+·Div(a)·+·3*Div(b)138 o11·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
139 o11·:·WeilDivisor·on·S</code></pre>139 o11·:·WeilDivisor·on·S</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i12·:·f^*·D142 <td>··············<pre><code·class="language-macaulay2">i12·:·f^*·D
  
143 o12·=·Div(a+1)·+·Div(a)·+·3*Div(b)143 o12·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
144 o12·:·WeilDivisor·on·S</code></pre>144 o12·:·WeilDivisor·on·S</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ········</table>146 ········</table>
147 ········<div>147 ········<div>
148 ··········<p>As·illustrated·by·the·previous·example,·the·same·functionality·can·also·be·accomplished·by·<span·class="tt">f^*</span>·(which·creates·a·function·which·sends·a·divisor·$D$·to·$f^*·D$).</p>148 ··········<p>As·illustrated·by·the·previous·example,·the·same·functionality·can·also·be·accomplished·by·<span·class="tt">f^*</span>·(which·creates·a·function·which·sends·a·divisor·$D$·to·$f^*·D$).</p>
149 ········</div>149 ········</div>
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30 i1·:·R·=·QQ[x,y,z,w]/ideal(z^2-y*w,y*z-x*w,y^2-x*z);30 i1·:·R·=·QQ[x,y,z,w]/ideal(z^2-y*w,y*z-x*w,y^2-x*z);
31 i2·:·T·=·QQ[a,b];31 i2·:·T·=·QQ[a,b];
32 i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});32 i3·:·f·=·map(T,·R,·{a^3,·a^2*b,·a*b^2,·b^3});
  
33 o3·:·RingMap·T·<--·R33 o3·:·RingMap·T·<--·R
34 i4·:·D·=·divisor(y*z)34 i4·:·D·=·divisor(y*z)
  
35 o4·=·3*Div(z,·y,·x)·+·3*Div(w,·z,·y)35 o4·=·3*Div(w,·z,·y)·+·3*Div(z,·y,·x)
  
36 o4·:·WeilDivisor·on·R36 o4·:·WeilDivisor·on·R
37 i5·:·pullback(f,·D,·Strategy=>Primes)37 i5·:·pullback(f,·D,·Strategy=>Primes)
  
38 o5·=·3*Div(a)·+·3*Div(b)38 o5·=·3*Div(b)·+·3*Div(a)
  
39 o5·:·WeilDivisor·on·T39 o5·:·WeilDivisor·on·T
40 i6·:·pullback(f,·D,·Strategy=>Sheaves)40 i6·:·pullback(f,·D,·Strategy=>Sheaves)
  
41 o6·=·3*Div(b)·+·3*Div(a)41 o6·=·3*Div(b)·+·3*Div(a)
  
42 o6·:·WeilDivisor·on·T42 o6·:·WeilDivisor·on·T
Offset 54, 20 lines modifiedOffset 54, 20 lines modified
  
54 o9·:·RingMap·S·<--·R54 o9·:·RingMap·S·<--·R
55 i10·:·D·=·divisor(x*y*(x+y));55 i10·:·D·=·divisor(x*y*(x+y));
  
56 o10·:·WeilDivisor·on·R56 o10·:·WeilDivisor·on·R
57 i11·:·D1·=·pullback(f,·D)57 i11·:·D1·=·pullback(f,·D)
  
58 o11·=·Div(a+1)·+·Div(a)·+·3*Div(b)58 o11·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
59 o11·:·WeilDivisor·on·S59 o11·:·WeilDivisor·on·S
60 i12·:·f^*·D60 i12·:·f^*·D
  
61 o12·=·Div(a+1)·+·Div(a)·+·3*Div(b)61 o12·=·Div(a)·+·3*Div(b)·+·Div(a+1)
  
62 o12·:·WeilDivisor·on·S62 o12·:·WeilDivisor·on·S
63 As·illustrated·by·the·previous·example,·the·same·functionality·can·also·be63 As·illustrated·by·the·previous·example,·the·same·functionality·can·also·be
64 accomplished·by·f^*·(which·creates·a·function·which·sends·a·divisor·$D$·to·$f^*64 accomplished·by·f^*·(which·creates·a·function·which·sends·a·divisor·$D$·to·$f^*
65 D$).65 D$).
66 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*66 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
67 ····*·_\x8P_\x8r_\x8i_\x8m_\x8e_\x8s·--·a·value·for·the·option·Strategy·for·the·pullback·method67 ····*·_\x8P_\x8r_\x8i_\x8m_\x8e_\x8s·--·a·value·for·the·option·Strategy·for·the·pullback·method
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Offset 229, 21 lines modifiedOffset 229, 21 lines modified
229 ··········<tr>229 ··········<tr>
230 <td>··············<pre><code·class="language-macaulay2">i22·:·J·=·I^21;230 <td>··············<pre><code·class="language-macaulay2">i22·:·J·=·I^21;
  
231 o22·:·Ideal·of·R</code></pre>231 o22·:·Ideal·of·R</code></pre>
232 </td>··········</tr>232 </td>··········</tr>
233 ··········<tr>233 ··········<tr>
234 <td>··············<pre><code·class="language-macaulay2">i23·:·time·reflexify(J);234 <td>··············<pre><code·class="language-macaulay2">i23·:·time·reflexify(J);
235 ·--·used·0.179189s·(cpu);·0.147909s·(thread);·0s·(gc)235 ·--·used·0.203596s·(cpu);·0.160719s·(thread);·0s·(gc)
  
236 o23·:·Ideal·of·R</code></pre>236 o23·:·Ideal·of·R</code></pre>
237 </td>··········</tr>237 </td>··········</tr>
238 ··········<tr>238 ··········<tr>
239 <td>··············<pre><code·class="language-macaulay2">i24·:·time·reflexify(J*R^1);239 <td>··············<pre><code·class="language-macaulay2">i24·:·time·reflexify(J*R^1);
240 ·--·used·0.309212s·(cpu);·0.273311s·(thread);·0s·(gc)</code></pre>240 ·--·used·0.372987s·(cpu);·0.327706s·(thread);·0s·(gc)</code></pre>
241 </td>··········</tr>241 </td>··········</tr>
242 ········</table>242 ········</table>
243 ········<div>243 ········<div>
244 ··········<p>Because·of·this,·there·are·two·strategies·for·computing·a·reflexification·(at·least·if·the·module·embeds·as·an·ideal).</p>244 ··········<p>Because·of·this,·there·are·two·strategies·for·computing·a·reflexification·(at·least·if·the·module·embeds·as·an·ideal).</p>
245 ········</div>245 ········</div>
246 ········<div>246 ········<div>
247 ··········<p><span·class="tt">IdealStrategy</span>.··In·the·case·that·$R$·is·a·domain,·and·our·module·is·isomorphic·to·an·ideal·$I$,·then·one·can·compute·the·reflexification·by·computing·colons.</p>247 ··········<p><span·class="tt">IdealStrategy</span>.··In·the·case·that·$R$·is·a·domain,·and·our·module·is·isomorphic·to·an·ideal·$I$,·then·one·can·compute·the·reflexification·by·computing·colons.</p>
Offset 269, 42 lines modifiedOffset 269, 42 lines modified
269 o27·:·Ideal·of·R</code></pre>269 o27·:·Ideal·of·R</code></pre>
270 </td>··········</tr>270 </td>··········</tr>
271 ··········<tr>271 ··········<tr>
272 <td>··············<pre><code·class="language-macaulay2">i28·:·M·=·J*R^1;</code></pre>272 <td>··············<pre><code·class="language-macaulay2">i28·:·M·=·J*R^1;</code></pre>
273 </td>··········</tr>273 </td>··········</tr>
274 ··········<tr>274 ··········<tr>
275 <td>··············<pre><code·class="language-macaulay2">i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)275 <td>··············<pre><code·class="language-macaulay2">i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)
276 ·--·used·0.196918s·(cpu);·0.123476s·(thread);·0s·(gc)276 ·--·used·0.205717s·(cpu);·0.113598s·(thread);·0s·(gc)
  
277 ··············2············2·····9·······9···11277 ··············2············2·····9·······9···11
278 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)278 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
279 o29·:·Ideal·of·R</code></pre>279 o29·:·Ideal·of·R</code></pre>
280 </td>··········</tr>280 </td>··········</tr>
281 ··········<tr>281 ··········<tr>
282 <td>··············<pre><code·class="language-macaulay2">i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)282 <td>··············<pre><code·class="language-macaulay2">i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)
283 ·--·used·4.72641s·(cpu);·3.73712s·(thread);·0s·(gc)283 ·--·used·5.42104s·(cpu);·4.32059s·(thread);·0s·(gc)
  
284 ··············2············2·····9·······9···11284 ··············2············2·····9·······9···11
285 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)285 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
286 o30·:·Ideal·of·R</code></pre>286 o30·:·Ideal·of·R</code></pre>
287 </td>··········</tr>287 </td>··········</tr>
288 ··········<tr>288 ··········<tr>
289 <td>··············<pre><code·class="language-macaulay2">i31·:·J1·==·J2289 <td>··············<pre><code·class="language-macaulay2">i31·:·J1·==·J2
  
290 o31·=·true</code></pre>290 o31·=·true</code></pre>
291 </td>··········</tr>291 </td>··········</tr>
292 ··········<tr>292 ··········<tr>
293 <td>··············<pre><code·class="language-macaulay2">i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);293 <td>··············<pre><code·class="language-macaulay2">i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
294 ·--·used·4.94131s·(cpu);·3.99555s·(thread);·0s·(gc)</code></pre>294 ·--·used·5.42293s·(cpu);·4.27129s·(thread);·0s·(gc)</code></pre>
295 </td>··········</tr>295 </td>··········</tr>
296 ··········<tr>296 ··········<tr>
297 <td>··············<pre><code·class="language-macaulay2">i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);297 <td>··············<pre><code·class="language-macaulay2">i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
298 ·--·used·0.428155s·(cpu);·0.313441s·(thread);·0s·(gc)</code></pre>298 ·--·used·0.450809s·(cpu);·0.330221s·(thread);·0s·(gc)</code></pre>
299 </td>··········</tr>299 </td>··········</tr>
300 ········</table>300 ········</table>
301 ········<div>301 ········<div>
302 ··········<p>However,·sometimes·<span·class="tt">ModuleStrategy</span>·is·faster,·especially·for·Monomial·ideals.</p>302 ··········<p>However,·sometimes·<span·class="tt">ModuleStrategy</span>·is·faster,·especially·for·Monomial·ideals.</p>
303 ········</div>303 ········</div>
304 ········<table·class="examples">304 ········<table·class="examples">
305 ··········<tr>305 ··········<tr>
Offset 321, 49 lines modifiedOffset 321, 49 lines modified
321 o36·:·Ideal·of·R</code></pre>321 o36·:·Ideal·of·R</code></pre>
322 </td>··········</tr>322 </td>··········</tr>
323 ··········<tr>323 ··········<tr>
324 <td>··············<pre><code·class="language-macaulay2">i37·:·M·=·I^20*R^1;</code></pre>324 <td>··············<pre><code·class="language-macaulay2">i37·:·M·=·I^20*R^1;</code></pre>
325 </td>··········</tr>325 </td>··········</tr>
326 ··········<tr>326 ··········<tr>
327 <td>··············<pre><code·class="language-macaulay2">i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)327 <td>··············<pre><code·class="language-macaulay2">i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)
328 ·--·used·0.79094s·(cpu);·0.303707s·(thread);·0s·(gc)328 ·--·used·0.924912s·(cpu);·0.306257s·(thread);·0s·(gc)
  
329 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·329 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·
330 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,330 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
331 ······-----------------------------------------------------------------------331 ······-----------------------------------------------------------------------
332 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·332 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·
333 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,333 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
334 ······-----------------------------------------------------------------------334 ······-----------------------------------------------------------------------
335 ·······19····20335 ·······19····20
336 ······x··u,·x··)336 ······x··u,·x··)
  
337 o38·:·Ideal·of·R</code></pre>337 o38·:·Ideal·of·R</code></pre>
338 </td>··········</tr>338 </td>··········</tr>
339 ··········<tr>339 ··········<tr>
340 <td>··············<pre><code·class="language-macaulay2">i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)340 <td>··············<pre><code·class="language-macaulay2">i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)
341 ·--·used·0.0145054s·(cpu);·0.0149865s·(thread);·0s·(gc)341 ·--·used·0.0133773s·(cpu);·0.0129823s·(thread);·0s·(gc)
  
342 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·342 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12·
343 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,343 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
344 ······-----------------------------------------------------------------------344 ······-----------------------------------------------------------------------
345 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·345 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2·
346 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,346 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
347 ······-----------------------------------------------------------------------347 ······-----------------------------------------------------------------------
348 ·······19····20348 ·······19····20
349 ······x··u,·x··)349 ······x··u,·x··)
  
350 o39·:·Ideal·of·R</code></pre>350 o39·:·Ideal·of·R</code></pre>
351 </td>··········</tr>351 </td>··········</tr>
352 ··········<tr>352 ··········<tr>
353 <td>··············<pre><code·class="language-macaulay2">i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);353 <td>··············<pre><code·class="language-macaulay2">i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
354 ·--·used·0.194438s·(cpu);·0.0762503s·(thread);·0s·(gc)</code></pre>354 ·--·used·0.223368s·(cpu);·0.0770611s·(thread);·0s·(gc)</code></pre>
355 </td>··········</tr>355 </td>··········</tr>
356 ··········<tr>356 ··········<tr>
357 <td>··············<pre><code·class="language-macaulay2">i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);357 <td>··············<pre><code·class="language-macaulay2">i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
358 ·--·used·0.00399938s·(cpu);·0.0057327s·(thread);·0s·(gc)</code></pre>358 ·--·used·0.00763461s·(cpu);·0.00806738s·(thread);·0s·(gc)</code></pre>
359 </td>··········</tr>359 </td>··········</tr>
360 ········</table>360 ········</table>
361 ········<div>361 ········<div>
362 ··········<p>For·ideals,·if·<span·class="tt">KnownDomain</span>·is·false·(default·value·is·<span·class="tt">true</span>),·then·the·function·will·check·whether·it·is·a·domain.··If·it·is·a·domain·(or·assumed·to·be·a·domain),·it·will·reflexify·using·a·strategy·which·can·speed·up·computation,·if·not·it·will·compute·using·a·sometimes·slower·method·which·is·essentially·reflexifying·it·as·a·module.</p>362 ··········<p>For·ideals,·if·<span·class="tt">KnownDomain</span>·is·false·(default·value·is·<span·class="tt">true</span>),·then·the·function·will·check·whether·it·is·a·domain.··If·it·is·a·domain·(or·assumed·to·be·a·domain),·it·will·reflexify·using·a·strategy·which·can·speed·up·computation,·if·not·it·will·compute·using·a·sometimes·slower·method·which·is·essentially·reflexifying·it·as·a·module.</p>
363 ········</div>363 ········</div>
364 ········<div>364 ········<div>
365 ··········<p>Consider·the·following·example·showing·the·importance·of·making·the·correct·assumption·about·the·ring·being·a·domain.</p>365 ··········<p>Consider·the·following·example·showing·the·importance·of·making·the·correct·assumption·about·the·ring·being·a·domain.</p>
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115 i21·:·I·=·ideal(x-z,y-2*z);115 i21·:·I·=·ideal(x-z,y-2*z);
  
116 o21·:·Ideal·of·R116 o21·:·Ideal·of·R
117 i22·:·J·=·I^21;117 i22·:·J·=·I^21;
  
118 o22·:·Ideal·of·R118 o22·:·Ideal·of·R
119 i23·:·time·reflexify(J);119 i23·:·time·reflexify(J);
120 ·--·used·0.179189s·(cpu);·0.147909s·(thread);·0s·(gc)120 ·--·used·0.203596s·(cpu);·0.160719s·(thread);·0s·(gc)
  
121 o23·:·Ideal·of·R121 o23·:·Ideal·of·R
122 i24·:·time·reflexify(J*R^1);122 i24·:·time·reflexify(J*R^1);
123 ·--·used·0.309212s·(cpu);·0.273311s·(thread);·0s·(gc)123 ·--·used·0.372987s·(cpu);·0.327706s·(thread);·0s·(gc)
124 Because·of·this,·there·are·two·strategies·for·computing·a·reflexification·(at124 Because·of·this,·there·are·two·strategies·for·computing·a·reflexification·(at
125 least·if·the·module·embeds·as·an·ideal).125 least·if·the·module·embeds·as·an·ideal).
126 IdealStrategy.·In·the·case·that·$R$·is·a·domain,·and·our·module·is·isomorphic126 IdealStrategy.·In·the·case·that·$R$·is·a·domain,·and·our·module·is·isomorphic
127 to·an·ideal·$I$,·then·one·can·compute·the·reflexification·by·computing·colons.127 to·an·ideal·$I$,·then·one·can·compute·the·reflexification·by·computing·colons.
128 ModuleStrategy.·This·computes·the·reflexification·simply·by·computing·$Hom$128 ModuleStrategy.·This·computes·the·reflexification·simply·by·computing·$Hom$
129 twice.129 twice.
130 ModuleStrategy·is·the·default·strategy·for·modules,·IdealStrategy·is·the130 ModuleStrategy·is·the·default·strategy·for·modules,·IdealStrategy·is·the
Offset 140, 73 lines modifiedOffset 140, 73 lines modified
  
140 o26·:·Ideal·of·R140 o26·:·Ideal·of·R
141 i27·:·J·=·I^20;141 i27·:·J·=·I^20;
  
142 o27·:·Ideal·of·R142 o27·:·Ideal·of·R
143 i28·:·M·=·J*R^1;143 i28·:·M·=·J*R^1;
144 i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)144 i29·:·J1·=·time·reflexify(·J,·Strategy=>IdealStrategy·)
145 ·--·used·0.196918s·(cpu);·0.123476s·(thread);·0s·(gc)145 ·--·used·0.205717s·(cpu);·0.113598s·(thread);·0s·(gc)
  
146 ··············2············2·····9·······9···11146 ··············2············2·····9·······9···11
147 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)147 o29·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
148 o29·:·Ideal·of·R148 o29·:·Ideal·of·R
149 i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)149 i30·:·J2·=·time·reflexify(·J,·Strategy=>ModuleStrategy·)
150 ·--·used·4.72641s·(cpu);·3.73712s·(thread);·0s·(gc)150 ·--·used·5.42104s·(cpu);·4.32059s·(thread);·0s·(gc)
  
151 ··············2············2·····9·······9···11151 ··············2············2·····9·······9···11
152 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)152 o30·=·ideal·(x··+·5x*y·+·3y·,·x*z··-·4y*z·,·z···+·x·-·4y)
  
153 o30·:·Ideal·of·R153 o30·:·Ideal·of·R
154 i31·:·J1·==·J2154 i31·:·J1·==·J2
  
155 o31·=·true155 o31·=·true
156 i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);156 i32·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
157 ·--·used·4.94131s·(cpu);·3.99555s·(thread);·0s·(gc)157 ·--·used·5.42293s·(cpu);·4.27129s·(thread);·0s·(gc)
158 i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);158 i33·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
159 ·--·used·0.428155s·(cpu);·0.313441s·(thread);·0s·(gc)159 ·--·used·0.450809s·(cpu);·0.330221s·(thread);·0s·(gc)
160 However,·sometimes·ModuleStrategy·is·faster,·especially·for·Monomial·ideals.160 However,·sometimes·ModuleStrategy·is·faster,·especially·for·Monomial·ideals.
161 i34·:·R·=·QQ[x,y,u,v]/ideal(x*y-u*v);161 i34·:·R·=·QQ[x,y,u,v]/ideal(x*y-u*v);
162 i35·:·I·=·ideal(x,u);162 i35·:·I·=·ideal(x,u);
  
163 o35·:·Ideal·of·R163 o35·:·Ideal·of·R
164 i36·:·J·=·I^20;164 i36·:·J·=·I^20;
  
165 o36·:·Ideal·of·R165 o36·:·Ideal·of·R
166 i37·:·M·=·I^20*R^1;166 i37·:·M·=·I^20*R^1;
167 i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)167 i38·:·time·reflexify(·J,·Strategy=>IdealStrategy·)
168 ·--·used·0.79094s·(cpu);·0.303707s·(thread);·0s·(gc)168 ·--·used·0.924912s·(cpu);·0.306257s·(thread);·0s·(gc)
  
169 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12169 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12
170 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,170 o38·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
171 ······-----------------------------------------------------------------------171 ······-----------------------------------------------------------------------
172 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2172 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2
173 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,173 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
174 ······-----------------------------------------------------------------------174 ······-----------------------------------------------------------------------
175 ·······19····20175 ·······19····20
176 ······x··u,·x··)176 ······x··u,·x··)
  
177 o38·:·Ideal·of·R177 o38·:·Ideal·of·R
178 i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)178 i39·:·time·reflexify(·J,·Strategy=>ModuleStrategy·)
179 ·--·used·0.0145054s·(cpu);·0.0149865s·(thread);·0s·(gc)179 ·--·used·0.0133773s·(cpu);·0.0129823s·(thread);·0s·(gc)
  
180 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12180 ··············20·····19···2·18···3·17···4·16···5·15···6·14···7·13···8·12
181 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,181 o39·=·ideal·(u··,·x*u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,·x·u··,
182 ······-----------------------------------------------------------------------182 ······-----------------------------------------------------------------------
183 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2183 ·······9·11···10·10···11·9···12·8···13·7···14·6···15·5···16·4···17·3···18·2
184 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,184 ······x·u··,·x··u··,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,·x··u·,
185 ······-----------------------------------------------------------------------185 ······-----------------------------------------------------------------------
186 ·······19····20186 ·······19····20
187 ······x··u,·x··)187 ······x··u,·x··)
  
188 o39·:·Ideal·of·R188 o39·:·Ideal·of·R
189 i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);189 i40·:·time·reflexify(·M,·Strategy=>IdealStrategy·);
190 ·--·used·0.194438s·(cpu);·0.0762503s·(thread);·0s·(gc)190 ·--·used·0.223368s·(cpu);·0.0770611s·(thread);·0s·(gc)
191 i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);191 i41·:·time·reflexify(·M,·Strategy=>ModuleStrategy·);
192 ·--·used·0.00399938s·(cpu);·0.0057327s·(thread);·0s·(gc)192 ·--·used·0.00763461s·(cpu);·0.00806738s·(thread);·0s·(gc)
193 For·ideals,·if·KnownDomain·is·false·(default·value·is·true),·then·the·function193 For·ideals,·if·KnownDomain·is·false·(default·value·is·true),·then·the·function
194 will·check·whether·it·is·a·domain.·If·it·is·a·domain·(or·assumed·to·be·a194 will·check·whether·it·is·a·domain.·If·it·is·a·domain·(or·assumed·to·be·a
195 domain),·it·will·reflexify·using·a·strategy·which·can·speed·up·computation,·if195 domain),·it·will·reflexify·using·a·strategy·which·can·speed·up·computation,·if
196 not·it·will·compute·using·a·sometimes·slower·method·which·is·essentially196 not·it·will·compute·using·a·sometimes·slower·method·which·is·essentially
197 reflexifying·it·as·a·module.197 reflexifying·it·as·a·module.
198 Consider·the·following·example·showing·the·importance·of·making·the·correct198 Consider·the·following·example·showing·the·importance·of·making·the·correct
199 assumption·about·the·ring·being·a·domain.199 assumption·about·the·ring·being·a·domain.
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114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·ideal(x-z,y-2*z);115 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·ideal(x-z,y-2*z);
  
116 o6·:·Ideal·of·R</code></pre>116 o6·:·Ideal·of·R</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i7·:·time·J20a·=·reflexivePower(20,·I);119 <td>··············<pre><code·class="language-macaulay2">i7·:·time·J20a·=·reflexivePower(20,·I);
120 ·--·used·0.0731817s·(cpu);·0.0389498s·(thread);·0s·(gc)120 ·--·used·0.0913967s·(cpu);·0.0405707s·(thread);·0s·(gc)
  
121 o7·:·Ideal·of·R</code></pre>121 o7·:·Ideal·of·R</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i8·:·I20·=·I^20;124 <td>··············<pre><code·class="language-macaulay2">i8·:·I20·=·I^20;
  
125 o8·:·Ideal·of·R</code></pre>125 o8·:·Ideal·of·R</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i9·:·time·J20b·=·reflexify(I20);128 <td>··············<pre><code·class="language-macaulay2">i9·:·time·J20b·=·reflexify(I20);
129 ·--·used·0.098693s·(cpu);·0.0988768s·(thread);·0s·(gc)129 ·--·used·0.125986s·(cpu);·0.125611s·(thread);·0s·(gc)
  
130 o9·:·Ideal·of·R</code></pre>130 o9·:·Ideal·of·R</code></pre>
131 </td>··········</tr>131 </td>··········</tr>
132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i10·:·J20a·==·J20b133 <td>··············<pre><code·class="language-macaulay2">i10·:·J20a·==·J20b
  
134 o10·=·true</code></pre>134 o10·=·true</code></pre>
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149 ··········<tr>149 ··········<tr>
150 <td>··············<pre><code·class="language-macaulay2">i12·:·I·=·ideal(x-z,y-2*z);150 <td>··············<pre><code·class="language-macaulay2">i12·:·I·=·ideal(x-z,y-2*z);
  
151 o12·:·Ideal·of·R</code></pre>151 o12·:·Ideal·of·R</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);154 <td>··············<pre><code·class="language-macaulay2">i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);
155 ·--·used·0.0232514s·(cpu);·0.0251434s·(thread);·0s·(gc)155 ·--·used·0.0346998s·(cpu);·0.0330071s·(thread);·0s·(gc)
  
156 o13·:·Ideal·of·R</code></pre>156 o13·:·Ideal·of·R</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);159 <td>··············<pre><code·class="language-macaulay2">i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);
160 ·--·used·0.0986717s·(cpu);·0.0788927s·(thread);·0s·(gc)160 ·--·used·0.137004s·(cpu);·0.097576s·(thread);·0s·(gc)
  
161 o14·:·Ideal·of·R</code></pre>161 o14·:·Ideal·of·R</code></pre>
162 </td>··········</tr>162 </td>··········</tr>
163 ··········<tr>163 ··········<tr>
164 <td>··············<pre><code·class="language-macaulay2">i15·:·J1·==·J2164 <td>··············<pre><code·class="language-macaulay2">i15·:·J1·==·J2
  
165 o15·=·true</code></pre>165 o15·=·true</code></pre>
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Offset 41, 39 lines modifiedOffset 41, 39 lines modified
41 of·the·generators·of·$I$.·Consider·the·example·of·a·cone·over·a·point·on·an41 of·the·generators·of·$I$.·Consider·the·example·of·a·cone·over·a·point·on·an
42 elliptic·curve.42 elliptic·curve.
43 i5·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);43 i5·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);
44 i6·:·I·=·ideal(x-z,y-2*z);44 i6·:·I·=·ideal(x-z,y-2*z);
  
45 o6·:·Ideal·of·R45 o6·:·Ideal·of·R
46 i7·:·time·J20a·=·reflexivePower(20,·I);46 i7·:·time·J20a·=·reflexivePower(20,·I);
47 ·--·used·0.0731817s·(cpu);·0.0389498s·(thread);·0s·(gc)47 ·--·used·0.0913967s·(cpu);·0.0405707s·(thread);·0s·(gc)
  
48 o7·:·Ideal·of·R48 o7·:·Ideal·of·R
49 i8·:·I20·=·I^20;49 i8·:·I20·=·I^20;
  
50 o8·:·Ideal·of·R50 o8·:·Ideal·of·R
51 i9·:·time·J20b·=·reflexify(I20);51 i9·:·time·J20b·=·reflexify(I20);
52 ·--·used·0.098693s·(cpu);·0.0988768s·(thread);·0s·(gc)52 ·--·used·0.125986s·(cpu);·0.125611s·(thread);·0s·(gc)
  
53 o9·:·Ideal·of·R53 o9·:·Ideal·of·R
54 i10·:·J20a·==·J20b54 i10·:·J20a·==·J20b
  
55 o10·=·true55 o10·=·true
56 This·passes·the·Strategy·option·to·a·reflexify·call.·Valid·options·are56 This·passes·the·Strategy·option·to·a·reflexify·call.·Valid·options·are
57 IdealStrategy·and·ModuleStrategy.57 IdealStrategy·and·ModuleStrategy.
58 i11·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);58 i11·:·R·=·QQ[x,y,z]/ideal(-y^2*z·+x^3·+·x^2*z·+·x*z^2+z^3);
59 i12·:·I·=·ideal(x-z,y-2*z);59 i12·:·I·=·ideal(x-z,y-2*z);
  
60 o12·:·Ideal·of·R60 o12·:·Ideal·of·R
61 i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);61 i13·:·time·J1·=·reflexivePower(20,·I,·Strategy=>IdealStrategy);
62 ·--·used·0.0232514s·(cpu);·0.0251434s·(thread);·0s·(gc)62 ·--·used·0.0346998s·(cpu);·0.0330071s·(thread);·0s·(gc)
  
63 o13·:·Ideal·of·R63 o13·:·Ideal·of·R
64 i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);64 i14·:·time·J2·=·reflexivePower(20,·I,·Strategy=>ModuleStrategy);
65 ·--·used·0.0986717s·(cpu);·0.0788927s·(thread);·0s·(gc)65 ·--·used·0.137004s·(cpu);·0.097576s·(thread);·0s·(gc)
  
66 o14·:·Ideal·of·R66 o14·:·Ideal·of·R
67 i15·:·J1·==·J267 i15·:·J1·==·J2
  
68 o15·=·true68 o15·=·true
69 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*69 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
70 ····*·_\x8r_\x8e_\x8f_\x8l_\x8e_\x8x_\x8i_\x8f_\x8y·--·calculate·the·double·dual·of·an·ideal·or·module·Hom(Hom(M,70 ····*·_\x8r_\x8e_\x8f_\x8l_\x8e_\x8x_\x8i_\x8f_\x8y·--·calculate·the·double·dual·of·an·ideal·or·module·Hom(Hom(M,
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11 dWx0IG9wdGlvbiBmb3IgcGljdHVyZSIsICJsaW5lbnVtIiA9PiAxMzUwLCBJbnB1dHMgPT4ge1NQ11 dWx0IG9wdGlvbiBmb3IgcGljdHVyZSIsICJsaW5lbnVtIiA9PiAxMzUwLCBJbnB1dHMgPT4ge1NQ
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4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=157 #:len=15
8 Y29tcGxlbWVudEdyYXBo8 Y29tcGxlbWVudEdyYXBo
9 #:len=20119 #:len=2011
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv
11 ZiBhIGdyYXBoIG9yIGh5cGVyZ3JhcGgiLCAibGluZW51bSIgPT4gMjA4MCwgSW5wdXRzID0+IHtT11 ZiBhIGdyYXBoIG9yIGh5cGVyZ3JhcGgiLCAibGluZW51bSIgPT4gMjA4MCwgSW5wdXRzID0+IHtT
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./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_complement__Graph.out
    
Offset 2, 15 lines modifiedOffset 2, 15 lines modified
  
2 i1·:·R·=·QQ[a,b,c,d,e];2 i1·:·R·=·QQ[a,b,c,d,e];
  
3 i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle3 i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle
  
4 i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle4 i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle
  
5 o3·=·Graph{"edges"·=>·{{a,·d},·{a,·c},·{b,·e},·{b,·d},·{c,·e}}}5 o3·=·Graph{"edges"·=>·{{a,·c},·{b,·e},·{b,·d},·{c,·e},·{a,·d}}}
6 ···········"ring"·=>·R6 ···········"ring"·=>·R
7 ···········"vertices"·=>·{a,·b,·c,·d,·e}7 ···········"vertices"·=>·{a,·b,·c,·d,·e}
  
8 o3·:·Graph8 o3·:·Graph
  
9 i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph9 i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph
  
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./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
    
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3 i1·:·R·=·QQ[x_1..x_5];3 i1·:·R·=·QQ[x_1..x_5];
  
4 i2·:·randomHyperGraph(R,{3,2,4})4 i2·:·randomHyperGraph(R,{3,2,4})
  
5 i3·:·randomHyperGraph(R,{3,2,4})5 i3·:·randomHyperGraph(R,{3,2,4})
  
6 o3·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}6 o3·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
7 ······························2···3···4·····1···2·····1···3···4···57 ······························1···4···5·····1···2·····2···3···4···5
8 ················"ring"·=>·R8 ················"ring"·=>·R
9 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}9 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}
10 ································1···2···3···4···510 ································1···2···3···4···5
  
11 o3·:·HyperGraph11 o3·:·HyperGraph
  
12 i4·:·randomHyperGraph(R,{3,2,4})12 i4·:·randomHyperGraph(R,{3,2,4})
  
13 o4·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}13 o4·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
14 ······························3···4···5·····1···3·····1···2···4···514 ······························3···4···5·····2···4·····1···2···3···5
15 ················"ring"·=>·R15 ················"ring"·=>·R
16 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}16 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}
17 ································1···2···3···4···517 ································1···2···3···4···5
  
18 o4·:·HyperGraph18 o4·:·HyperGraph
  
19 i5·:·randomHyperGraph(R,{4,4,2,2})·--·impossible,·returns·null·when·time/branch·limit·reached19 i5·:·randomHyperGraph(R,{4,4,2,2})·--·impossible,·returns·null·when·time/branch·limit·reached
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./usr/share/doc/Macaulay2/EdgeIdeals/html/_complement__Graph.html
    
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81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle</code></pre>83 <td>··············<pre><code·class="language-macaulay2">i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle86 <td>··············<pre><code·class="language-macaulay2">i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle
  
87 o3·=·Graph{&quot;edges&quot;·=>·{{a,·d},·{a,·c},·{b,·e},·{b,·d},·{c,·e}}}87 o3·=·Graph{&quot;edges&quot;·=>·{{a,·c},·{b,·e},·{b,·d},·{c,·e},·{a,·d}}}
88 ···········&quot;ring&quot;·=>·R88 ···········&quot;ring&quot;·=>·R
89 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}89 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}
  
90 o3·:·Graph</code></pre>90 o3·:·Graph</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph93 <td>··············<pre><code·class="language-macaulay2">i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph
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23 When·applied·to·a·graph,·complementGraph·returns·the·graph·whose·edge·set·is23 When·applied·to·a·graph,·complementGraph·returns·the·graph·whose·edge·set·is
24 the·set·of·edges·not·in·G.·When·applied·to·a·hypergraph,·the·edge·set·is·found24 the·set·of·edges·not·in·G.·When·applied·to·a·hypergraph,·the·edge·set·is·found
25 by·taking·the·complement·of·each·edge·of·H·in·the·vertex·set.25 by·taking·the·complement·of·each·edge·of·H·in·the·vertex·set.
26 i1·:·R·=·QQ[a,b,c,d,e];26 i1·:·R·=·QQ[a,b,c,d,e];
27 i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle27 i2·:·c5·=·graph·{a*b,b*c,c*d,d*e,e*a};·--·graph·of·the·5-cycle
28 i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle28 i3·:·complementGraph·c5·--·the·graph·complement·of·the·5-cycle
  
29 o3·=·Graph{"edges"·=>·{{a,·d},·{a,·c},·{b,·e},·{b,·d},·{c,·e}}}29 o3·=·Graph{"edges"·=>·{{a,·c},·{b,·e},·{b,·d},·{c,·e},·{a,·d}}}
30 ···········"ring"·=>·R30 ···········"ring"·=>·R
31 ···········"vertices"·=>·{a,·b,·c,·d,·e}31 ···········"vertices"·=>·{a,·b,·c,·d,·e}
  
32 o3·:·Graph32 o3·:·Graph
33 i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph33 i4·:·c5hypergraph·=·hyperGraph·c5·--·the·5-cycle,·but·viewed·as·a·hypergraph
  
34 o4·=·HyperGraph{"edges"·=>·{{a,·b},·{b,·c},·{c,·d},·{a,·e},·{d,·e}}}34 o4·=·HyperGraph{"edges"·=>·{{a,·b},·{b,·c},·{c,·d},·{a,·e},·{d,·e}}}
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./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
    
Offset 87, 26 lines modifiedOffset 87, 26 lines modified
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·randomHyperGraph(R,{3,2,4})</code></pre>88 <td>··············<pre><code·class="language-macaulay2">i2·:·randomHyperGraph(R,{3,2,4})</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·randomHyperGraph(R,{3,2,4})91 <td>··············<pre><code·class="language-macaulay2">i3·:·randomHyperGraph(R,{3,2,4})
  
92 o3·=·HyperGraph{&quot;edges&quot;·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}92 o3·=·HyperGraph{&quot;edges&quot;·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
93 ······························2···3···4·····1···2·····1···3···4···593 ······························1···4···5·····1···2·····2···3···4···5
94 ················&quot;ring&quot;·=>·R94 ················&quot;ring&quot;·=>·R
95 ················&quot;vertices&quot;·=>·{x·,·x·,·x·,·x·,·x·}95 ················&quot;vertices&quot;·=>·{x·,·x·,·x·,·x·,·x·}
96 ································1···2···3···4···596 ································1···2···3···4···5
  
97 o3·:·HyperGraph</code></pre>97 o3·:·HyperGraph</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·randomHyperGraph(R,{3,2,4})100 <td>··············<pre><code·class="language-macaulay2">i4·:·randomHyperGraph(R,{3,2,4})
  
101 o4·=·HyperGraph{&quot;edges&quot;·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}101 o4·=·HyperGraph{&quot;edges&quot;·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
102 ······························3···4···5·····1···3·····1···2···4···5102 ······························3···4···5·····2···4·····1···2···3···5
103 ················&quot;ring&quot;·=>·R103 ················&quot;ring&quot;·=>·R
104 ················&quot;vertices&quot;·=>·{x·,·x·,·x·,·x·,·x·}104 ················&quot;vertices&quot;·=>·{x·,·x·,·x·,·x·,·x·}
105 ································1···2···3···4···5105 ································1···2···3···4···5
  
106 o4·:·HyperGraph</code></pre>106 o4·:·HyperGraph</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
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26 _\x8T_\x8i_\x8m_\x8e_\x8L_\x8i_\x8m_\x8i_\x8t).·The·method·will·return·null·if·it·cannot·find·a·hypergraph·within26 _\x8T_\x8i_\x8m_\x8e_\x8L_\x8i_\x8m_\x8i_\x8t).·The·method·will·return·null·if·it·cannot·find·a·hypergraph·within
27 the·branch·and·time·limits.27 the·branch·and·time·limits.
28 i1·:·R·=·QQ[x_1..x_5];28 i1·:·R·=·QQ[x_1..x_5];
29 i2·:·randomHyperGraph(R,{3,2,4})29 i2·:·randomHyperGraph(R,{3,2,4})
30 i3·:·randomHyperGraph(R,{3,2,4})30 i3·:·randomHyperGraph(R,{3,2,4})
  
31 o3·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}31 o3·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
32 ······························2···3···4·····1···2·····1···3···4···532 ······························1···4···5·····1···2·····2···3···4···5
33 ················"ring"·=>·R33 ················"ring"·=>·R
34 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}34 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}
35 ································1···2···3···4···535 ································1···2···3···4···5
  
36 o3·:·HyperGraph36 o3·:·HyperGraph
37 i4·:·randomHyperGraph(R,{3,2,4})37 i4·:·randomHyperGraph(R,{3,2,4})
  
38 o4·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}38 o4·=·HyperGraph{"edges"·=>·{{x·,·x·,·x·},·{x·,·x·},·{x·,·x·,·x·,·x·}}}
39 ······························3···4···5·····1···3·····1···2···4···539 ······························3···4···5·····2···4·····1···2···3···5
40 ················"ring"·=>·R40 ················"ring"·=>·R
41 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}41 ················"vertices"·=>·{x·,·x·,·x·,·x·,·x·}
42 ································1···2···3···4···542 ································1···2···3···4···5
  
43 o4·:·HyperGraph43 o4·:·HyperGraph
44 i5·:·randomHyperGraph(R,{4,4,2,2})·--·impossible,·returns·null·when·time/branch44 i5·:·randomHyperGraph(R,{4,4,2,2})·--·impossible,·returns·null·when·time/branch
45 limit·reached45 limit·reached
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./usr/share/doc/Macaulay2/EigenSolver/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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6 #·End·of·header6 #·End·of·header
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s
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./usr/share/doc/Macaulay2/EigenSolver/example-output/___Eigen__Solver.out
    
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15 ·····a*b*e*f·+·a*d*e*f·+·c*d*e*f,·a*b*c*d*e·+·a*b*c*d*f·+·a*b*c*e*f·+15 ·····a*b*e*f·+·a*d*e*f·+·c*d*e*f,·a*b*c*d*e·+·a*b*c*d*f·+·a*b*c*e*f·+
16 ·····------------------------------------------------------------------------16 ·····------------------------------------------------------------------------
17 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)17 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)
  
18 o2·:·Ideal·of·QQ[a..f]18 o2·:·Ideal·of·QQ[a..f]
  
19 i3·:·elapsedTime·sols·=·zeroDimSolve·I;19 i3·:·elapsedTime·sols·=·zeroDimSolve·I;
20 ·--·.167363s·elapsed20 ·--·.322044s·elapsed
  
21 i4·:·#sols·--·156·solutions21 i4·:·#sols·--·156·solutions
  
22 o4·=·15622 o4·=·156
  
23 i5·:·23 i5·:·
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./usr/share/doc/Macaulay2/EigenSolver/html/index.html
    
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68 ·····------------------------------------------------------------------------68 ·····------------------------------------------------------------------------
69 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)69 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)
  
70 o2·:·Ideal·of·QQ[a..f]</code></pre>70 o2·:·Ideal·of·QQ[a..f]</code></pre>
71 </td>··········</tr>71 </td>··········</tr>
72 ··········<tr>72 ··········<tr>
73 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·sols·=·zeroDimSolve·I;73 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·sols·=·zeroDimSolve·I;
74 ·--·.167363s·elapsed</code></pre>74 ·--·.322044s·elapsed</code></pre>
75 </td>··········</tr>75 </td>··········</tr>
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i4·:·#sols·--·156·solutions77 <td>··············<pre><code·class="language-macaulay2">i4·:·#sols·--·156·solutions
  
78 o4·=·156</code></pre>78 o4·=·156</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ········</table>80 ········</table>
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29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
30 ·····a*b*e*f·+·a*d*e*f·+·c*d*e*f,·a*b*c*d*e·+·a*b*c*d*f·+·a*b*c*e*f·+30 ·····a*b*e*f·+·a*d*e*f·+·c*d*e*f,·a*b*c*d*e·+·a*b*c*d*f·+·a*b*c*e*f·+
31 ·····------------------------------------------------------------------------31 ·····------------------------------------------------------------------------
32 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)32 ·····a*b*d*e*f·+·a*c*d*e*f·+·b*c*d*e*f,·a*b*c*d*e*f·-·1)
  
33 o2·:·Ideal·of·QQ[a..f]33 o2·:·Ideal·of·QQ[a..f]
34 i3·:·elapsedTime·sols·=·zeroDimSolve·I;34 i3·:·elapsedTime·sols·=·zeroDimSolve·I;
35 ·--·.167363s·elapsed35 ·--·.322044s·elapsed
36 i4·:·#sols·--·156·solutions36 i4·:·#sols·--·156·solutions
  
37 o4·=·15637 o4·=·156
38 The·authors·would·like·to·acknowledge·the·June·2020·Macaulay2·workshop·held38 The·authors·would·like·to·acknowledge·the·June·2020·Macaulay2·workshop·held
39 virtually·at·Warwick,·where·this·package·was·first·developed.39 virtually·at·Warwick,·where·this·package·was·first·developed.
40 R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s:40 R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s:
41 ····*·[1]·Sturmfels,·Bernd.·Solving·systems·of·polynomial·equations.·No.·97.41 ····*·[1]·Sturmfels,·Bernd.·Solving·systems·of·polynomial·equations.·No.·97.
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9 #:len=2009 #:len=200
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./usr/share/doc/Macaulay2/Elimination/example-output/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.out
    
Offset 17, 23 lines modifiedOffset 17, 23 lines modified
  
17 ······217 ······2
18 o3·=·x··+·x*c·+·d18 o3·=·x··+·x*c·+·d
  
19 o3·:·R19 o3·:·R
  
20 i4·:·time·eliminate(x,ideal(f,g))20 i4·:·time·eliminate(x,ideal(f,g))
21 ·--·used·0.00400159s·(cpu);·0.00200992s·(thread);·0s·(gc)21 ·--·used·0.00399867s·(cpu);·0.0031114s·(thread);·0s·(gc)
  
22 ······················2····2·············2···········222 ······················2····2·············2···········2
23 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)23 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
24 o4·:·Ideal·of·R24 o4·:·Ideal·of·R
  
25 i5·:·time·ideal·resultant(f,g,x)25 i5·:·time·ideal·resultant(f,g,x)
26 ·--·used·0.000336236s·(cpu);·0.00116058s·(thread);·0s·(gc)26 ·--·used·0.000376225s·(cpu);·0.00176721s·(thread);·0s·(gc)
  
27 ························2····2·············2···········227 ························2····2·············2···········2
28 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)28 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
29 o5·:·Ideal·of·R29 o5·:·Ideal·of·R
  
30 i6·:·sylvesterMatrix(f,g,x)30 i6·:·sylvesterMatrix(f,g,x)
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808 B
./usr/share/doc/Macaulay2/Elimination/example-output/_eliminate.out
    
Offset 17, 23 lines modifiedOffset 17, 23 lines modified
  
17 ······217 ······2
18 o3·=·x··+·x*c·+·d18 o3·=·x··+·x*c·+·d
  
19 o3·:·R19 o3·:·R
  
20 i4·:·time·eliminate(x,ideal(f,g))20 i4·:·time·eliminate(x,ideal(f,g))
21 ·--·used·0.00371003s·(cpu);·0.00208006s·(thread);·0s·(gc)21 ·--·used·0.00400961s·(cpu);·0.00222181s·(thread);·0s·(gc)
  
22 ······················2····2·············2···········222 ······················2····2·············2···········2
23 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)23 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
24 o4·:·Ideal·of·R24 o4·:·Ideal·of·R
  
25 i5·:·time·ideal·resultant(f,g,x)25 i5·:·time·ideal·resultant(f,g,x)
26 ·--·used·0.000295395s·(cpu);·0.00112741s·(thread);·0s·(gc)26 ·--·used·0.000313869s·(cpu);·0.00129816s·(thread);·0s·(gc)
  
27 ························2····2·············2···········227 ························2····2·············2···········2
28 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)28 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
29 o5·:·Ideal·of·R29 o5·:·Ideal·of·R
  
30 i6·:·sylvesterMatrix(f,g,x)30 i6·:·sylvesterMatrix(f,g,x)
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./usr/share/doc/Macaulay2/Elimination/example-output/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.out
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
  
17 ······8····517 ······8····5
18 o3·=·x··+·x··+·x*c·+·d18 o3·=·x··+·x··+·x*c·+·d
  
19 o3·:·R19 o3·:·R
  
20 i4·:·time·eliminate(ideal(f,g),x)20 i4·:·time·eliminate(ideal(f,g),x)
21 ·--·used·1.24333s·(cpu);·1.12073s·(thread);·0s·(gc)21 ·--·used·1.3537s·(cpu);·1.21733s·(thread);·0s·(gc)
  
22 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··22 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
23 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-23 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
24 ·····------------------------------------------------------------------------24 ·····------------------------------------------------------------------------
25 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··25 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
26 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+26 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
27 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
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73 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
74 ···············3··········4········474 ···············3··········4········4
75 ·····-·216b*c*d··+·2052a*d··-·1944d·)75 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
76 o4·:·Ideal·of·R76 o4·:·Ideal·of·R
  
77 i5·:·time·ideal·resultant(f,g,x)77 i5·:·time·ideal·resultant(f,g,x)
78 ·--·used·0.0105181s·(cpu);·0.012554s·(thread);·0s·(gc)78 ·--·used·0.0131768s·(cpu);·0.0125064s·(thread);·0s·(gc)
  
79 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··79 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
80 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+80 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
81 ·····------------------------------------------------------------------------81 ·····------------------------------------------------------------------------
82 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··82 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
83 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-83 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
84 ·····------------------------------------------------------------------------84 ·····------------------------------------------------------------------------
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19 ······8····519 ······8····5
20 o4·=·x··+·x··+·x*c·+·d20 o4·=·x··+·x··+·x*c·+·d
  
21 o4·:·R21 o4·:·R
  
22 i5·:·time·eliminate(ideal(f,g),x)22 i5·:·time·eliminate(ideal(f,g),x)
23 ·--·used·1.33118s·(cpu);·1.20904s·(thread);·0s·(gc)23 ·--·used·1.42832s·(cpu);·1.25722s·(thread);·0s·(gc)
  
24 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··24 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
25 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-25 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
26 ·····------------------------------------------------------------------------26 ·····------------------------------------------------------------------------
27 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··27 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
28 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+28 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
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75 ·····------------------------------------------------------------------------75 ·····------------------------------------------------------------------------
76 ···············3··········4········476 ···············3··········4········4
77 ·····-·216b*c*d··+·2052a*d··-·1944d·)77 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
78 o5·:·Ideal·of·R78 o5·:·Ideal·of·R
  
79 i6·:·time·ideal·resultant(f,g,x)79 i6·:·time·ideal·resultant(f,g,x)
80 ·--·used·0.0120001s·(cpu);·0.0121949s·(thread);·0s·(gc)80 ·--·used·0.0120011s·(cpu);·0.0131536s·(thread);·0s·(gc)
  
81 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··81 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
82 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+82 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
83 ·····------------------------------------------------------------------------83 ·····------------------------------------------------------------------------
84 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··84 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
85 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-85 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
86 ·····------------------------------------------------------------------------86 ·····------------------------------------------------------------------------
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./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
    
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100 ······2100 ······2
101 o3·=·x··+·x*c·+·d101 o3·=·x··+·x*c·+·d
  
102 o3·:·R</code></pre>102 o3·:·R</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(x,ideal(f,g))105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(x,ideal(f,g))
106 ·--·used·0.00400159s·(cpu);·0.00200992s·(thread);·0s·(gc)106 ·--·used·0.00399867s·(cpu);·0.0031114s·(thread);·0s·(gc)
  
107 ······················2····2·············2···········2107 ······················2····2·············2···········2
108 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)108 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
109 o4·:·Ideal·of·R</code></pre>109 o4·:·Ideal·of·R</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)112 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)
113 ·--·used·0.000336236s·(cpu);·0.00116058s·(thread);·0s·(gc)113 ·--·used·0.000376225s·(cpu);·0.00176721s·(thread);·0s·(gc)
  
114 ························2····2·············2···········2114 ························2····2·············2···········2
115 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)115 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
116 o5·:·Ideal·of·R</code></pre>116 o5·:·Ideal·of·R</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
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30 i3·:·g·=·x^2+c*x+d30 i3·:·g·=·x^2+c*x+d
  
31 ······231 ······2
32 o3·=·x··+·x*c·+·d32 o3·=·x··+·x*c·+·d
  
33 o3·:·R33 o3·:·R
34 i4·:·time·eliminate(x,ideal(f,g))34 i4·:·time·eliminate(x,ideal(f,g))
35 ·--·used·0.00400159s·(cpu);·0.00200992s·(thread);·0s·(gc)35 ·--·used·0.00399867s·(cpu);·0.0031114s·(thread);·0s·(gc)
  
36 ······················2····2·············2···········236 ······················2····2·············2···········2
37 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)37 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
38 o4·:·Ideal·of·R38 o4·:·Ideal·of·R
39 i5·:·time·ideal·resultant(f,g,x)39 i5·:·time·ideal·resultant(f,g,x)
40 ·--·used·0.000336236s·(cpu);·0.00116058s·(thread);·0s·(gc)40 ·--·used·0.000376225s·(cpu);·0.00176721s·(thread);·0s·(gc)
  
41 ························2····2·············2···········241 ························2····2·············2···········2
42 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)42 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
43 o5·:·Ideal·of·R43 o5·:·Ideal·of·R
44 i6·:·sylvesterMatrix(f,g,x)44 i6·:·sylvesterMatrix(f,g,x)
  
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91 ······291 ······2
92 o3·=·x··+·x*c·+·d92 o3·=·x··+·x*c·+·d
  
93 o3·:·R</code></pre>93 o3·:·R</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(x,ideal(f,g))96 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(x,ideal(f,g))
97 ·--·used·0.00371003s·(cpu);·0.00208006s·(thread);·0s·(gc)97 ·--·used·0.00400961s·(cpu);·0.00222181s·(thread);·0s·(gc)
  
98 ······················2····2·············2···········298 ······················2····2·············2···········2
99 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)99 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
100 o4·:·Ideal·of·R</code></pre>100 o4·:·Ideal·of·R</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)103 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)
104 ·--·used·0.000295395s·(cpu);·0.00112741s·(thread);·0s·(gc)104 ·--·used·0.000313869s·(cpu);·0.00129816s·(thread);·0s·(gc)
  
105 ························2····2·············2···········2105 ························2····2·············2···········2
106 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)106 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
107 o5·:·Ideal·of·R</code></pre>107 o5·:·Ideal·of·R</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
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31 i3·:·g·=·x^2+c*x+d31 i3·:·g·=·x^2+c*x+d
  
32 ······232 ······2
33 o3·=·x··+·x*c·+·d33 o3·=·x··+·x*c·+·d
  
34 o3·:·R34 o3·:·R
35 i4·:·time·eliminate(x,ideal(f,g))35 i4·:·time·eliminate(x,ideal(f,g))
36 ·--·used·0.00371003s·(cpu);·0.00208006s·(thread);·0s·(gc)36 ·--·used·0.00400961s·(cpu);·0.00222181s·(thread);·0s·(gc)
  
37 ······················2····2·············2···········237 ······················2····2·············2···········2
38 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)38 o4·=·ideal(a*b*c·-·b*c··-·a·d·+·a*c*d·-·b··+·2b*d·-·d·)
  
39 o4·:·Ideal·of·R39 o4·:·Ideal·of·R
40 i5·:·time·ideal·resultant(f,g,x)40 i5·:·time·ideal·resultant(f,g,x)
41 ·--·used·0.000295395s·(cpu);·0.00112741s·(thread);·0s·(gc)41 ·--·used·0.000313869s·(cpu);·0.00129816s·(thread);·0s·(gc)
  
42 ························2····2·············2···········242 ························2····2·············2···········2
43 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)43 o5·=·ideal(-·a*b*c·+·b*c··+·a·d·-·a*c*d·+·b··-·2b*d·+·d·)
  
44 o5·:·Ideal·of·R44 o5·:·Ideal·of·R
45 i6·:·sylvesterMatrix(f,g,x)45 i6·:·sylvesterMatrix(f,g,x)
  
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103 ······8····5103 ······8····5
104 o3·=·x··+·x··+·x*c·+·d104 o3·=·x··+·x··+·x*c·+·d
  
105 o3·:·R</code></pre>105 o3·:·R</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(ideal(f,g),x)108 <td>··············<pre><code·class="language-macaulay2">i4·:·time·eliminate(ideal(f,g),x)
109 ·--·used·1.24333s·(cpu);·1.12073s·(thread);·0s·(gc)109 ·--·used·1.3537s·(cpu);·1.21733s·(thread);·0s·(gc)
  
110 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··110 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
111 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-111 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
112 ·····------------------------------------------------------------------------112 ·····------------------------------------------------------------------------
113 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··113 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
114 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+114 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
115 ·····------------------------------------------------------------------------115 ·····------------------------------------------------------------------------
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160 ···············3··········4········4160 ···············3··········4········4
161 ·····-·216b*c*d··+·2052a*d··-·1944d·)161 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
162 o4·:·Ideal·of·R</code></pre>162 o4·:·Ideal·of·R</code></pre>
163 </td>··········</tr>163 </td>··········</tr>
164 ··········<tr>164 ··········<tr>
165 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)165 <td>··············<pre><code·class="language-macaulay2">i5·:·time·ideal·resultant(f,g,x)
166 ·--·used·0.0105181s·(cpu);·0.012554s·(thread);·0s·(gc)166 ·--·used·0.0131768s·(cpu);·0.0125064s·(thread);·0s·(gc)
  
167 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··167 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
168 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+168 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
169 ·····------------------------------------------------------------------------169 ·····------------------------------------------------------------------------
170 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··170 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
171 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-171 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
172 ·····------------------------------------------------------------------------172 ·····------------------------------------------------------------------------
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36 i3·:·g·=·x^8+x^5+c*x+d36 i3·:·g·=·x^8+x^5+c*x+d
  
37 ······8····537 ······8····5
38 o3·=·x··+·x··+·x*c·+·d38 o3·=·x··+·x··+·x*c·+·d
  
39 o3·:·R39 o3·:·R
40 i4·:·time·eliminate(ideal(f,g),x)40 i4·:·time·eliminate(ideal(f,g),x)
41 ·--·used·1.24333s·(cpu);·1.12073s·(thread);·0s·(gc)41 ·--·used·1.3537s·(cpu);·1.21733s·(thread);·0s·(gc)
  
42 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·242 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2
43 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-43 o4·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
44 ·····------------------------------------------------------------------------44 ·····------------------------------------------------------------------------
45 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······745 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7
46 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+46 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
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91 ·····+·792a*b·c*d·-·1512a*b*c·d·+·648a*c·d·-·360a·b*d··+·648a·c*d··-·504b·d91 ·····+·792a*b·c*d·-·1512a*b*c·d·+·648a*c·d·-·360a·b*d··+·648a·c*d··-·504b·d
92 ·····------------------------------------------------------------------------92 ·····------------------------------------------------------------------------
93 ···············3··········4········493 ···············3··········4········4
94 ·····-·216b*c*d··+·2052a*d··-·1944d·)94 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
95 o4·:·Ideal·of·R95 o4·:·Ideal·of·R
96 i5·:·time·ideal·resultant(f,g,x)96 i5·:·time·ideal·resultant(f,g,x)
97 ·--·used·0.0105181s·(cpu);·0.012554s·(thread);·0s·(gc)97 ·--·used·0.0131768s·(cpu);·0.0125064s·(thread);·0s·(gc)
  
98 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·298 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2
99 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+99 o5·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
101 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7101 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7
102 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-102 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
103 ·····------------------------------------------------------------------------103 ·····------------------------------------------------------------------------
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98 ······8····598 ······8····5
99 o4·=·x··+·x··+·x*c·+·d99 o4·=·x··+·x··+·x*c·+·d
  
100 o4·:·R</code></pre>100 o4·:·R</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i5·:·time·eliminate(ideal(f,g),x)103 <td>··············<pre><code·class="language-macaulay2">i5·:·time·eliminate(ideal(f,g),x)
104 ·--·used·1.33118s·(cpu);·1.20904s·(thread);·0s·(gc)104 ·--·used·1.42832s·(cpu);·1.25722s·(thread);·0s·(gc)
  
105 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··105 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
106 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-106 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
107 ·····------------------------------------------------------------------------107 ·····------------------------------------------------------------------------
108 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··108 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
109 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+109 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
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155 ···············3··········4········4155 ···············3··········4········4
156 ·····-·216b*c*d··+·2052a*d··-·1944d·)156 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
157 o5·:·Ideal·of·R</code></pre>157 o5·:·Ideal·of·R</code></pre>
158 </td>··········</tr>158 </td>··········</tr>
159 ··········<tr>159 ··········<tr>
160 <td>··············<pre><code·class="language-macaulay2">i6·:·time·ideal·resultant(f,g,x)160 <td>··············<pre><code·class="language-macaulay2">i6·:·time·ideal·resultant(f,g,x)
161 ·--·used·0.0120001s·(cpu);·0.0121949s·(thread);·0s·(gc)161 ·--·used·0.0120011s·(cpu);·0.0131536s·(thread);·0s·(gc)
  
162 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··162 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2··
163 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+163 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
164 ·····------------------------------------------------------------------------164 ·····------------------------------------------------------------------------
165 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··165 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7··
166 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-166 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
167 ·····------------------------------------------------------------------------167 ·····------------------------------------------------------------------------
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31 i4·:·g·=·x^8+x^5+c*x+d31 i4·:·g·=·x^8+x^5+c*x+d
  
32 ······8····532 ······8····5
33 o4·=·x··+·x··+·x*c·+·d33 o4·=·x··+·x··+·x*c·+·d
  
34 o4·:·R34 o4·:·R
35 i5·:·time·eliminate(ideal(f,g),x)35 i5·:·time·eliminate(ideal(f,g),x)
36 ·--·used·1.33118s·(cpu);·1.20904s·(thread);·0s·(gc)36 ·--·used·1.42832s·(cpu);·1.25722s·(thread);·0s·(gc)
  
37 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·237 ············7·······8·····3·5····8·····6·········3·4······7·······3·3·2
38 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-38 o5·=·ideal(a·b*c·-·a·d·+·a·b··-·b··-·6a·b*c·-·18a·b·c·+·7b·c·+·48a·b·c··-
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······740 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7
41 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+41 ·····21b·c··-·46a·b·c··+·35b·c··+·15a·b*c··-·35b·c··+·21b·c··-·7b·c··+·b*c··+
42 ·····------------------------------------------------------------------------42 ·····------------------------------------------------------------------------
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86 ·····+·792a*b·c*d·-·1512a*b*c·d·+·648a*c·d·-·360a·b*d··+·648a·c*d··-·504b·d86 ·····+·792a*b·c*d·-·1512a*b*c·d·+·648a*c·d·-·360a·b*d··+·648a·c*d··-·504b·d
87 ·····------------------------------------------------------------------------87 ·····------------------------------------------------------------------------
88 ···············3··········4········488 ···············3··········4········4
89 ·····-·216b*c*d··+·2052a*d··-·1944d·)89 ·····-·216b*c*d··+·2052a*d··-·1944d·)
  
90 o5·:·Ideal·of·R90 o5·:·Ideal·of·R
91 i6·:·time·ideal·resultant(f,g,x)91 i6·:·time·ideal·resultant(f,g,x)
92 ·--·used·0.0120001s·(cpu);·0.0121949s·(thread);·0s·(gc)92 ·--·used·0.0120011s·(cpu);·0.0131536s·(thread);·0s·(gc)
  
93 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·293 ··············7·······8·····3·5····8·····6·········3·4······7·······3·3·2
94 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+94 o6·=·ideal(-·a·b*c·+·a·d·-·a·b··+·b··+·6a·b*c·+·18a·b·c·-·7b·c·-·48a·b·c··+
95 ·····------------------------------------------------------------------------95 ·····------------------------------------------------------------------------
96 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······796 ········6·2······3·2·3······5·3······3···4······4·4······3·5·····2·6······7
97 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-97 ·····21b·c··+·46a·b·c··-·35b·c··-·15a·b*c··+·35b·c··-·21b·c··+·7b·c··-·b*c··-
98 ·····------------------------------------------------------------------------98 ·····------------------------------------------------------------------------
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743 B
./usr/share/doc/Macaulay2/EllipticIntegrals/dump/rawdocumentation.dump
    
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723 B
./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
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⊟
743 B
./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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613 B
./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_lines__Hypersurface.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·13319756731771 --·-*-·M2-comint·-*-·hash:·1331975673177
  
2 i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)2 i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)
3 ·--·used·0.0205368s·(cpu);·0.0196737s·(thread);·0s·(gc)3 ·--·used·0.0263001s·(cpu);·0.0285928s·(thread);·0s·(gc)
  
4 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,4 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
5 ·····------------------------------------------------------------------------5 ·····------------------------------------------------------------------------
6 ·····289139638632755625,·520764738758073845321}6 ·····289139638632755625,·520764738758073845321}
  
7 o1·:·List7 o1·:·List
  
⊟
2.41 KB
./usr/share/doc/Macaulay2/EnumerationCurves/example-output/_rational__Curve.out
    
Offset 37, 83 lines modifiedOffset 37, 83 lines modified
37 i6·:·rationalCurve(2)·-·rationalCurve(1)/837 i6·:·rationalCurve(2)·-·rationalCurve(1)/8
  
38 o6·=·60925038 o6·=·609250
  
39 o6·:·QQ39 o6·:·QQ
  
40 i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/840 i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/8
41 ·--·used·0.243856s·(cpu);·0.218697s·(thread);·0s·(gc)41 ·--·used·0.314917s·(cpu);·0.272947s·(thread);·0s·(gc)
  
42 o7·=·{609250,·92288,·52812,·22428,·9728}42 o7·=·{609250,·92288,·52812,·22428,·9728}
  
43 o7·:·List43 o7·:·List
  
44 i8·:·time·rationalCurve(3)44 i8·:·time·rationalCurve(3)
45 ·--·used·0.0943056s·(cpu);·0.0963087s·(thread);·0s·(gc)45 ·--·used·0.126037s·(cpu);·0.125924s·(thread);·0s·(gc)
  
46 ·····856457500046 ·····8564575000
47 o8·=·----------47 o8·=·----------
48 ·········2748 ·········27
  
49 o8·:·QQ49 o8·:·QQ
  
50 i9·:·time·for·D·in·T·list·rationalCurve(3,D)50 i9·:·time·for·D·in·T·list·rationalCurve(3,D)
51 ·--·used·3.48976s·(cpu);·3.17909s·(thread);·0s·(gc)51 ·--·used·3.99634s·(cpu);·3.63959s·(thread);·0s·(gc)
  
52 ······8564575000··422690816···········4834592··1123942452 ······8564575000··422690816···········4834592··11239424
53 o9·=·{----------,·---------,·6424365,·-------,·--------}53 o9·=·{----------,·---------,·6424365,·-------,·--------}
54 ··········27··········27·················3········2754 ··········27··········27·················3········27
  
55 o9·:·List55 o9·:·List
  
56 i10·:·time·rationalCurve(3)·-·rationalCurve(1)/2756 i10·:·time·rationalCurve(3)·-·rationalCurve(1)/27
57 ·--·used·0.0962841s·(cpu);·0.0980935s·(thread);·0s·(gc)57 ·--·used·0.102511s·(cpu);·0.105267s·(thread);·0s·(gc)
  
58 o10·=·31720637558 o10·=·317206375
  
59 o10·:·QQ59 o10·:·QQ
  
60 i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/2760 i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/27
61 ·--·used·3.49569s·(cpu);·3.20644s·(thread);·0s·(gc)61 ·--·used·3.94711s·(cpu);·3.53343s·(thread);·0s·(gc)
  
62 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}62 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}
  
63 o11·:·List63 o11·:·List
  
64 i12·:·time·rationalCurve(4)64 i12·:·time·rationalCurve(4)
65 ·--·used·1.15666s·(cpu);·1.02455s·(thread);·0s·(gc)65 ·--·used·1.27414s·(cpu);·1.09605s·(thread);·0s·(gc)
  
66 ······1551792679687566 ······15517926796875
67 o12·=·--------------67 o12·=·--------------
68 ············6468 ············64
  
69 o12·:·QQ69 o12·:·QQ
  
70 i13·:·time·rationalCurve(4,{4,2})70 i13·:·time·rationalCurve(4,{4,2})
71 ·--·used·4.96943s·(cpu);·4.16153s·(thread);·0s·(gc)71 ·--·used·6.02459s·(cpu);·4.85168s·(thread);·0s·(gc)
  
72 o13·=·388391408472 o13·=·3883914084
  
73 o13·:·QQ73 o13·:·QQ
  
74 i14·:·time·rationalCurve(4)·-·rationalCurve(2)/874 i14·:·time·rationalCurve(4)·-·rationalCurve(2)/8
75 ·--·used·1.10808s·(cpu);·1.00616s·(thread);·0s·(gc)75 ·--·used·1.2177s·(cpu);·1.09919s·(thread);·0s·(gc)
  
76 o14·=·24246753000076 o14·=·242467530000
  
77 o14·:·QQ77 o14·:·QQ
  
78 i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/878 i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/8
79 ·--·used·4.92355s·(cpu);·4.18521s·(thread);·0s·(gc)79 ·--·used·5.94291s·(cpu);·4.80478s·(thread);·0s·(gc)
  
80 o15·=·388390252880 o15·=·3883902528
  
81 o15·:·QQ81 o15·:·QQ
  
82 i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/882 i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/8
83 ·--·used·4.83904s·(cpu);·4.12416s·(thread);·0s·(gc)83 ·--·used·5.70355s·(cpu);·4.76614s·(thread);·0s·(gc)
  
84 o16·=·113944838484 o16·=·1139448384
  
85 o16·:·QQ85 o16·:·QQ
  
86 i17·:·86 i17·:·
⊟
1.66 KB
./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
    
Offset 70, 15 lines modifiedOffset 70, 15 lines modified
70 ········<div>70 ········<div>
71 ··········<p>Computes·the·number·of·lines·on·a·general·hypersurface·of·degree·2n·-·3·in·\mathbb·P^n.</p>71 ··········<p>Computes·the·number·of·lines·on·a·general·hypersurface·of·degree·2n·-·3·in·\mathbb·P^n.</p>
72 ··········<p></p>72 ··········<p></p>
73 ········</div>73 ········</div>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)76 <td>··············<pre><code·class="language-macaulay2">i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)
77 ·--·used·0.0205368s·(cpu);·0.0196737s·(thread);·0s·(gc)77 ·--·used·0.0263001s·(cpu);·0.0285928s·(thread);·0s·(gc)
  
78 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,78 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
79 ·····------------------------------------------------------------------------79 ·····------------------------------------------------------------------------
80 ·····289139638632755625,·520764738758073845321}80 ·····289139638632755625,·520764738758073845321}
  
81 o1·:·List</code></pre>81 o1·:·List</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
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848 B
html2text {}
    
Offset 12, 15 lines modifiedOffset 12, 15 lines modified
12 ····*·Outputs:12 ····*·Outputs:
13 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·number·of·lines·on·a·general·hypersurface·of·degree13 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·number·of·lines·on·a·general·hypersurface·of·degree
14 ············2n·-·3·in·\mathbb·P^n14 ············2n·-·3·in·\mathbb·P^n
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 Computes·the·number·of·lines·on·a·general·hypersurface·of·degree·2n·-·3·in16 Computes·the·number·of·lines·on·a·general·hypersurface·of·degree·2n·-·3·in
17 \mathbb·P^n.17 \mathbb·P^n.
18 i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)18 i1·:·time·for·n·from·2·to·10·list·linesHypersurface(n)
19 ·--·used·0.0205368s·(cpu);·0.0196737s·(thread);·0s·(gc)19 ·--·used·0.0263001s·(cpu);·0.0285928s·(thread);·0s·(gc)
  
20 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,20 o1·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
21 ·····------------------------------------------------------------------------21 ·····------------------------------------------------------------------------
22 ·····289139638632755625,·520764738758073845321}22 ·····289139638632755625,·520764738758073845321}
  
23 o1·:·List23 o1·:·List
24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·l\x8li\x8in\x8ne\x8es\x8sH\x8Hy\x8yp\x8pe\x8er\x8rs\x8su\x8ur\x8rf\x8fa\x8ac\x8ce\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·l\x8li\x8in\x8ne\x8es\x8sH\x8Hy\x8yp\x8pe\x8er\x8rs\x8su\x8ur\x8rf\x8fa\x8ac\x8ce\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*
⊟
7.8 KB
./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
    
Offset 140, 39 lines modifiedOffset 140, 39 lines modified
140 ········<div>140 ········<div>
141 ··········<p>The·numbers·of·conics·on·general·complete·intersection·Calabi-Yau·threefolds·can·be·computed·as·follows:</p>141 ··········<p>The·numbers·of·conics·on·general·complete·intersection·Calabi-Yau·threefolds·can·be·computed·as·follows:</p>
142 ··········<p></p>142 ··········<p></p>
143 ········</div>143 ········</div>
144 ········<table·class="examples">144 ········<table·class="examples">
145 ··········<tr>145 ··········<tr>
146 <td>··············<pre><code·class="language-macaulay2">i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/8146 <td>··············<pre><code·class="language-macaulay2">i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/8
147 ·--·used·0.243856s·(cpu);·0.218697s·(thread);·0s·(gc)147 ·--·used·0.314917s·(cpu);·0.272947s·(thread);·0s·(gc)
  
148 o7·=·{609250,·92288,·52812,·22428,·9728}148 o7·=·{609250,·92288,·52812,·22428,·9728}
  
149 o7·:·List</code></pre>149 o7·:·List</code></pre>
150 </td>··········</tr>150 </td>··········</tr>
151 ········</table>151 ········</table>
152 ········<div>152 ········<div>
153 ··········<p>For·rational·curves·of·degree·3:</p>153 ··········<p>For·rational·curves·of·degree·3:</p>
154 ··········<p></p>154 ··········<p></p>
155 ········</div>155 ········</div>
156 ········<table·class="examples">156 ········<table·class="examples">
157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i8·:·time·rationalCurve(3)158 <td>··············<pre><code·class="language-macaulay2">i8·:·time·rationalCurve(3)
159 ·--·used·0.0943056s·(cpu);·0.0963087s·(thread);·0s·(gc)159 ·--·used·0.126037s·(cpu);·0.125924s·(thread);·0s·(gc)
  
160 ·····8564575000160 ·····8564575000
161 o8·=·----------161 o8·=·----------
162 ·········27162 ·········27
  
163 o8·:·QQ</code></pre>163 o8·:·QQ</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ··········<tr>165 ··········<tr>
166 <td>··············<pre><code·class="language-macaulay2">i9·:·time·for·D·in·T·list·rationalCurve(3,D)166 <td>··············<pre><code·class="language-macaulay2">i9·:·time·for·D·in·T·list·rationalCurve(3,D)
167 ·--·used·3.48976s·(cpu);·3.17909s·(thread);·0s·(gc)167 ·--·used·3.99634s·(cpu);·3.63959s·(thread);·0s·(gc)
  
168 ······8564575000··422690816···········4834592··11239424168 ······8564575000··422690816···········4834592··11239424
169 o9·=·{----------,·---------,·6424365,·-------,·--------}169 o9·=·{----------,·---------,·6424365,·-------,·--------}
170 ··········27··········27·················3········27170 ··········27··········27·················3········27
  
171 o9·:·List</code></pre>171 o9·:·List</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
Offset 180, 89 lines modifiedOffset 180, 89 lines modified
180 ········<div>180 ········<div>
181 ··········<p>The·number·of·rational·curves·of·degree·3·on·a·general·quintic·threefold·can·be·computed·as·follows:</p>181 ··········<p>The·number·of·rational·curves·of·degree·3·on·a·general·quintic·threefold·can·be·computed·as·follows:</p>
182 ··········<p></p>182 ··········<p></p>
183 ········</div>183 ········</div>
184 ········<table·class="examples">184 ········<table·class="examples">
185 ··········<tr>185 ··········<tr>
186 <td>··············<pre><code·class="language-macaulay2">i10·:·time·rationalCurve(3)·-·rationalCurve(1)/27186 <td>··············<pre><code·class="language-macaulay2">i10·:·time·rationalCurve(3)·-·rationalCurve(1)/27
187 ·--·used·0.0962841s·(cpu);·0.0980935s·(thread);·0s·(gc)187 ·--·used·0.102511s·(cpu);·0.105267s·(thread);·0s·(gc)
  
188 o10·=·317206375188 o10·=·317206375
  
189 o10·:·QQ</code></pre>189 o10·:·QQ</code></pre>
190 </td>··········</tr>190 </td>··········</tr>
191 ········</table>191 ········</table>
192 ········<div>192 ········<div>
193 ··········<p>The·numbers·of·rational·curves·of·degree·3·on·general·complete·intersection·Calabi-Yau·threefolds·can·be·computed·as·follows:</p>193 ··········<p>The·numbers·of·rational·curves·of·degree·3·on·general·complete·intersection·Calabi-Yau·threefolds·can·be·computed·as·follows:</p>
194 ··········<p></p>194 ··········<p></p>
195 ········</div>195 ········</div>
196 ········<table·class="examples">196 ········<table·class="examples">
197 ··········<tr>197 ··········<tr>
198 <td>··············<pre><code·class="language-macaulay2">i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/27198 <td>··············<pre><code·class="language-macaulay2">i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/27
199 ·--·used·3.49569s·(cpu);·3.20644s·(thread);·0s·(gc)199 ·--·used·3.94711s·(cpu);·3.53343s·(thread);·0s·(gc)
  
200 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}200 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}
  
201 o11·:·List</code></pre>201 o11·:·List</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ········</table>203 ········</table>
204 ········<div>204 ········<div>
205 ··········<p>For·rational·curves·of·degree·4:</p>205 ··········<p>For·rational·curves·of·degree·4:</p>
206 ··········<p></p>206 ··········<p></p>
207 ········</div>207 ········</div>
208 ········<table·class="examples">208 ········<table·class="examples">
209 ··········<tr>209 ··········<tr>
210 <td>··············<pre><code·class="language-macaulay2">i12·:·time·rationalCurve(4)210 <td>··············<pre><code·class="language-macaulay2">i12·:·time·rationalCurve(4)
211 ·--·used·1.15666s·(cpu);·1.02455s·(thread);·0s·(gc)211 ·--·used·1.27414s·(cpu);·1.09605s·(thread);·0s·(gc)
  
212 ······15517926796875212 ······15517926796875
213 o12·=·--------------213 o12·=·--------------
214 ············64214 ············64
  
215 o12·:·QQ</code></pre>215 o12·:·QQ</code></pre>
216 </td>··········</tr>216 </td>··········</tr>
217 ··········<tr>217 ··········<tr>
218 <td>··············<pre><code·class="language-macaulay2">i13·:·time·rationalCurve(4,{4,2})218 <td>··············<pre><code·class="language-macaulay2">i13·:·time·rationalCurve(4,{4,2})
219 ·--·used·4.96943s·(cpu);·4.16153s·(thread);·0s·(gc)219 ·--·used·6.02459s·(cpu);·4.85168s·(thread);·0s·(gc)
  
220 o13·=·3883914084220 o13·=·3883914084
  
221 o13·:·QQ</code></pre>221 o13·:·QQ</code></pre>
222 </td>··········</tr>222 </td>··········</tr>
223 ········</table>223 ········</table>
224 ········<div>224 ········<div>
225 ··········<p>The·number·of·rational·curves·of·degree·4·on·a·general·quintic·threefold·can·be·computed·as·follows:</p>225 ··········<p>The·number·of·rational·curves·of·degree·4·on·a·general·quintic·threefold·can·be·computed·as·follows:</p>
226 ··········<p></p>226 ··········<p></p>
227 ········</div>227 ········</div>
228 ········<table·class="examples">228 ········<table·class="examples">
229 ··········<tr>229 ··········<tr>
230 <td>··············<pre><code·class="language-macaulay2">i14·:·time·rationalCurve(4)·-·rationalCurve(2)/8230 <td>··············<pre><code·class="language-macaulay2">i14·:·time·rationalCurve(4)·-·rationalCurve(2)/8
231 ·--·used·1.10808s·(cpu);·1.00616s·(thread);·0s·(gc)231 ·--·used·1.2177s·(cpu);·1.09919s·(thread);·0s·(gc)
  
232 o14·=·242467530000232 o14·=·242467530000
  
233 o14·:·QQ</code></pre>233 o14·:·QQ</code></pre>
234 </td>··········</tr>234 </td>··········</tr>
235 ········</table>235 ········</table>
236 ········<div>236 ········<div>
237 ··········<p>The·numbers·of·rational·curves·of·degree·4·on·general·complete·intersections·of·types·(4,2)·and·(3,3)·in·\mathbb·P^5·can·be·computed·as·follows:</p>237 ··········<p>The·numbers·of·rational·curves·of·degree·4·on·general·complete·intersections·of·types·(4,2)·and·(3,3)·in·\mathbb·P^5·can·be·computed·as·follows:</p>
238 ··········<p></p>238 ··········<p></p>
239 ········</div>239 ········</div>
240 ········<table·class="examples">240 ········<table·class="examples">
241 ··········<tr>241 ··········<tr>
242 <td>··············<pre><code·class="language-macaulay2">i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/8242 <td>··············<pre><code·class="language-macaulay2">i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/8
243 ·--·used·4.92355s·(cpu);·4.18521s·(thread);·0s·(gc)243 ·--·used·5.94291s·(cpu);·4.80478s·(thread);·0s·(gc)
  
244 o15·=·3883902528244 o15·=·3883902528
  
245 o15·:·QQ</code></pre>245 o15·:·QQ</code></pre>
246 </td>··········</tr>246 </td>··········</tr>
247 ··········<tr>247 ··········<tr>
248 <td>··············<pre><code·class="language-macaulay2">i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/8248 <td>··············<pre><code·class="language-macaulay2">i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/8
249 ·--·used·4.83904s·(cpu);·4.12416s·(thread);·0s·(gc)249 ·--·used·5.70355s·(cpu);·4.76614s·(thread);·0s·(gc)
  
250 o16·=·1139448384250 o16·=·1139448384
  
251 o16·:·QQ</code></pre>251 o16·:·QQ</code></pre>
252 </td>··········</tr>252 </td>··········</tr>
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Offset 60, 85 lines modifiedOffset 60, 85 lines modified
  
60 o6·=·60925060 o6·=·609250
  
61 o6·:·QQ61 o6·:·QQ
62 The·numbers·of·conics·on·general·complete·intersection·Calabi-Yau·threefolds62 The·numbers·of·conics·on·general·complete·intersection·Calabi-Yau·threefolds
63 can·be·computed·as·follows:63 can·be·computed·as·follows:
64 i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/864 i7·:·time·for·D·in·T·list·rationalCurve(2,D)·-·rationalCurve(1,D)/8
65 ·--·used·0.243856s·(cpu);·0.218697s·(thread);·0s·(gc)65 ·--·used·0.314917s·(cpu);·0.272947s·(thread);·0s·(gc)
  
66 o7·=·{609250,·92288,·52812,·22428,·9728}66 o7·=·{609250,·92288,·52812,·22428,·9728}
  
67 o7·:·List67 o7·:·List
68 For·rational·curves·of·degree·3:68 For·rational·curves·of·degree·3:
69 i8·:·time·rationalCurve(3)69 i8·:·time·rationalCurve(3)
70 ·--·used·0.0943056s·(cpu);·0.0963087s·(thread);·0s·(gc)70 ·--·used·0.126037s·(cpu);·0.125924s·(thread);·0s·(gc)
  
71 ·····856457500071 ·····8564575000
72 o8·=·----------72 o8·=·----------
73 ·········2773 ·········27
  
74 o8·:·QQ74 o8·:·QQ
75 i9·:·time·for·D·in·T·list·rationalCurve(3,D)75 i9·:·time·for·D·in·T·list·rationalCurve(3,D)
76 ·--·used·3.48976s·(cpu);·3.17909s·(thread);·0s·(gc)76 ·--·used·3.99634s·(cpu);·3.63959s·(thread);·0s·(gc)
  
77 ······8564575000··422690816···········4834592··1123942477 ······8564575000··422690816···········4834592··11239424
78 o9·=·{----------,·---------,·6424365,·-------,·--------}78 o9·=·{----------,·---------,·6424365,·-------,·--------}
79 ··········27··········27·················3········2779 ··········27··········27·················3········27
  
80 o9·:·List80 o9·:·List
81 The·number·of·rational·curves·of·degree·3·on·a·general·quintic·threefold·can·be81 The·number·of·rational·curves·of·degree·3·on·a·general·quintic·threefold·can·be
82 computed·as·follows:82 computed·as·follows:
83 i10·:·time·rationalCurve(3)·-·rationalCurve(1)/2783 i10·:·time·rationalCurve(3)·-·rationalCurve(1)/27
84 ·--·used·0.0962841s·(cpu);·0.0980935s·(thread);·0s·(gc)84 ·--·used·0.102511s·(cpu);·0.105267s·(thread);·0s·(gc)
  
85 o10·=·31720637585 o10·=·317206375
  
86 o10·:·QQ86 o10·:·QQ
87 The·numbers·of·rational·curves·of·degree·3·on·general·complete·intersection87 The·numbers·of·rational·curves·of·degree·3·on·general·complete·intersection
88 Calabi-Yau·threefolds·can·be·computed·as·follows:88 Calabi-Yau·threefolds·can·be·computed·as·follows:
89 i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/2789 i11·:·time·for·D·in·T·list·rationalCurve(3,D)·-·rationalCurve(1,D)/27
90 ·--·used·3.49569s·(cpu);·3.20644s·(thread);·0s·(gc)90 ·--·used·3.94711s·(cpu);·3.53343s·(thread);·0s·(gc)
  
91 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}91 o11·=·{317206375,·15655168,·6424326,·1611504,·416256}
  
92 o11·:·List92 o11·:·List
93 For·rational·curves·of·degree·4:93 For·rational·curves·of·degree·4:
94 i12·:·time·rationalCurve(4)94 i12·:·time·rationalCurve(4)
95 ·--·used·1.15666s·(cpu);·1.02455s·(thread);·0s·(gc)95 ·--·used·1.27414s·(cpu);·1.09605s·(thread);·0s·(gc)
  
96 ······1551792679687596 ······15517926796875
97 o12·=·--------------97 o12·=·--------------
98 ············6498 ············64
  
99 o12·:·QQ99 o12·:·QQ
100 i13·:·time·rationalCurve(4,{4,2})100 i13·:·time·rationalCurve(4,{4,2})
101 ·--·used·4.96943s·(cpu);·4.16153s·(thread);·0s·(gc)101 ·--·used·6.02459s·(cpu);·4.85168s·(thread);·0s·(gc)
  
102 o13·=·3883914084102 o13·=·3883914084
  
103 o13·:·QQ103 o13·:·QQ
104 The·number·of·rational·curves·of·degree·4·on·a·general·quintic·threefold·can·be104 The·number·of·rational·curves·of·degree·4·on·a·general·quintic·threefold·can·be
105 computed·as·follows:105 computed·as·follows:
106 i14·:·time·rationalCurve(4)·-·rationalCurve(2)/8106 i14·:·time·rationalCurve(4)·-·rationalCurve(2)/8
107 ·--·used·1.10808s·(cpu);·1.00616s·(thread);·0s·(gc)107 ·--·used·1.2177s·(cpu);·1.09919s·(thread);·0s·(gc)
  
108 o14·=·242467530000108 o14·=·242467530000
  
109 o14·:·QQ109 o14·:·QQ
110 The·numbers·of·rational·curves·of·degree·4·on·general·complete·intersections·of110 The·numbers·of·rational·curves·of·degree·4·on·general·complete·intersections·of
111 types·(4,2)·and·(3,3)·in·\mathbb·P^5·can·be·computed·as·follows:111 types·(4,2)·and·(3,3)·in·\mathbb·P^5·can·be·computed·as·follows:
112 i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/8112 i15·:·time·rationalCurve(4,{4,2})·-·rationalCurve(2,{4,2})/8
113 ·--·used·4.92355s·(cpu);·4.18521s·(thread);·0s·(gc)113 ·--·used·5.94291s·(cpu);·4.80478s·(thread);·0s·(gc)
  
114 o15·=·3883902528114 o15·=·3883902528
  
115 o15·:·QQ115 o15·:·QQ
116 i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/8116 i16·:·time·rationalCurve(4,{3,3})·-·rationalCurve(2,{3,3})/8
117 ·--·used·4.83904s·(cpu);·4.12416s·(thread);·0s·(gc)117 ·--·used·5.70355s·(cpu);·4.76614s·(thread);·0s·(gc)
  
118 o16·=·1139448384118 o16·=·1139448384
  
119 o16·:·QQ119 o16·:·QQ
120 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lC\x8Cu\x8ur\x8rv\x8ve\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*120 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lC\x8Cu\x8ur\x8rv\x8ve\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*
121 ····*·rationalCurve(ZZ)121 ····*·rationalCurve(ZZ)
122 ····*·rationalCurve(ZZ,List)122 ····*·rationalCurve(ZZ,List)
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11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp
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./usr/share/doc/Macaulay2/EquivariantGB/example-output/_egb__Toric.out
    
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10 o3·=·map·(R,·S,·{x·,·x·x·,·x·x·,·x·})10 o3·=·map·(R,·S,·{x·,·x·x·,·x·x·,·x·})
11 ··················1···1·0···1·0···011 ··················1···1·0···1·0···0
  
12 o3·:·RingMap·R·<--·S12 o3·:·RingMap·R·<--·S
  
13 i4·:·G·=·egbToric(m,·OutFile=>stdio)13 i4·:·G·=·egbToric(m,·OutFile=>stdio)
14 314 3
15 ·····--·used·.00395949·seconds15 ·····--·used·.00536995·seconds
16 ·····--·used·.00401057·seconds16 ·····--·used·0·seconds
17 (9,·9)17 (9,·9)
18 new·stuff·found18 new·stuff·found
19 419 4
20 ·····--·used·.001595·seconds 
21 ·····--·used·.00396808·seconds20 ·····--·used·.00284708·seconds
 21 ·····--·used·.00265212·seconds
22 (16,·26)22 (16,·26)
23 new·stuff·found23 new·stuff·found
24 524 5
25 ·····--·used·.00601227·seconds25 ·····--·used·.00677539·seconds
26 ·····--·used·.0199947·seconds26 ·····--·used·.0245966·seconds
27 (25,·60)27 (25,·60)
28 628 6
29 ·····--·used·.010128·seconds29 ·····--·used·.012985·seconds
30 ·····--·used·.152499·seconds30 ·····--·used·.170545·seconds
31 (36,·120)31 (36,·120)
32 732 7
33 ·····--·used·.0323028·seconds33 ·····--·used·.0387461·seconds
34 ·····--·used·.613835·seconds34 ·····--·used·.744076·seconds
35 (49,·217)35 (49,·217)
  
36 ···································236 ···································2
37 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+37 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+
38 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,0··38 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,0··
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+40 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+
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./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
    
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96 ··················1···1·0···1·0···096 ··················1···1·0···1·0···0
  
97 o3·:·RingMap·R·&lt;--·S</code></pre>97 o3·:·RingMap·R·&lt;--·S</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·G·=·egbToric(m,·OutFile=>stdio)100 <td>··············<pre><code·class="language-macaulay2">i4·:·G·=·egbToric(m,·OutFile=>stdio)
101 3101 3
102 ·····--·used·.00395949·seconds102 ·····--·used·.00536995·seconds
103 ·····--·used·.00401057·seconds103 ·····--·used·0·seconds
104 (9,·9)104 (9,·9)
105 new·stuff·found105 new·stuff·found
106 4106 4
107 ·····--·used·.001595·seconds 
108 ·····--·used·.00396808·seconds107 ·····--·used·.00284708·seconds
 108 ·····--·used·.00265212·seconds
109 (16,·26)109 (16,·26)
110 new·stuff·found110 new·stuff·found
111 5111 5
112 ·····--·used·.00601227·seconds112 ·····--·used·.00677539·seconds
113 ·····--·used·.0199947·seconds113 ·····--·used·.0245966·seconds
114 (25,·60)114 (25,·60)
115 6115 6
116 ·····--·used·.010128·seconds116 ·····--·used·.012985·seconds
117 ·····--·used·.152499·seconds117 ·····--·used·.170545·seconds
118 (36,·120)118 (36,·120)
119 7119 7
120 ·····--·used·.0323028·seconds120 ·····--·used·.0387461·seconds
121 ·····--·used·.613835·seconds121 ·····--·used·.744076·seconds
122 (49,·217)122 (49,·217)
  
123 ···································2123 ···································2
124 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+124 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+
125 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,0··125 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,0··
126 ·····------------------------------------------------------------------------126 ·····------------------------------------------------------------------------
127 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+127 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+
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34 ··················2···············234 ··················2···············2
35 o3·=·map·(R,·S,·{x·,·x·x·,·x·x·,·x·})35 o3·=·map·(R,·S,·{x·,·x·x·,·x·x·,·x·})
36 ··················1···1·0···1·0···036 ··················1···1·0···1·0···0
  
37 o3·:·RingMap·R·<--·S37 o3·:·RingMap·R·<--·S
38 i4·:·G·=·egbToric(m,·OutFile=>stdio)38 i4·:·G·=·egbToric(m,·OutFile=>stdio)
39 339 3
40 ·····--·used·.00395949·seconds40 ·····--·used·.00536995·seconds
41 ·····--·used·.00401057·seconds41 ·····--·used·0·seconds
42 (9,·9)42 (9,·9)
43 new·stuff·found43 new·stuff·found
44 444 4
45 ·····--·used·.001595·seconds 
46 ·····--·used·.00396808·seconds45 ·····--·used·.00284708·seconds
 46 ·····--·used·.00265212·seconds
47 (16,·26)47 (16,·26)
48 new·stuff·found48 new·stuff·found
49 549 5
50 ·····--·used·.00601227·seconds50 ·····--·used·.00677539·seconds
51 ·····--·used·.0199947·seconds51 ·····--·used·.0245966·seconds
52 (25,·60)52 (25,·60)
53 653 6
54 ·····--·used·.010128·seconds54 ·····--·used·.012985·seconds
55 ·····--·used·.152499·seconds55 ·····--·used·.170545·seconds
56 (36,·120)56 (36,·120)
57 757 7
58 ·····--·used·.0323028·seconds58 ·····--·used·.0387461·seconds
59 ·····--·used·.613835·seconds59 ·····--·used·.744076·seconds
60 (49,·217)60 (49,·217)
  
61 ···································261 ···································2
62 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+62 o4·=·{-·y····+·y···,·-·y···y····+·y···,·-·y···y····+·y···y···,·-·y···y····+
63 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,063 ·········1,0····0,1·····1,1·0,0····1,0·····2,1·0,0····2,0·1,0·····2,1·1,0
64 ·····------------------------------------------------------------------------64 ·····------------------------------------------------------------------------
65 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+65 ·····y···y···,·-·y···y····+·y···y···,·-·y···y····+·y···y···,·-·y···y····+
⊟
721 B
./usr/share/doc/Macaulay2/ExampleSystems/dump/rawdocumentation.dump
    
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9 #:len=15169 #:len=1516
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753 B
./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
    
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZU1hY2F1bGF5RXhwYW5zaW9uLExpc3QpLCJz11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzb2x2ZU1hY2F1bGF5RXhwYW5zaW9uLExpc3QpLCJz
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751 B
./usr/share/doc/Macaulay2/ExteriorModules/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=267 #:len=26
8 YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=8 YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=
9 #:len=2929 #:len=292
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAyLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt
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717 B
./usr/share/doc/Macaulay2/FGLM/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=167 #:len=16
8 ZmdsbShJZGVhbCxSaW5nKQ==8 ZmdsbShJZGVhbCxSaW5nKQ==
9 #:len=2119 #:len=211
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp
⊟
741 B
./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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9 #:len=2729 #:len=272
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11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl
⊟
5.35 KB
./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
    
Offset 462, 50 lines modifiedOffset 462, 50 lines modified
462 ···············3·2·4·····3·6462 ···············3·2·4·····3·6
463 o27·=·ideal(12x·x·x··-·4x·x·)463 o27·=·ideal(12x·x·x··-·4x·x·)
464 ···············3·7·9·····3·9464 ···············3·7·9·····3·9
  
465 o27·:·Ideal·of·S465 o27·:·Ideal·of·S
  
466 i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))466 i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))
467 ·--·used·0.0990349s·(cpu);·0.0986288s·(thread);·0s·(gc)467 ·--·used·0.10298s·(cpu);·0.102107s·(thread);·0s·(gc)
  
468 o28·=·2468 o28·=·2
  
469 i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))469 i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))
470 ·--·used·0.240733s·(cpu);·0.176896s·(thread);·0s·(gc)470 ·--·used·0.245805s·(cpu);·0.171742s·(thread);·0s·(gc)
  
471 o29·=·3471 o29·=·3
  
472 i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))472 i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))
473 ·--·used·0.314859s·(cpu);·0.24812s·(thread);·0s·(gc)473 ·--·used·0.335311s·(cpu);·0.267543s·(thread);·0s·(gc)
  
474 o30·=·1474 o30·=·1
  
475 i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))475 i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))
476 ·--·used·0.22722s·(cpu);·0.161955s·(thread);·0s·(gc)476 ·--·used·0.220887s·(cpu);·0.170257s·(thread);·0s·(gc)
  
477 o31·=·2477 o31·=·2
  
478 i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))478 i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))
479 ·--·used·0.218217s·(cpu);·0.154621s·(thread);·0s·(gc)479 ·--·used·0.242895s·(cpu);·0.146413s·(thread);·0s·(gc)
  
480 o32·=·3480 o32·=·3
  
481 i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallestTerm))481 i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallestTerm))
482 ·--·used·0.253356s·(cpu);·0.18669s·(thread);·0s·(gc)482 ·--·used·0.249652s·(cpu);·0.1774s·(thread);·0s·(gc)
  
483 o33·=·3483 o33·=·3
  
484 i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))484 i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))
485 ·--·used·0.225233s·(cpu);·0.169651s·(thread);·0s·(gc)485 ·--·used·0.24987s·(cpu);·0.164533s·(thread);·0s·(gc)
  
486 o34·=·3486 o34·=·3
  
487 i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))487 i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))
488 ·--·used·9.72612s·(cpu);·7.03107s·(thread);·0s·(gc)488 ·--·used·11.0167s·(cpu);·7.71193s·(thread);·0s·(gc)
  
489 o35·=·1489 o35·=·1
  
490 i36·:·peek·StrategyDefault490 i36·:·peek·StrategyDefault
  
491 o36·=·OptionTable{GRevLexLargest·=>·0······}491 o36·=·OptionTable{GRevLexLargest·=>·0······}
492 ··················GRevLexSmallest·=>·16492 ··················GRevLexSmallest·=>·16
Offset 514, 15 lines modifiedOffset 514, 15 lines modified
514 ··················LexSmallest·=>·16514 ··················LexSmallest·=>·16
515 ··················LexSmallestTerm·=>·16515 ··················LexSmallestTerm·=>·16
516 ··················Points·=>·0516 ··················Points·=>·0
517 ··················Random·=>·16517 ··················Random·=>·16
518 ··················RandomNonzero·=>·16518 ··················RandomNonzero·=>·16
  
519 i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,·Verbose=>true);519 i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,·Verbose=>true);
520 ·--·used·0.407857s·(cpu);·0.316004s·(thread);·0s·(gc)520 ·--·used·0.485201s·(cpu);·0.346923s·(thread);·0s·(gc)
521 internalChooseMinor:·Choosing·Random521 internalChooseMinor:·Choosing·Random
522 internalChooseMinor:·Choosing·LexSmallest522 internalChooseMinor:·Choosing·LexSmallest
523 internalChooseMinor:·Choosing·Random523 internalChooseMinor:·Choosing·Random
524 internalChooseMinor:·Choosing·GRevLexSmallestTerm524 internalChooseMinor:·Choosing·GRevLexSmallestTerm
525 internalChooseMinor:·Choosing·RandomNonZero525 internalChooseMinor:·Choosing·RandomNonZero
526 internalChooseMinor:·Choosing·RandomNonZero526 internalChooseMinor:·Choosing·RandomNonZero
527 internalChooseMinor:·Choosing·LexSmallest527 internalChooseMinor:·Choosing·LexSmallest
Offset 582, 15 lines modifiedOffset 582, 15 lines modified
582 i41·:·ptsStratGeometric·=·new·OptionTable·from·(options·chooseGoodMinors)#PointOptions;582 i41·:·ptsStratGeometric·=·new·OptionTable·from·(options·chooseGoodMinors)#PointOptions;
  
583 i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value583 i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value
  
584 o42·=·true584 o42·=·true
  
585 i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratGeometric))585 i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratGeometric))
586 ·--·used·0.638168s·(cpu);·0.443521s·(thread);·0s·(gc)586 ·--·used·0.827591s·(cpu);·0.517732s·(thread);·0s·(gc)
  
587 o43·=·2587 o43·=·2
  
588 i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that·value588 i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that·value
  
589 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}589 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}
590 ··················DimensionFunction·=>·dim590 ··················DimensionFunction·=>·dim
Offset 605, 47 lines modifiedOffset 605, 47 lines modified
605 o44·:·OptionTable605 o44·:·OptionTable
  
606 i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value606 i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value
  
607 o45·=·false607 o45·=·false
  
608 i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratRational))608 i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratRational))
609 ·--·used·0.520675s·(cpu);·0.353044s·(thread);·0s·(gc)609 ·--·used·0.648394s·(cpu);·0.403013s·(thread);·0s·(gc)
  
610 o46·=·2610 o46·=·2
  
611 i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefault)611 i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefault)
612 ·--·used·3.51988s·(cpu);·2.84854s·(thread);·0s·(gc)612 ·--·used·4.50417s·(cpu);·3.50025s·(thread);·0s·(gc)
  
613 i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultNonRandom)613 i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultNonRandom)
614 ·--·used·0.812059s·(cpu);·0.606774s·(thread);·0s·(gc)614 ·--·used·0.933099s·(cpu);·0.636156s·(thread);·0s·(gc)
  
615 o48·=·true615 o48·=·true
  
616 i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)616 i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)
617 ·--·used·2.71639s·(cpu);·2.44142s·(thread);·0s·(gc)617 ·--·used·3.0767s·(cpu);·2.73986s·(thread);·0s·(gc)
  
618 i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallest)618 i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallest)
619 ·--·used·2.6533s·(cpu);·1.89165s·(thread);·0s·(gc)619 ·--·used·3.33268s·(cpu);·2.22841s·(thread);·0s·(gc)
  
620 i51·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallestTerm)620 i51·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallestTerm)
621 ·--·used·0.905182s·(cpu);·0.729877s·(thread);·0s·(gc)621 ·--·used·0.998581s·(cpu);·0.774542s·(thread);·0s·(gc)
  
622 o51·=·true622 o51·=·true
  
623 i52·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>GRevLexSmallest)623 i52·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>GRevLexSmallest)
624 ·--·used·2.95097s·(cpu);·2.17025s·(thread);·0s·(gc)624 ·--·used·3.22386s·(cpu);·2.20576s·(thread);·0s·(gc)
  
625 i53·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>GRevLexSmallestTerm)625 i53·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>GRevLexSmallestTerm)
626 ·--·used·3.39347s·(cpu);·2.56847s·(thread);·0s·(gc)626 ·--·used·3.68377s·(cpu);·2.58807s·(thread);·0s·(gc)
  
627 i54·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Points)627 i54·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Points)
628 ·--·used·10.6283s·(cpu);·7.36659s·(thread);·0s·(gc)628 ·--·used·13.2967s·(cpu);·8.58915s·(thread);·0s·(gc)
  
629 o54·=·true629 o54·=·true
  
630 i55·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultWithPoints)630 i55·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultWithPoints)
631 ·--·used·7.84088s·(cpu);·5.52707s·(thread);·0s·(gc)631 ·--·used·9.90555s·(cpu);·6.44207s·(thread);·0s·(gc)
  
Max diff block lines reached; 10/5300 bytes (0.19%) of diff not shown.
⊟
6.45 KB
./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
    
Offset 7, 20 lines modifiedOffset 7, 20 lines modified
7 o2·:·Ideal·of·S7 o2·:·Ideal·of·S
  
8 i3·:·dim·(S/J)8 i3·:·dim·(S/J)
  
9 o3·=·49 o3·=·4
  
10 i4·:·time·regularInCodimension(1,·S/J)10 i4·:·time·regularInCodimension(1,·S/J)
11 ·--·used·0.659643s·(cpu);·0.520517s·(thread);·0s·(gc)11 ·--·used·0.905996s·(cpu);·0.62782s·(thread);·0s·(gc)
  
12 o4·=·true12 o4·=·true
  
13 i5·:·time·regularInCodimension(2,·S/J)13 i5·:·time·regularInCodimension(2,·S/J)
14 ·--·used·7.82665s·(cpu);·6.35155s·(thread);·0s·(gc)14 ·--·used·9.39668s·(cpu);·7.19171s·(thread);·0s·(gc)
  
15 i6·:·time·regularInCodimension(1,·S/J,·Verbose=>true)15 i6·:·time·regularInCodimension(1,·S/J,·Verbose=>true)
16 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·452.908·of·them.16 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·452.908·of·them.
17 regularInCodimension:·About·to·enter·loop17 regularInCodimension:·About·to·enter·loop
18 internalChooseMinor:·Choosing·LexSmallestTerm18 internalChooseMinor:·Choosing·LexSmallestTerm
19 internalChooseMinor:·Choosing·Random19 internalChooseMinor:·Choosing·Random
20 internalChooseMinor:·Choosing·GRevLexSmallest20 internalChooseMinor:·Choosing·GRevLexSmallest
Offset 87, 21 lines modifiedOffset 87, 21 lines modified
87 internalChooseMinor:·Choosing·LexSmallest87 internalChooseMinor:·Choosing·LexSmallest
88 internalChooseMinor:·Choosing·LexSmallestTerm88 internalChooseMinor:·Choosing·LexSmallestTerm
89 internalChooseMinor:·Choosing·LexSmallest89 internalChooseMinor:·Choosing·LexSmallest
90 internalChooseMinor:·Choosing·Random90 internalChooseMinor:·Choosing·Random
91 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·49,·and·computed·=·3991 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·49,·and·computed·=·39
92 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast92 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast
93 regularInCodimension:··partial·singular·locus·dimension·computed,·=·293 regularInCodimension:··partial·singular·locus·dimension·computed,·=·2
94 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute·--·used·1.01371s·(cpu);·0.813721s·(thread);·0s·(gc)94 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute·--·used·1.18226s·(cpu);·0.90525s·(thread);·0s·(gc)
95 d·=·39.··singular·locus·dimension·appears·to·be·=·295 d·=·39.··singular·locus·dimension·appears·to·be·=·2
  
96 o6·=·true96 o6·=·true
  
97 i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)97 i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)
98 ·--·used·0.13374s·(cpu);·0.0997883s·(thread);·0s·(gc)98 ·--·used·0.162761s·(cpu);·0.121774s·(thread);·0s·(gc)
99 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.99 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
100 regularInCodimension:·About·to·enter·loop100 regularInCodimension:·About·to·enter·loop
101 internalChooseMinor:·Choosing·Random101 internalChooseMinor:·Choosing·Random
102 internalChooseMinor:·Choosing·RandomNonZero102 internalChooseMinor:·Choosing·RandomNonZero
103 internalChooseMinor:·Choosing·GRevLexSmallestTerm103 internalChooseMinor:·Choosing·GRevLexSmallestTerm
104 internalChooseMinor:·Choosing·Random104 internalChooseMinor:·Choosing·Random
105 internalChooseMinor:·Choosing·Random105 internalChooseMinor:·Choosing·Random
Offset 115, 15 lines modifiedOffset 115, 15 lines modified
115 internalChooseMinor:·Choosing·LexSmallest115 internalChooseMinor:·Choosing·LexSmallest
116 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10116 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10
117 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.117 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
118 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3118 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3
119 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3119 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3
  
120 i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,·Verbose=>true)120 i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,·Verbose=>true)
121 ·--·used·0.120882s·(cpu);·0.0963134s·(thread);·0s·(gc)121 ·--·used·0.13672s·(cpu);·0.102678s·(thread);·0s·(gc)
122 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.122 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
123 regularInCodimension:·About·to·enter·loop123 regularInCodimension:·About·to·enter·loop
124 internalChooseMinor:·Choosing·Random124 internalChooseMinor:·Choosing·Random
125 internalChooseMinor:·Choosing·Random125 internalChooseMinor:·Choosing·Random
126 internalChooseMinor:·Choosing·Random126 internalChooseMinor:·Choosing·Random
127 internalChooseMinor:·Choosing·Random127 internalChooseMinor:·Choosing·Random
128 internalChooseMinor:·Choosing·Random128 internalChooseMinor:·Choosing·Random
Offset 137, 15 lines modifiedOffset 137, 15 lines modified
137 internalChooseMinor:·Choosing·Random137 internalChooseMinor:·Choosing·Random
138 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10138 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10
139 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.139 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
140 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3140 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3
141 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3141 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3
  
142 i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t->3,·Verbose=>true)142 i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t->3,·Verbose=>true)
143 ·--·used·0.473747s·(cpu);·0.383319s·(thread);·0s·(gc)143 ·--·used·0.566642s·(cpu);·0.425168s·(thread);·0s·(gc)
144 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.144 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
145 regularInCodimension:·About·to·enter·loop145 regularInCodimension:·About·to·enter·loop
146 internalChooseMinor:·Choosing·RandomNonZero146 internalChooseMinor:·Choosing·RandomNonZero
147 internalChooseMinor:·Choosing·Random147 internalChooseMinor:·Choosing·Random
148 internalChooseMinor:·Choosing·LexSmallest148 internalChooseMinor:·Choosing·LexSmallest
149 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·3,·and·computed·=·3149 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·3,·and·computed·=·3
150 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.150 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
Offset 165, 15 lines modifiedOffset 165, 15 lines modified
165 internalChooseMinor:·Choosing·GRevLexSmallestTerm165 internalChooseMinor:·Choosing·GRevLexSmallestTerm
166 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10166 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·10,·and·computed·=·10
167 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.167 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
168 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3168 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3
169 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3169 regularInCodimension:··Loop·completed,·submatrices·considered·=·10,·and·computed·=·10.··singular·locus·dimension·appears·to·be·=·3
  
170 i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t->t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)170 i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t->t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)
171 ·--·used·0.541436s·(cpu);·0.420353s·(thread);·0s·(gc)171 ·--·used·0.561127s·(cpu);·0.428435s·(thread);·0s·(gc)
172 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.172 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.
173 regularInCodimension:·About·to·enter·loop173 regularInCodimension:·About·to·enter·loop
174 internalChooseMinor:·Choosing·GRevLexSmallestTerm174 internalChooseMinor:·Choosing·GRevLexSmallestTerm
175 internalChooseMinor:·Choosing·GRevLexSmallestTerm175 internalChooseMinor:·Choosing·GRevLexSmallestTerm
176 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·2,·and·computed·=·2176 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·2,·and·computed·=·2
177 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.177 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
178 regularInCodimension:··partial·singular·locus·dimension·computed,·=·4178 regularInCodimension:··partial·singular·locus·dimension·computed,·=·4
Offset 214, 15 lines modifiedOffset 214, 15 lines modified
214 internalChooseMinor:·Choosing·GRevLexSmallestTerm214 internalChooseMinor:·Choosing·GRevLexSmallestTerm
215 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·25,·and·computed·=·23215 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·25,·and·computed·=·23
216 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.216 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
217 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3217 regularInCodimension:··partial·singular·locus·dimension·computed,·=·3
218 regularInCodimension:··Loop·completed,·submatrices·considered·=·25,·and·computed·=·23.··singular·locus·dimension·appears·to·be·=·3218 regularInCodimension:··Loop·completed,·submatrices·considered·=·25,·and·computed·=·23.··singular·locus·dimension·appears·to·be·=·3
  
219 i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>·true,·Verbose=>true)219 i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>·true,·Verbose=>true)
220 ·--·used·0.322447s·(cpu);·0.247392s·(thread);·0s·(gc)220 ·--·used·0.413418s·(cpu);·0.288524s·(thread);·0s·(gc)
221 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.221 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.
222 regularInCodimension:·About·to·enter·loop222 regularInCodimension:·About·to·enter·loop
223 internalChooseMinor:·Choosing·GRevLexSmallest223 internalChooseMinor:·Choosing·GRevLexSmallest
224 internalChooseMinor:·Choosing·LexSmallest224 internalChooseMinor:·Choosing·LexSmallest
225 internalChooseMinor:·Choosing·RandomNonZero225 internalChooseMinor:·Choosing·RandomNonZero
226 internalChooseMinor:·Choosing·RandomNonZero226 internalChooseMinor:·Choosing·RandomNonZero
227 internalChooseMinor:·Choosing·GRevLexSmallestTerm227 internalChooseMinor:·Choosing·GRevLexSmallestTerm
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1.21 KB
./usr/share/doc/Macaulay2/FastMinors/example-output/___Strategy__Default.out
    
Offset 1, 13 lines modifiedOffset 1, 13 lines modified
1 --·-*-·M2-comint·-*-·hash:·55092798754059419991 --·-*-·M2-comint·-*-·hash:·5509279875405941999
  
2 i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);2 i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
  
3 i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)3 i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)
4 ·--·1.95279s·elapsed4 ·--·1.48846s·elapsed
  
5 o2·=·true5 o2·=·true
  
6 i3·:·peek·StrategyDefault6 i3·:·peek·StrategyDefault
  
7 o3·=·OptionTable{GRevLexLargest·=>·0······}7 o3·=·OptionTable{GRevLexLargest·=>·0······}
8 ·················GRevLexSmallest·=>·168 ·················GRevLexSmallest·=>·16
Offset 16, 12 lines modifiedOffset 16, 12 lines modified
16 ·················LexSmallest·=>·1616 ·················LexSmallest·=>·16
17 ·················LexSmallestTerm·=>·1617 ·················LexSmallestTerm·=>·16
18 ·················Points·=>·018 ·················Points·=>·0
19 ·················Random·=>·1619 ·················Random·=>·16
20 ·················RandomNonzero·=>·1620 ·················RandomNonzero·=>·16
  
21 i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)21 i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)
22 ·--·1.14197s·elapsed22 ·--·.963819s·elapsed
  
23 o4·=·true23 o4·=·true
  
24 i5·:·24 i5·:·
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1.88 KB
./usr/share/doc/Macaulay2/FastMinors/example-output/_is__Codim__At__Least.out
    
Offset 16, 29 lines modifiedOffset 16, 29 lines modified
16 i5·:·r·=·rank·myDiff;16 i5·:·r·=·rank·myDiff;
  
17 i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);17 i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);
  
18 o6·:·Ideal·of·R18 o6·:·Ideal·of·R
  
19 i7·:·time·isCodimAtLeast(3,·J)19 i7·:·time·isCodimAtLeast(3,·J)
20 ·--·used·0.00399435s·(cpu);·0.00202445s·(thread);·0s·(gc)20 ·--·used·0.00400086s·(cpu);·0.00223807s·(thread);·0s·(gc)
  
21 o7·=·true21 o7·=·true
  
22 i8·:·I·=·ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);22 i8·:·I·=·ideal(x_2^8*x_10^3-3*x_1*x_2^7*x_10^2*x_11+3*x_1^2*x_2^6*x_10*x_11^2-x_1^3*x_2^5*x_11^3,x_5^5*x_6^3*x_11^3-3*x_5^6*x_6^2*x_11^2*x_12+3*x_5^7*x_6*x_11*x_12^2-x_5^8*x_12^3,x_1^5*x_2^3*x_4^3-3*x_1^6*x_2^2*x_4^2*x_5+3*x_1^7*x_2*x_4*x_5^2-x_1^8*x_5^3,x_6^8*x_11^3-3*x_5*x_6^7*x_11^2*x_12+3*x_5^2*x_6^6*x_11*x_12^2-x_5^3*x_6^5*x_12^3,x_8^3*x_10^8-3*x_7*x_8^2*x_10^7*x_11+3*x_7^2*x_8*x_10^6*x_11^2-x_7^3*x_10^5*x_11^3,x_2^8*x_4^3-3*x_1*x_2^7*x_4^2*x_5+3*x_1^2*x_2^6*x_4*x_5^2-x_1^3*x_2^5*x_5^3,-x_6^3*x_11^8+3*x_5*x_6^2*x_11^7*x_12-3*x_5^2*x_6*x_11^6*x_12^2+x_5^3*x_11^5*x_12^3,-x_6^3*x_7^3*x_9^5+3*x_4*x_6^2*x_7^2*x_9^6-3*x_4^2*x_6*x_7*x_9^7+x_4^3*x_9^8,x_8^8*x_10^3-3*x_7*x_8^7*x_10^2*x_11+3*x_7^2*x_8^6*x_10*x_11^2-x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
  
23 ···············ZZ23 ···············ZZ
24 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]24 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]
25 ··············127··11···8···1···9···12···6···5···10···2···4···3···725 ··············127··11···8···1···9···12···6···5···10···2···4···3···7
  
26 i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)26 i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)
27 ·--·used·0.00134704s·(cpu);·0.00196404s·(thread);·0s·(gc)27 ·--·used·0.00172096s·(cpu);·0.00230666s·(thread);·0s·(gc)
28 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.28 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.
  
29 o9·=·true29 o9·=·true
  
30 i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)30 i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)
31 ·--·used·0.000208464s·(cpu);·0.00199842s·(thread);·0s·(gc)31 ·--·used·0.000912411s·(cpu);·0.00280648s·(thread);·0s·(gc)
  
32 o10·=·true32 o10·=·true
  
33 i11·:·33 i11·:·
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586 B
./usr/share/doc/Macaulay2/FastMinors/example-output/_proj__Dim.out
    
Offset 7, 17 lines modifiedOffset 7, 17 lines modified
7 o2·:·Ideal·of·R7 o2·:·Ideal·of·R
  
8 i3·:·pdim(module·I)8 i3·:·pdim(module·I)
  
9 o3·=·29 o3·=·2
  
10 i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)10 i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)
11 ·--·used·0.190172s·(cpu);·0.134312s·(thread);·0s·(gc)11 ·--·used·0.244968s·(cpu);·0.144192s·(thread);·0s·(gc)
  
12 o4·=·112 o4·=·1
  
13 i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)13 i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)
14 ·--·used·0.0109934s·(cpu);·0.0123083s·(thread);·0s·(gc)14 ·--·used·0.0126589s·(cpu);·0.0133201s·(thread);·0s·(gc)
  
15 o5·=·115 o5·=·1
  
16 i6·:·16 i6·:·
⊟
644 B
./usr/share/doc/Macaulay2/FastMinors/example-output/_recursive__Minors.out
    
Offset 4, 20 lines modifiedOffset 4, 20 lines modified
  
4 i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);4 i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);
  
5 ·············6······75 ·············6······7
6 o2·:·Matrix·R··<--·R6 o2·:·Matrix·R··<--·R
  
7 i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);7 i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);
8 ·--·used·0.321843s·(cpu);·0.318977s·(thread);·0s·(gc)8 ·--·used·0.434706s·(cpu);·0.435866s·(thread);·0s·(gc)
  
9 o3·:·Ideal·of·R9 o3·:·Ideal·of·R
  
10 i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);10 i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);
11 ·--·used·1.08533s·(cpu);·0.999458s·(thread);·0s·(gc)11 ·--·used·1.297s·(cpu);·1.1431s·(thread);·0s·(gc)
  
12 o4·:·Ideal·of·R12 o4·:·Ideal·of·R
  
13 i5·:·I1·==·I213 i5·:·I1·==·I2
  
14 o5·=·true14 o5·=·true
  
⊟
5.67 KB
./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
    
Offset 17, 44 lines modifiedOffset 17, 44 lines modified
17 i6·:·S·=·T/I;17 i6·:·S·=·T/I;
  
18 i7·:·dim·S18 i7·:·dim·S
  
19 o7·=·319 o7·=·3
  
20 i8·:·time·regularInCodimension(1,·S)20 i8·:·time·regularInCodimension(1,·S)
21 ·--·used·1.18412s·(cpu);·0.96706s·(thread);·0s·(gc)21 ·--·used·1.65731s·(cpu);·1.32984s·(thread);·0s·(gc)
  
22 o8·=·true22 o8·=·true
  
23 i9·:·time·regularInCodimension(2,·S)23 i9·:·time·regularInCodimension(2,·S)
24 ·--·used·5.06537s·(cpu);·4.21399s·(thread);·0s·(gc)24 ·--·used·5.95441s·(cpu);·4.74918s·(thread);·0s·(gc)
  
25 i10·:·R·=·QQ[c,·f,·g,·h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);25 i10·:·R·=·QQ[c,·f,·g,·h]/ideal(g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
  
26 i11·:·dim(R)26 i11·:·dim(R)
  
27 o11·=·227 o11·=·2
  
28 i12·:·time·(dim·singularLocus·(R))28 i12·:·time·(dim·singularLocus·(R))
29 ·--·used·0.0157777s·(cpu);·0.0148119s·(thread);·0s·(gc)29 ·--·used·0.0279973s·(cpu);·0.0254638s·(thread);·0s·(gc)
  
30 o12·=·-130 o12·=·-1
  
31 i13·:·time·regularInCodimension(2,·R)31 i13·:·time·regularInCodimension(2,·R)
32 ·--·used·0.369174s·(cpu);·0.287095s·(thread);·0s·(gc)32 ·--·used·0.516182s·(cpu);·0.383168s·(thread);·0s·(gc)
  
33 o13·=·true33 o13·=·true
  
34 i14·:·time·regularInCodimension(2,·R)34 i14·:·time·regularInCodimension(2,·R)
35 ·--·used·0.649844s·(cpu);·0.490454s·(thread);·0s·(gc)35 ·--·used·0.812796s·(cpu);·0.626848s·(thread);·0s·(gc)
  
36 o14·=·true36 o14·=·true
  
37 i15·:·time·regularInCodimension(2,·R)37 i15·:·time·regularInCodimension(2,·R)
38 ·--·used·0.260796s·(cpu);·0.201669s·(thread);·0s·(gc)38 ·--·used·0.353608s·(cpu);·0.256535s·(thread);·0s·(gc)
  
39 o15·=·true39 o15·=·true
  
40 i16·:·time·regularInCodimension(2,·S,·Verbose=>true)40 i16·:·time·regularInCodimension(2,·S,·Verbose=>true)
41 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·327.599·of·them.41 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·327.599·of·them.
42 regularInCodimension:·About·to·enter·loop42 regularInCodimension:·About·to·enter·loop
43 internalChooseMinor:·Choosing·GRevLexSmallestTerm43 internalChooseMinor:·Choosing·GRevLexSmallestTerm
Offset 386, 15 lines modifiedOffset 386, 15 lines modified
386 internalChooseMinor:·Choosing·GRevLexSmallestTerm386 internalChooseMinor:·Choosing·GRevLexSmallestTerm
387 internalChooseMinor:·Choosing·RandomNonZero387 internalChooseMinor:·Choosing·RandomNonZero
388 internalChooseMinor:·Choosing·LexSmallest388 internalChooseMinor:·Choosing·LexSmallest
389 internalChooseMinor:·Choosing·Random389 internalChooseMinor:·Choosing·Random
390 internalChooseMinor:·Choosing·Random390 internalChooseMinor:·Choosing·Random
391 internalChooseMinor:·Choosing·LexSmallestTerm391 internalChooseMinor:·Choosing·LexSmallestTerm
392 internalChooseMinor:·Choosing·GRevLexSmallestTerm392 internalChooseMinor:·Choosing·GRevLexSmallestTerm
393 internalChooseMinor:·Choosing·--·used·5.05989s·(cpu);·4.23214s·(thread);·0s·(gc)393 internalChooseMinor:·Choosing·--·used·6.09346s·(cpu);·4.79091s·(thread);·0s·(gc)
394 ·LexSmallestTerm394 ·LexSmallestTerm
395 internalChooseMinor:·Choosing·Random395 internalChooseMinor:·Choosing·Random
396 internalChooseMinor:·Choosing·Random396 internalChooseMinor:·Choosing·Random
397 internalChooseMinor:·Choosing·GRevLexSmallest397 internalChooseMinor:·Choosing·GRevLexSmallest
398 internalChooseMinor:·Choosing·RandomNonZero398 internalChooseMinor:·Choosing·RandomNonZero
399 internalChooseMinor:·Choosing·GRevLexSmallest399 internalChooseMinor:·Choosing·GRevLexSmallest
400 internalChooseMinor:·Choosing·GRevLexSmallestTerm400 internalChooseMinor:·Choosing·GRevLexSmallestTerm
Offset 430, 15 lines modifiedOffset 430, 15 lines modified
430 internalChooseMinor:·Choosing·Random430 internalChooseMinor:·Choosing·Random
431 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·328,·and·computed·=·186431 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·328,·and·computed·=·186
432 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.432 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
433 regularInCodimension:··partial·singular·locus·dimension·computed,·=·1433 regularInCodimension:··partial·singular·locus·dimension·computed,·=·1
434 regularInCodimension:··Loop·completed,·submatrices·considered·=·328,·and·computed·=·186.··singular·locus·dimension·appears·to·be·=·1434 regularInCodimension:··Loop·completed,·submatrices·considered·=·328,·and·computed·=·186.··singular·locus·dimension·appears·to·be·=·1
  
435 i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)435 i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)
436 ·--·used·1.04176s·(cpu);·0.888601s·(thread);·0s·(gc)436 ·--·used·1.44101s·(cpu);·1.19387s·(thread);·0s·(gc)
437 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·30·of·them.437 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·30·of·them.
438 regularInCodimension:·About·to·enter·loop438 regularInCodimension:·About·to·enter·loop
439 internalChooseMinor:·Choosing·Random439 internalChooseMinor:·Choosing·Random
440 internalChooseMinor:·Choosing·Random440 internalChooseMinor:·Choosing·Random
441 internalChooseMinor:·Choosing·RandomNonZero441 internalChooseMinor:·Choosing·RandomNonZero
442 internalChooseMinor:·Choosing·LexSmallestTerm442 internalChooseMinor:·Choosing·LexSmallestTerm
443 internalChooseMinor:·Choosing·RandomNonZero443 internalChooseMinor:·Choosing·RandomNonZero
Offset 490, 59 lines modifiedOffset 490, 59 lines modified
490 i18·:·StrategyCurrent#Random·=·0;490 i18·:·StrategyCurrent#Random·=·0;
  
491 i19·:·StrategyCurrent#LexSmallest·=·100;491 i19·:·StrategyCurrent#LexSmallest·=·100;
  
492 i20·:·StrategyCurrent#LexSmallestTerm·=·0;492 i20·:·StrategyCurrent#LexSmallestTerm·=·0;
  
493 i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)493 i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
494 ·--·used·0.101049s·(cpu);·0.073258s·(thread);·0s·(gc)494 ·--·used·0.160646s·(cpu);·0.107785s·(thread);·0s·(gc)
  
495 o21·=·true495 o21·=·true
  
496 i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)496 i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
497 ·--·used·0.14821s·(cpu);·0.12367s·(thread);·0s·(gc)497 ·--·used·0.253229s·(cpu);·0.168654s·(thread);·0s·(gc)
  
498 o22·=·true498 o22·=·true
  
499 i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)499 i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
500 ·--·used·1.15046s·(cpu);·0.964828s·(thread);·0s·(gc)500 ·--·used·1.34427s·(cpu);·1.03428s·(thread);·0s·(gc)
  
501 o23·=·true501 o23·=·true
  
502 i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)502 i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
503 ·--·used·0.964293s·(cpu);·0.782207s·(thread);·0s·(gc)503 ·--·used·1.10973s·(cpu);·0.865713s·(thread);·0s·(gc)
  
504 o24·=·true504 o24·=·true
  
505 i25·:·StrategyCurrent#LexSmallest·=·0;505 i25·:·StrategyCurrent#LexSmallest·=·0;
  
506 i26·:·StrategyCurrent#LexSmallestTerm·=·100;506 i26·:·StrategyCurrent#LexSmallestTerm·=·100;
  
507 i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)507 i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
508 ·--·used·0.30488s·(cpu);·0.264081s·(thread);·0s·(gc)508 ·--·used·0.445197s·(cpu);·0.282256s·(thread);·0s·(gc)
  
509 o27·=·true509 o27·=·true
  
510 i28·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)510 i28·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
511 ·--·used·1.70445s·(cpu);·1.34525s·(thread);·0s·(gc)511 ·--·used·2.08877s·(cpu);·1.52933s·(thread);·0s·(gc)
  
512 i29·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)512 i29·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
513 ·--·used·0.129454s·(cpu);·0.102854s·(thread);·0s·(gc)513 ·--·used·0.149107s·(cpu);·0.12124s·(thread);·0s·(gc)
  
514 o29·=·true514 o29·=·true
  
515 i30·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)515 i30·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
516 ·--·used·0.16017s·(cpu);·0.135024s·(thread);·0s·(gc)516 ·--·used·0.202604s·(cpu);·0.144515s·(thread);·0s·(gc)
  
517 o30·=·true517 o30·=·true
  
518 i31·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)518 i31·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)
519 ·--·used·1.17756s·(cpu);·0.999373s·(thread);·0s·(gc)519 ·--·used·1.55939s·(cpu);·1.30429s·(thread);·0s·(gc)
Max diff block lines reached; 209/5646 bytes (3.70%) of diff not shown.
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17.8 KB
./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
    
Offset 558, 57 lines modifiedOffset 558, 57 lines modified
558 ········</table>558 ········</table>
559 ········<div>559 ········<div>
560 ··········<p>Here·the·$1$·passed·to·the·function·says·how·many·minors·to·compute.··For·instance,·let's·compute·8·minors·for·each·of·these·strategies·and·see·if·that·was·enough·to·verify·that·the·ring·is·regular·in·codimension·1.··In·other·words,·if·the·dimension·of·$J$·plus·the·ideal·of·partial·minors·is·$\leq·1$·(since·$S/J$·has·dimension·3).</p>560 ··········<p>Here·the·$1$·passed·to·the·function·says·how·many·minors·to·compute.··For·instance,·let's·compute·8·minors·for·each·of·these·strategies·and·see·if·that·was·enough·to·verify·that·the·ring·is·regular·in·codimension·1.··In·other·words,·if·the·dimension·of·$J$·plus·the·ideal·of·partial·minors·is·$\leq·1$·(since·$S/J$·has·dimension·3).</p>
561 ········</div>561 ········</div>
562 ········<table·class="examples">562 ········<table·class="examples">
563 ··········<tr>563 ··········<tr>
564 <td>··············<pre><code·class="language-macaulay2">i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))564 <td>··············<pre><code·class="language-macaulay2">i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))
565 ·--·used·0.0990349s·(cpu);·0.0986288s·(thread);·0s·(gc)565 ·--·used·0.10298s·(cpu);·0.102107s·(thread);·0s·(gc)
  
566 o28·=·2</code></pre>566 o28·=·2</code></pre>
567 </td>··········</tr>567 </td>··········</tr>
568 ··········<tr>568 ··········<tr>
569 <td>··············<pre><code·class="language-macaulay2">i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))569 <td>··············<pre><code·class="language-macaulay2">i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))
570 ·--·used·0.240733s·(cpu);·0.176896s·(thread);·0s·(gc)570 ·--·used·0.245805s·(cpu);·0.171742s·(thread);·0s·(gc)
  
571 o29·=·3</code></pre>571 o29·=·3</code></pre>
572 </td>··········</tr>572 </td>··········</tr>
573 ··········<tr>573 ··········<tr>
574 <td>··············<pre><code·class="language-macaulay2">i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))574 <td>··············<pre><code·class="language-macaulay2">i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))
575 ·--·used·0.314859s·(cpu);·0.24812s·(thread);·0s·(gc)575 ·--·used·0.335311s·(cpu);·0.267543s·(thread);·0s·(gc)
  
576 o30·=·1</code></pre>576 o30·=·1</code></pre>
577 </td>··········</tr>577 </td>··········</tr>
578 ··········<tr>578 ··········<tr>
579 <td>··············<pre><code·class="language-macaulay2">i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))579 <td>··············<pre><code·class="language-macaulay2">i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))
580 ·--·used·0.22722s·(cpu);·0.161955s·(thread);·0s·(gc)580 ·--·used·0.220887s·(cpu);·0.170257s·(thread);·0s·(gc)
  
581 o31·=·2</code></pre>581 o31·=·2</code></pre>
582 </td>··········</tr>582 </td>··········</tr>
583 ··········<tr>583 ··········<tr>
584 <td>··············<pre><code·class="language-macaulay2">i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))584 <td>··············<pre><code·class="language-macaulay2">i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))
585 ·--·used·0.218217s·(cpu);·0.154621s·(thread);·0s·(gc)585 ·--·used·0.242895s·(cpu);·0.146413s·(thread);·0s·(gc)
  
586 o32·=·3</code></pre>586 o32·=·3</code></pre>
587 </td>··········</tr>587 </td>··········</tr>
588 ··········<tr>588 ··········<tr>
589 <td>··············<pre><code·class="language-macaulay2">i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallestTerm))589 <td>··············<pre><code·class="language-macaulay2">i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallestTerm))
590 ·--·used·0.253356s·(cpu);·0.18669s·(thread);·0s·(gc)590 ·--·used·0.249652s·(cpu);·0.1774s·(thread);·0s·(gc)
  
591 o33·=·3</code></pre>591 o33·=·3</code></pre>
592 </td>··········</tr>592 </td>··········</tr>
593 ··········<tr>593 ··········<tr>
594 <td>··············<pre><code·class="language-macaulay2">i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))594 <td>··············<pre><code·class="language-macaulay2">i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))
595 ·--·used·0.225233s·(cpu);·0.169651s·(thread);·0s·(gc)595 ·--·used·0.24987s·(cpu);·0.164533s·(thread);·0s·(gc)
  
596 o34·=·3</code></pre>596 o34·=·3</code></pre>
597 </td>··········</tr>597 </td>··········</tr>
598 ··········<tr>598 ··········<tr>
599 <td>··············<pre><code·class="language-macaulay2">i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))599 <td>··············<pre><code·class="language-macaulay2">i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))
600 ·--·used·9.72612s·(cpu);·7.03107s·(thread);·0s·(gc)600 ·--·used·11.0167s·(cpu);·7.71193s·(thread);·0s·(gc)
  
601 o35·=·1</code></pre>601 o35·=·1</code></pre>
602 </td>··········</tr>602 </td>··········</tr>
603 ········</table>603 ········</table>
604 ········<div>604 ········<div>
605 ··········<p>Indeed,·in·this·example,·even·computing·determinants·of·1,000·random·submatrices·is·not·typically·enough·to·verify·that·$V(J)$·is·regular·in·codimension·1.··On·the·other·hand,·<span·class="tt">Points</span>·is·almost·always·quite·effective·at·finding·valuable·submatrices,·but·can·be·quite·slow.··In·this·particular·example,·we·can·see·that·<span·class="tt">LexSmallestTerm</span>·also·performs·very·well·(and·does·it·quickly).·Since·different·strategies·work·better·or·worse·on·different·examples,·the·default·strategy·actually·mixes·and·matches·various·strategies.··The·default·strategy,·which·we·now·elucidate,</p>605 ··········<p>Indeed,·in·this·example,·even·computing·determinants·of·1,000·random·submatrices·is·not·typically·enough·to·verify·that·$V(J)$·is·regular·in·codimension·1.··On·the·other·hand,·<span·class="tt">Points</span>·is·almost·always·quite·effective·at·finding·valuable·submatrices,·but·can·be·quite·slow.··In·this·particular·example,·we·can·see·that·<span·class="tt">LexSmallestTerm</span>·also·performs·very·well·(and·does·it·quickly).·Since·different·strategies·work·better·or·worse·on·different·examples,·the·default·strategy·actually·mixes·and·matches·various·strategies.··The·default·strategy,·which·we·now·elucidate,</p>
606 ········</div>606 ········</div>
Offset 629, 15 lines modifiedOffset 629, 15 lines modified
629 ········</table>629 ········</table>
630 ········<div>630 ········<div>
631 ··········<p>says·that·we·should·use·<span·class="tt">GRevLexSmallest,·GRevLexSmallestTerm,·LexSmallest,·LexSmallestTerm,·Random,·RandomNonzero</span>·all·with·equal·probability·(note·<span·class="tt">RandomNonzero</span>,·which·we·have·not·yet·discussed·chooses·random·submatrices·where·no·row·or·column·is·zero,·which·is·good·for·working·in·sparse·matrices).··For·instance,·if·we·run:</p>631 ··········<p>says·that·we·should·use·<span·class="tt">GRevLexSmallest,·GRevLexSmallestTerm,·LexSmallest,·LexSmallestTerm,·Random,·RandomNonzero</span>·all·with·equal·probability·(note·<span·class="tt">RandomNonzero</span>,·which·we·have·not·yet·discussed·chooses·random·submatrices·where·no·row·or·column·is·zero,·which·is·good·for·working·in·sparse·matrices).··For·instance,·if·we·run:</p>
632 ········</div>632 ········</div>
633 ········<table·class="examples">633 ········<table·class="examples">
634 ··········<tr>634 ··········<tr>
635 <td>··············<pre><code·class="language-macaulay2">i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,·Verbose=>true);635 <td>··············<pre><code·class="language-macaulay2">i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,·Verbose=>true);
636 ·--·used·0.407857s·(cpu);·0.316004s·(thread);·0s·(gc)636 ·--·used·0.485201s·(cpu);·0.346923s·(thread);·0s·(gc)
637 internalChooseMinor:·Choosing·Random637 internalChooseMinor:·Choosing·Random
638 internalChooseMinor:·Choosing·LexSmallest638 internalChooseMinor:·Choosing·LexSmallest
639 internalChooseMinor:·Choosing·Random639 internalChooseMinor:·Choosing·Random
640 internalChooseMinor:·Choosing·GRevLexSmallestTerm640 internalChooseMinor:·Choosing·GRevLexSmallestTerm
641 internalChooseMinor:·Choosing·RandomNonZero641 internalChooseMinor:·Choosing·RandomNonZero
642 internalChooseMinor:·Choosing·RandomNonZero642 internalChooseMinor:·Choosing·RandomNonZero
643 internalChooseMinor:·Choosing·LexSmallest643 internalChooseMinor:·Choosing·LexSmallest
Offset 728, 15 lines modifiedOffset 728, 15 lines modified
728 ··········<tr>728 ··········<tr>
729 <td>··············<pre><code·class="language-macaulay2">i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value729 <td>··············<pre><code·class="language-macaulay2">i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value
  
730 o42·=·true</code></pre>730 o42·=·true</code></pre>
731 </td>··········</tr>731 </td>··········</tr>
732 ··········<tr>732 ··········<tr>
733 <td>··············<pre><code·class="language-macaulay2">i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratGeometric))733 <td>··············<pre><code·class="language-macaulay2">i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratGeometric))
734 ·--·used·0.638168s·(cpu);·0.443521s·(thread);·0s·(gc)734 ·--·used·0.827591s·(cpu);·0.517732s·(thread);·0s·(gc)
  
735 o43·=·2</code></pre>735 o43·=·2</code></pre>
736 </td>··········</tr>736 </td>··········</tr>
737 ··········<tr>737 ··········<tr>
738 <td>··············<pre><code·class="language-macaulay2">i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that·value738 <td>··············<pre><code·class="language-macaulay2">i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that·value
  
739 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}739 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}
Offset 754, 67 lines modifiedOffset 754, 67 lines modified
754 ··········<tr>754 ··········<tr>
755 <td>··············<pre><code·class="language-macaulay2">i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value755 <td>··············<pre><code·class="language-macaulay2">i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value
  
756 o45·=·false</code></pre>756 o45·=·false</code></pre>
757 </td>··········</tr>757 </td>··········</tr>
758 ··········<tr>758 ··········<tr>
759 <td>··············<pre><code·class="language-macaulay2">i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratRational))759 <td>··············<pre><code·class="language-macaulay2">i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,·PointOptions=>ptsStratRational))
760 ·--·used·0.520675s·(cpu);·0.353044s·(thread);·0s·(gc)760 ·--·used·0.648394s·(cpu);·0.403013s·(thread);·0s·(gc)
  
761 o46·=·2</code></pre>761 o46·=·2</code></pre>
762 </td>··········</tr>762 </td>··········</tr>
763 ········</table>763 ········</table>
764 ········<div>764 ········<div>
765 ··········<p>Other·options·may·also·be·passed·to·the·<a·title="Obtain·random·points·in·a·variety"·href="../../RandomPoints/html/index.html">RandomPoints</a>·package·via·the·<a·title="options·to·pass·to·functions·in·the·package·RandomPoints"·href="___Point__Options.html">PointOptions</a>·option.</p>765 ··········<p>Other·options·may·also·be·passed·to·the·<a·title="Obtain·random·points·in·a·variety"·href="../../RandomPoints/html/index.html">RandomPoints</a>·package·via·the·<a·title="options·to·pass·to·functions·in·the·package·RandomPoints"·href="___Point__Options.html">PointOptions</a>·option.</p>
766 ········</div>766 ········</div>
767 ········<div>767 ········<div>
768 ··········<p><b>regularInCodimension:</b>··It·is·reasonable·to·think·that·you·should·find·a·few·minors·(with·one·strategy·or·another),·and·see·if·perhaps·the·minors·you·have·computed·so·far·are·enough·to·verify·our·ring·is·regular·in·codimension·1.··This·is·exactly·what·<span·class="tt">regularInCodimension</span>·does.··One·can·control·at·a·fine·level·how·frequently·new·minors·are·computed,·and·how·frequently·the·dimension·of·what·we·have·computed·so·far·is·checked,·by·the·option·<span·class="tt">codimCheckFunction</span>.··For·more·on·that,·see·<a·title="A·tutorial·for·how·to·use·the·advanced·options·of·the·regularInCodimension·function"·href="___Regular__In__Codimension__Tutorial.html">RegularInCodimensionTutorial</a>·and·<a·title="attempts·to·show·that·the·ring·is·regular·in·codimension·n"·href="_regular__In__Codimension.html">regularInCodimension</a>.··Let·us·finish·running·<span·class="tt">regularInCodimension</span>·on·our·example·with·several·different·strategies.</p>768 ··········<p><b>regularInCodimension:</b>··It·is·reasonable·to·think·that·you·should·find·a·few·minors·(with·one·strategy·or·another),·and·see·if·perhaps·the·minors·you·have·computed·so·far·are·enough·to·verify·our·ring·is·regular·in·codimension·1.··This·is·exactly·what·<span·class="tt">regularInCodimension</span>·does.··One·can·control·at·a·fine·level·how·frequently·new·minors·are·computed,·and·how·frequently·the·dimension·of·what·we·have·computed·so·far·is·checked,·by·the·option·<span·class="tt">codimCheckFunction</span>.··For·more·on·that,·see·<a·title="A·tutorial·for·how·to·use·the·advanced·options·of·the·regularInCodimension·function"·href="___Regular__In__Codimension__Tutorial.html">RegularInCodimensionTutorial</a>·and·<a·title="attempts·to·show·that·the·ring·is·regular·in·codimension·n"·href="_regular__In__Codimension.html">regularInCodimension</a>.··Let·us·finish·running·<span·class="tt">regularInCodimension</span>·on·our·example·with·several·different·strategies.</p>
769 ········</div>769 ········</div>
770 ········<table·class="examples">770 ········<table·class="examples">
771 ··········<tr>771 ··········<tr>
772 <td>··············<pre><code·class="language-macaulay2">i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefault)772 <td>··············<pre><code·class="language-macaulay2">i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefault)
773 ·--·used·3.51988s·(cpu);·2.84854s·(thread);·0s·(gc)</code></pre>773 ·--·used·4.50417s·(cpu);·3.50025s·(thread);·0s·(gc)</code></pre>
774 </td>··········</tr>774 </td>··········</tr>
775 ··········<tr>775 ··········<tr>
776 <td>··············<pre><code·class="language-macaulay2">i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultNonRandom)776 <td>··············<pre><code·class="language-macaulay2">i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>StrategyDefaultNonRandom)
777 ·--·used·0.812059s·(cpu);·0.606774s·(thread);·0s·(gc)777 ·--·used·0.933099s·(cpu);·0.636156s·(thread);·0s·(gc)
  
778 o48·=·true</code></pre>778 o48·=·true</code></pre>
779 </td>··········</tr>779 </td>··········</tr>
780 ··········<tr>780 ··········<tr>
781 <td>··············<pre><code·class="language-macaulay2">i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)781 <td>··············<pre><code·class="language-macaulay2">i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)
782 ·--·used·2.71639s·(cpu);·2.44142s·(thread);·0s·(gc)</code></pre>782 ·--·used·3.0767s·(cpu);·2.73986s·(thread);·0s·(gc)</code></pre>
783 </td>··········</tr>783 </td>··········</tr>
784 ··········<tr>784 ··········<tr>
785 <td>··············<pre><code·class="language-macaulay2">i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallest)785 <td>··············<pre><code·class="language-macaulay2">i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>LexSmallest)
786 ·--·used·2.6533s·(cpu);·1.89165s·(thread);·0s·(gc)</code></pre>786 ·--·used·3.33268s·(cpu);·2.22841s·(thread);·0s·(gc)</code></pre>
787 </td>··········</tr>787 </td>··········</tr>
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Offset 486, 44 lines modifiedOffset 486, 44 lines modified
486 o27·:·Ideal·of·S486 o27·:·Ideal·of·S
487 Here·the·$1$·passed·to·the·function·says·how·many·minors·to·compute.·For487 Here·the·$1$·passed·to·the·function·says·how·many·minors·to·compute.·For
488 instance,·let's·compute·8·minors·for·each·of·these·strategies·and·see·if·that488 instance,·let's·compute·8·minors·for·each·of·these·strategies·and·see·if·that
489 was·enough·to·verify·that·the·ring·is·regular·in·codimension·1.·In·other·words,489 was·enough·to·verify·that·the·ring·is·regular·in·codimension·1.·In·other·words,
490 if·the·dimension·of·$J$·plus·the·ideal·of·partial·minors·is·$\leq·1$·(since·$S/490 if·the·dimension·of·$J$·plus·the·ideal·of·partial·minors·is·$\leq·1$·(since·$S/
491 J$·has·dimension·3).491 J$·has·dimension·3).
492 i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))492 i28·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Random))
493 ·--·used·0.0990349s·(cpu);·0.0986288s·(thread);·0s·(gc)493 ·--·used·0.10298s·(cpu);·0.102107s·(thread);·0s·(gc)
  
494 o28·=·2494 o28·=·2
495 i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))495 i29·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallest))
496 ·--·used·0.240733s·(cpu);·0.176896s·(thread);·0s·(gc)496 ·--·used·0.245805s·(cpu);·0.171742s·(thread);·0s·(gc)
  
497 o29·=·3497 o29·=·3
498 i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))498 i30·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexSmallestTerm))
499 ·--·used·0.314859s·(cpu);·0.24812s·(thread);·0s·(gc)499 ·--·used·0.335311s·(cpu);·0.267543s·(thread);·0s·(gc)
  
500 o30·=·1500 o30·=·1
501 i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))501 i31·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>LexLargest))
502 ·--·used·0.22722s·(cpu);·0.161955s·(thread);·0s·(gc)502 ·--·used·0.220887s·(cpu);·0.170257s·(thread);·0s·(gc)
  
503 o31·=·2503 o31·=·2
504 i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))504 i32·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexSmallest))
505 ·--·used·0.218217s·(cpu);·0.154621s·(thread);·0s·(gc)505 ·--·used·0.242895s·(cpu);·0.146413s·(thread);·0s·(gc)
  
506 o32·=·3506 o32·=·3
507 i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,507 i33·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,
508 Strategy=>GRevLexSmallestTerm))508 Strategy=>GRevLexSmallestTerm))
509 ·--·used·0.253356s·(cpu);·0.18669s·(thread);·0s·(gc)509 ·--·used·0.249652s·(cpu);·0.1774s·(thread);·0s·(gc)
  
510 o33·=·3510 o33·=·3
511 i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))511 i34·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>GRevLexLargest))
512 ·--·used·0.225233s·(cpu);·0.169651s·(thread);·0s·(gc)512 ·--·used·0.24987s·(cpu);·0.164533s·(thread);·0s·(gc)
  
513 o34·=·3513 o34·=·3
514 i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))514 i35·:·time·dim·(J·+·chooseGoodMinors(8,·6,·M,·J,·Strategy=>Points))
515 ·--·used·9.72612s·(cpu);·7.03107s·(thread);·0s·(gc)515 ·--·used·11.0167s·(cpu);·7.71193s·(thread);·0s·(gc)
  
516 o35·=·1516 o35·=·1
517 Indeed,·in·this·example,·even·computing·determinants·of·1,000·random517 Indeed,·in·this·example,·even·computing·determinants·of·1,000·random
518 submatrices·is·not·typically·enough·to·verify·that·$V(J)$·is·regular·in518 submatrices·is·not·typically·enough·to·verify·that·$V(J)$·is·regular·in
519 codimension·1.·On·the·other·hand,·Points·is·almost·always·quite·effective·at519 codimension·1.·On·the·other·hand,·Points·is·almost·always·quite·effective·at
520 finding·valuable·submatrices,·but·can·be·quite·slow.·In·this·particular520 finding·valuable·submatrices,·but·can·be·quite·slow.·In·this·particular
521 example,·we·can·see·that·LexSmallestTerm·also·performs·very·well·(and·does·it521 example,·we·can·see·that·LexSmallestTerm·also·performs·very·well·(and·does·it
Offset 544, 15 lines modifiedOffset 544, 15 lines modified
544 says·that·we·should·use·GRevLexSmallest,·GRevLexSmallestTerm,·LexSmallest,544 says·that·we·should·use·GRevLexSmallest,·GRevLexSmallestTerm,·LexSmallest,
545 LexSmallestTerm,·Random,·RandomNonzero·all·with·equal·probability·(note545 LexSmallestTerm,·Random,·RandomNonzero·all·with·equal·probability·(note
546 RandomNonzero,·which·we·have·not·yet·discussed·chooses·random·submatrices·where546 RandomNonzero,·which·we·have·not·yet·discussed·chooses·random·submatrices·where
547 no·row·or·column·is·zero,·which·is·good·for·working·in·sparse·matrices).·For547 no·row·or·column·is·zero,·which·is·good·for·working·in·sparse·matrices).·For
548 instance,·if·we·run:548 instance,·if·we·run:
549 i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,549 i37·:·time·chooseGoodMinors(20,·6,·M,·J,·Strategy=>StrategyDefault,
550 Verbose=>true);550 Verbose=>true);
551 ·--·used·0.407857s·(cpu);·0.316004s·(thread);·0s·(gc)551 ·--·used·0.485201s·(cpu);·0.346923s·(thread);·0s·(gc)
552 internalChooseMinor:·Choosing·Random552 internalChooseMinor:·Choosing·Random
553 internalChooseMinor:·Choosing·LexSmallest553 internalChooseMinor:·Choosing·LexSmallest
554 internalChooseMinor:·Choosing·Random554 internalChooseMinor:·Choosing·Random
555 internalChooseMinor:·Choosing·GRevLexSmallestTerm555 internalChooseMinor:·Choosing·GRevLexSmallestTerm
556 internalChooseMinor:·Choosing·RandomNonZero556 internalChooseMinor:·Choosing·RandomNonZero
557 internalChooseMinor:·Choosing·RandomNonZero557 internalChooseMinor:·Choosing·RandomNonZero
558 internalChooseMinor:·Choosing·LexSmallest558 internalChooseMinor:·Choosing·LexSmallest
Offset 633, 15 lines modifiedOffset 633, 15 lines modified
633 i41·:·ptsStratGeometric·=·new·OptionTable·from·(options633 i41·:·ptsStratGeometric·=·new·OptionTable·from·(options
634 chooseGoodMinors)#PointOptions;634 chooseGoodMinors)#PointOptions;
635 i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value635 i42·:·ptsStratGeometric#ExtendField·--look·at·the·default·value
  
636 o42·=·true636 o42·=·true
637 i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,637 i43·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,
638 PointOptions=>ptsStratGeometric))638 PointOptions=>ptsStratGeometric))
639 ·--·used·0.638168s·(cpu);·0.443521s·(thread);·0s·(gc)639 ·--·used·0.827591s·(cpu);·0.517732s·(thread);·0s·(gc)
  
640 o43·=·2640 o43·=·2
641 i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that641 i44·:·ptsStratRational·=·ptsStratGeometric++{ExtendField=>false}·--change·that
642 value642 value
  
643 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}643 o44·=·OptionTable{DecompositionStrategy·=>·Decompose}
644 ··················DimensionFunction·=>·dim644 ··················DimensionFunction·=>·dim
Offset 655, 58 lines modifiedOffset 655, 58 lines modified
  
655 o44·:·OptionTable655 o44·:·OptionTable
656 i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value656 i45·:·ptsStratRational.ExtendField·--look·at·our·changed·value
  
657 o45·=·false657 o45·=·false
658 i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,658 i46·:·time·dim·(J·+·chooseGoodMinors(1,·6,·M,·J,·Strategy=>Points,
659 PointOptions=>ptsStratRational))659 PointOptions=>ptsStratRational))
660 ·--·used·0.520675s·(cpu);·0.353044s·(thread);·0s·(gc)660 ·--·used·0.648394s·(cpu);·0.403013s·(thread);·0s·(gc)
  
661 o46·=·2661 o46·=·2
662 Other·options·may·also·be·passed·to·the·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s·package·via·the662 Other·options·may·also·be·passed·to·the·_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s·package·via·the
663 _\x8P_\x8o_\x8i_\x8n_\x8t_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s·option.663 _\x8P_\x8o_\x8i_\x8n_\x8t_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s·option.
664 r\x8re\x8eg\x8gu\x8ul\x8la\x8ar\x8rI\x8In\x8nC\x8Co\x8od\x8di\x8im\x8me\x8en\x8ns\x8si\x8io\x8on\x8n:\x8:·It·is·reasonable·to·think·that·you·should·find·a·few664 r\x8re\x8eg\x8gu\x8ul\x8la\x8ar\x8rI\x8In\x8nC\x8Co\x8od\x8di\x8im\x8me\x8en\x8ns\x8si\x8io\x8on\x8n:\x8:·It·is·reasonable·to·think·that·you·should·find·a·few
665 minors·(with·one·strategy·or·another),·and·see·if·perhaps·the·minors·you·have665 minors·(with·one·strategy·or·another),·and·see·if·perhaps·the·minors·you·have
666 computed·so·far·are·enough·to·verify·our·ring·is·regular·in·codimension·1.·This666 computed·so·far·are·enough·to·verify·our·ring·is·regular·in·codimension·1.·This
667 is·exactly·what·regularInCodimension·does.·One·can·control·at·a·fine·level·how667 is·exactly·what·regularInCodimension·does.·One·can·control·at·a·fine·level·how
668 frequently·new·minors·are·computed,·and·how·frequently·the·dimension·of·what·we668 frequently·new·minors·are·computed,·and·how·frequently·the·dimension·of·what·we
669 have·computed·so·far·is·checked,·by·the·option·codimCheckFunction.·For·more·on669 have·computed·so·far·is·checked,·by·the·option·codimCheckFunction.·For·more·on
670 that,·see·_\x8R_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8T_\x8u_\x8t_\x8o_\x8r_\x8i_\x8a_\x8l·and·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n.·Let·us·finish670 that,·see·_\x8R_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8T_\x8u_\x8t_\x8o_\x8r_\x8i_\x8a_\x8l·and·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n.·Let·us·finish
671 running·regularInCodimension·on·our·example·with·several·different·strategies.671 running·regularInCodimension·on·our·example·with·several·different·strategies.
672 i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,672 i47·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
673 Strategy=>StrategyDefault)673 Strategy=>StrategyDefault)
674 ·--·used·3.51988s·(cpu);·2.84854s·(thread);·0s·(gc)674 ·--·used·4.50417s·(cpu);·3.50025s·(thread);·0s·(gc)
675 i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,675 i48·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
676 Strategy=>StrategyDefaultNonRandom)676 Strategy=>StrategyDefaultNonRandom)
677 ·--·used·0.812059s·(cpu);·0.606774s·(thread);·0s·(gc)677 ·--·used·0.933099s·(cpu);·0.636156s·(thread);·0s·(gc)
  
678 o48·=·true678 o48·=·true
679 i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)679 i49·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Random)
680 ·--·used·2.71639s·(cpu);·2.44142s·(thread);·0s·(gc)680 ·--·used·3.0767s·(cpu);·2.73986s·(thread);·0s·(gc)
681 i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,681 i50·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
682 Strategy=>LexSmallest)682 Strategy=>LexSmallest)
683 ·--·used·2.6533s·(cpu);·1.89165s·(thread);·0s·(gc)683 ·--·used·3.33268s·(cpu);·2.22841s·(thread);·0s·(gc)
684 i51·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,684 i51·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
685 Strategy=>LexSmallestTerm)685 Strategy=>LexSmallestTerm)
686 ·--·used·0.905182s·(cpu);·0.729877s·(thread);·0s·(gc)686 ·--·used·0.998581s·(cpu);·0.774542s·(thread);·0s·(gc)
  
687 o51·=·true687 o51·=·true
688 i52·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,688 i52·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
689 Strategy=>GRevLexSmallest)689 Strategy=>GRevLexSmallest)
690 ·--·used·2.95097s·(cpu);·2.17025s·(thread);·0s·(gc)690 ·--·used·3.22386s·(cpu);·2.20576s·(thread);·0s·(gc)
691 i53·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,691 i53·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
692 Strategy=>GRevLexSmallestTerm)692 Strategy=>GRevLexSmallestTerm)
693 ·--·used·3.39347s·(cpu);·2.56847s·(thread);·0s·(gc)693 ·--·used·3.68377s·(cpu);·2.58807s·(thread);·0s·(gc)
694 i54·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Points)694 i54·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,·Strategy=>Points)
695 ·--·used·10.6283s·(cpu);·7.36659s·(thread);·0s·(gc)695 ·--·used·13.2967s·(cpu);·8.58915s·(thread);·0s·(gc)
  
696 o54·=·true696 o54·=·true
697 i55·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,697 i55·:·time·regularInCodimension(1,·S/J,·MaxMinors·=>·100,
698 Strategy=>StrategyDefaultWithPoints)698 Strategy=>StrategyDefaultWithPoints)
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Offset 67, 21 lines modifiedOffset 67, 21 lines modified
67 ········</table>67 ········</table>
68 ········<div>68 ········<div>
69 ··········<p>It·is·the·cone·over·$P^2·\times·E$·where·$E$·is·an·elliptic·curve.··We·have·embedded·it·with·a·Segre·embedding·inside·$P^8$.··In·particular,·this·example·is·even·regular·in·codimension·3.</p>69 ··········<p>It·is·the·cone·over·$P^2·\times·E$·where·$E$·is·an·elliptic·curve.··We·have·embedded·it·with·a·Segre·embedding·inside·$P^8$.··In·particular,·this·example·is·even·regular·in·codimension·3.</p>
70 ········</div>70 ········</div>
71 ········<table·class="examples">71 ········<table·class="examples">
72 ··········<tr>72 ··········<tr>
73 <td>··············<pre><code·class="language-macaulay2">i4·:·time·regularInCodimension(1,·S/J)73 <td>··············<pre><code·class="language-macaulay2">i4·:·time·regularInCodimension(1,·S/J)
74 ·--·used·0.659643s·(cpu);·0.520517s·(thread);·0s·(gc)74 ·--·used·0.905996s·(cpu);·0.62782s·(thread);·0s·(gc)
  
75 o4·=·true</code></pre>75 o4·=·true</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i5·:·time·regularInCodimension(2,·S/J)78 <td>··············<pre><code·class="language-macaulay2">i5·:·time·regularInCodimension(2,·S/J)
79 ·--·used·7.82665s·(cpu);·6.35155s·(thread);·0s·(gc)</code></pre>79 ·--·used·9.39668s·(cpu);·7.19171s·(thread);·0s·(gc)</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ········</table>81 ········</table>
82 ········<div>82 ········<div>
83 ··········<p>We·try·to·verify·that·$S/J$·is·regular·in·codimension·1·or·2·by·computing·the·ideal·made·up·of·a·small·number·of·minors·of·the·Jacobian·matrix.·In·this·example,·instead·of·computing·all·relevant·1465128·minors·to·compute·the·singular·locus,·and·then·trying·to·compute·the·dimension·of·the·ideal·they·generate,·we·instead·compute·a·few·of·them.··<span·class="tt">regularInCodimension</span>·returns·<span·class="tt">true</span>·if·it·verified·that·the·ring·is·regular·in·codim·1·or·2·(respectively)·and·<span·class="tt">null</span>·if·not.··Because·of·the·randomness·that·exists·in·terms·of·selecting·minors,·the·execution·time·can·actually·vary·quite·a·bit.···Let's·take·a·look·at·what·is·occurring·by·using·the·<span·class="tt">Verbose</span>·option.··We·go·through·the·output·and·explain·what·each·line·is·telling·us.</p>83 ··········<p>We·try·to·verify·that·$S/J$·is·regular·in·codimension·1·or·2·by·computing·the·ideal·made·up·of·a·small·number·of·minors·of·the·Jacobian·matrix.·In·this·example,·instead·of·computing·all·relevant·1465128·minors·to·compute·the·singular·locus,·and·then·trying·to·compute·the·dimension·of·the·ideal·they·generate,·we·instead·compute·a·few·of·them.··<span·class="tt">regularInCodimension</span>·returns·<span·class="tt">true</span>·if·it·verified·that·the·ring·is·regular·in·codim·1·or·2·(respectively)·and·<span·class="tt">null</span>·if·not.··Because·of·the·randomness·that·exists·in·terms·of·selecting·minors,·the·execution·time·can·actually·vary·quite·a·bit.···Let's·take·a·look·at·what·is·occurring·by·using·the·<span·class="tt">Verbose</span>·option.··We·go·through·the·output·and·explain·what·each·line·is·telling·us.</p>
84 ········</div>84 ········</div>
85 ········<table·class="examples">85 ········<table·class="examples">
86 ··········<tr>86 ··········<tr>
Offset 154, 27 lines modifiedOffset 154, 27 lines modified
154 internalChooseMinor:·Choosing·LexSmallest154 internalChooseMinor:·Choosing·LexSmallest
155 internalChooseMinor:·Choosing·LexSmallestTerm155 internalChooseMinor:·Choosing·LexSmallestTerm
156 internalChooseMinor:·Choosing·LexSmallest156 internalChooseMinor:·Choosing·LexSmallest
157 internalChooseMinor:·Choosing·Random157 internalChooseMinor:·Choosing·Random
158 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·49,·and·computed·=·39158 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·49,·and·computed·=·39
159 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast159 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast
160 regularInCodimension:··partial·singular·locus·dimension·computed,·=·2160 regularInCodimension:··partial·singular·locus·dimension·computed,·=·2
161 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute·--·used·1.01371s·(cpu);·0.813721s·(thread);·0s·(gc)161 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute·--·used·1.18226s·(cpu);·0.90525s·(thread);·0s·(gc)
162 d·=·39.··singular·locus·dimension·appears·to·be·=·2162 d·=·39.··singular·locus·dimension·appears·to·be·=·2
  
163 o6·=·true</code></pre>163 o6·=·true</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ········</table>165 ········</table>
166 ········<div>166 ········<div>
167 ··········<p><b>MaxMinors.</b>··The·first·output·says·that·we·will·compute·up·to·452.9·minors·before·giving·up.··We·can·control·that·by·setting·the·option·<span·class="tt">MaxMinors</span>.</p>167 ··········<p><b>MaxMinors.</b>··The·first·output·says·that·we·will·compute·up·to·452.9·minors·before·giving·up.··We·can·control·that·by·setting·the·option·<span·class="tt">MaxMinors</span>.</p>
168 ········</div>168 ········</div>
169 ········<table·class="examples">169 ········<table·class="examples">
170 ··········<tr>170 ··········<tr>
171 <td>··············<pre><code·class="language-macaulay2">i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)171 <td>··············<pre><code·class="language-macaulay2">i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)
172 ·--·used·0.13374s·(cpu);·0.0997883s·(thread);·0s·(gc)172 ·--·used·0.162761s·(cpu);·0.121774s·(thread);·0s·(gc)
173 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.173 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
174 regularInCodimension:·About·to·enter·loop174 regularInCodimension:·About·to·enter·loop
175 internalChooseMinor:·Choosing·Random175 internalChooseMinor:·Choosing·Random
176 internalChooseMinor:·Choosing·RandomNonZero176 internalChooseMinor:·Choosing·RandomNonZero
177 internalChooseMinor:·Choosing·GRevLexSmallestTerm177 internalChooseMinor:·Choosing·GRevLexSmallestTerm
178 internalChooseMinor:·Choosing·Random178 internalChooseMinor:·Choosing·Random
179 internalChooseMinor:·Choosing·Random179 internalChooseMinor:·Choosing·Random
Offset 197, 15 lines modifiedOffset 197, 15 lines modified
197 ········</div>197 ········</div>
198 ········<div>198 ········<div>
199 ··········<p><b>Selecting·submatrices·of·the·Jacobian.</b>··We·also·see·output·like:·``Choosing·LexSmallest''·or·``Choosing·Random''.··This·is·saying·how·we·are·selecting·a·given·submatrix.··For·instance,·we·can·run:</p>199 ··········<p><b>Selecting·submatrices·of·the·Jacobian.</b>··We·also·see·output·like:·``Choosing·LexSmallest''·or·``Choosing·Random''.··This·is·saying·how·we·are·selecting·a·given·submatrix.··For·instance,·we·can·run:</p>
200 ········</div>200 ········</div>
201 ········<table·class="examples">201 ········<table·class="examples">
202 ··········<tr>202 ··········<tr>
203 <td>··············<pre><code·class="language-macaulay2">i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,·Verbose=>true)203 <td>··············<pre><code·class="language-macaulay2">i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,·Verbose=>true)
204 ·--·used·0.120882s·(cpu);·0.0963134s·(thread);·0s·(gc)204 ·--·used·0.13672s·(cpu);·0.102678s·(thread);·0s·(gc)
205 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.205 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
206 regularInCodimension:·About·to·enter·loop206 regularInCodimension:·About·to·enter·loop
207 internalChooseMinor:·Choosing·Random207 internalChooseMinor:·Choosing·Random
208 internalChooseMinor:·Choosing·Random208 internalChooseMinor:·Choosing·Random
209 internalChooseMinor:·Choosing·Random209 internalChooseMinor:·Choosing·Random
210 internalChooseMinor:·Choosing·Random210 internalChooseMinor:·Choosing·Random
211 internalChooseMinor:·Choosing·Random211 internalChooseMinor:·Choosing·Random
Offset 228, 15 lines modifiedOffset 228, 15 lines modified
228 ········</div>228 ········</div>
229 ········<div>229 ········<div>
230 ··········<p><b>Computing·minors·vs·considering·the·dimension·of·what·has·been·computed.</b>··Periodically·we·compute·the·codimension·of·the·partial·ideal·of·minors·we·have·computed·so·far.··There·are·two·options·to·control·this.··First,·we·can·tell·the·function·when·to·first·compute·the·dimension·of·the·working·partial·ideal·of·minors.</p>230 ··········<p><b>Computing·minors·vs·considering·the·dimension·of·what·has·been·computed.</b>··Periodically·we·compute·the·codimension·of·the·partial·ideal·of·minors·we·have·computed·so·far.··There·are·two·options·to·control·this.··First,·we·can·tell·the·function·when·to·first·compute·the·dimension·of·the·working·partial·ideal·of·minors.</p>
231 ········</div>231 ········</div>
232 ········<table·class="examples">232 ········<table·class="examples">
233 ··········<tr>233 ··········<tr>
234 <td>··············<pre><code·class="language-macaulay2">i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t->3,·Verbose=>true)234 <td>··············<pre><code·class="language-macaulay2">i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t->3,·Verbose=>true)
235 ·--·used·0.473747s·(cpu);·0.383319s·(thread);·0s·(gc)235 ·--·used·0.566642s·(cpu);·0.425168s·(thread);·0s·(gc)
236 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.236 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·10·of·them.
237 regularInCodimension:·About·to·enter·loop237 regularInCodimension:·About·to·enter·loop
238 internalChooseMinor:·Choosing·RandomNonZero238 internalChooseMinor:·Choosing·RandomNonZero
239 internalChooseMinor:·Choosing·Random239 internalChooseMinor:·Choosing·Random
240 internalChooseMinor:·Choosing·LexSmallest240 internalChooseMinor:·Choosing·LexSmallest
241 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·3,·and·computed·=·3241 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·3,·and·computed·=·3
242 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.242 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
Offset 265, 15 lines modifiedOffset 265, 15 lines modified
265 ········</div>265 ········</div>
266 ········<div>266 ········<div>
267 ··········<p><b>CodimCheckFunction.</b>·The·option·<span·class="tt">CodimCheckFunction</span>·controls·how·frequently·the·dimension·of·the·partial·ideal·of·minors·is·computed.··For·instance,·setting·<span·class="tt">CodimCheckFunction·=>·t·->·t/5</span>·will·say·it·should·compute·dimension·after·every·5·minors·are·examined.··In·general,·after·the·output·of·the·CodimCheckFunction·increases·by·an·integer·we·compute·the·codimension·again.··The·default·function·has·the·space·between·computations·grow·exponentially.</p>267 ··········<p><b>CodimCheckFunction.</b>·The·option·<span·class="tt">CodimCheckFunction</span>·controls·how·frequently·the·dimension·of·the·partial·ideal·of·minors·is·computed.··For·instance,·setting·<span·class="tt">CodimCheckFunction·=>·t·->·t/5</span>·will·say·it·should·compute·dimension·after·every·5·minors·are·examined.··In·general,·after·the·output·of·the·CodimCheckFunction·increases·by·an·integer·we·compute·the·codimension·again.··The·default·function·has·the·space·between·computations·grow·exponentially.</p>
268 ········</div>268 ········</div>
269 ········<table·class="examples">269 ········<table·class="examples">
270 ··········<tr>270 ··········<tr>
271 <td>··············<pre><code·class="language-macaulay2">i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t->t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)271 <td>··············<pre><code·class="language-macaulay2">i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t->t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)
272 ·--·used·0.541436s·(cpu);·0.420353s·(thread);·0s·(gc)272 ·--·used·0.561127s·(cpu);·0.428435s·(thread);·0s·(gc)
273 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.273 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.
274 regularInCodimension:·About·to·enter·loop274 regularInCodimension:·About·to·enter·loop
275 internalChooseMinor:·Choosing·GRevLexSmallestTerm275 internalChooseMinor:·Choosing·GRevLexSmallestTerm
276 internalChooseMinor:·Choosing·GRevLexSmallestTerm276 internalChooseMinor:·Choosing·GRevLexSmallestTerm
277 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·2,·and·computed·=·2277 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices·considered:·2,·and·computed·=·2
278 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.278 regularInCodimension:··isCodimAtLeast·failed,·computing·codim.
279 regularInCodimension:··partial·singular·locus·dimension·computed,·=·4279 regularInCodimension:··partial·singular·locus·dimension·computed,·=·4
Offset 320, 15 lines modifiedOffset 320, 15 lines modified
320 ········</table>320 ········</table>
321 ········<div>321 ········<div>
322 ··········<p><b>isCodimAtLeast·and·dim</b>.··We·see·the·lines·about·the·``isCodimAtLeast·failed''.··This·means·that·<span·class="tt">isCodimAtLeast</span>·was·not·enough·on·its·own·to·verify·that·our·ring·is·regular·in·codimension·1.··After·this,·``partial·singular·locus·dimension·computed''·indicates·we·did·a·complete·dimension·computation·of·the·partial·ideal·defining·the·singular·locus.··How·<span·class="tt">isCodimAtLeast</span>·is·called·can·be·controlled·via·the·options·<span·class="tt">SPairsFunction</span>·and·<span·class="tt">PairLimit</span>,·which·are·simply·passed·to·<a·title="returns·true·if·we·can·quickly·see·whether·the·codim·is·at·least·a·given·number"·href="_is__Codim__At__Least.html">isCodimAtLeast</a>.··You·can·force·the·function·to·only·use·<span·class="tt">isCodimAtLeast</span>·and·not·call·dimension·by·setting·<span·class="tt">UseOnlyFastCodim·=>·true</span>.</p>322 ··········<p><b>isCodimAtLeast·and·dim</b>.··We·see·the·lines·about·the·``isCodimAtLeast·failed''.··This·means·that·<span·class="tt">isCodimAtLeast</span>·was·not·enough·on·its·own·to·verify·that·our·ring·is·regular·in·codimension·1.··After·this,·``partial·singular·locus·dimension·computed''·indicates·we·did·a·complete·dimension·computation·of·the·partial·ideal·defining·the·singular·locus.··How·<span·class="tt">isCodimAtLeast</span>·is·called·can·be·controlled·via·the·options·<span·class="tt">SPairsFunction</span>·and·<span·class="tt">PairLimit</span>,·which·are·simply·passed·to·<a·title="returns·true·if·we·can·quickly·see·whether·the·codim·is·at·least·a·given·number"·href="_is__Codim__At__Least.html">isCodimAtLeast</a>.··You·can·force·the·function·to·only·use·<span·class="tt">isCodimAtLeast</span>·and·not·call·dimension·by·setting·<span·class="tt">UseOnlyFastCodim·=>·true</span>.</p>
323 ········</div>323 ········</div>
324 ········<table·class="examples">324 ········<table·class="examples">
325 ··········<tr>325 ··········<tr>
326 <td>··············<pre><code·class="language-macaulay2">i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>·true,·Verbose=>true)326 <td>··············<pre><code·class="language-macaulay2">i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>·true,·Verbose=>true)
327 ·--·used·0.322447s·(cpu);·0.247392s·(thread);·0s·(gc)327 ·--·used·0.413418s·(cpu);·0.288524s·(thread);·0s·(gc)
328 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.328 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5·minors,·we·will·compute·up·to·25·of·them.
329 regularInCodimension:·About·to·enter·loop329 regularInCodimension:·About·to·enter·loop
330 internalChooseMinor:·Choosing·GRevLexSmallest330 internalChooseMinor:·Choosing·GRevLexSmallest
331 internalChooseMinor:·Choosing·LexSmallest331 internalChooseMinor:·Choosing·LexSmallest
332 internalChooseMinor:·Choosing·RandomNonZero332 internalChooseMinor:·Choosing·RandomNonZero
333 internalChooseMinor:·Choosing·RandomNonZero333 internalChooseMinor:·Choosing·RandomNonZero
334 internalChooseMinor:·Choosing·GRevLexSmallestTerm334 internalChooseMinor:·Choosing·GRevLexSmallestTerm
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Offset 24, 19 lines modifiedOffset 24, 19 lines modified
24 i3·:·dim·(S/J)24 i3·:·dim·(S/J)
  
25 o3·=·425 o3·=·4
26 It·is·the·cone·over·$P^2·\times·E$·where·$E$·is·an·elliptic·curve.·We·have26 It·is·the·cone·over·$P^2·\times·E$·where·$E$·is·an·elliptic·curve.·We·have
27 embedded·it·with·a·Segre·embedding·inside·$P^8$.·In·particular,·this·example·is27 embedded·it·with·a·Segre·embedding·inside·$P^8$.·In·particular,·this·example·is
28 even·regular·in·codimension·3.28 even·regular·in·codimension·3.
29 i4·:·time·regularInCodimension(1,·S/J)29 i4·:·time·regularInCodimension(1,·S/J)
30 ·--·used·0.659643s·(cpu);·0.520517s·(thread);·0s·(gc)30 ·--·used·0.905996s·(cpu);·0.62782s·(thread);·0s·(gc)
  
31 o4·=·true31 o4·=·true
32 i5·:·time·regularInCodimension(2,·S/J)32 i5·:·time·regularInCodimension(2,·S/J)
33 ·--·used·7.82665s·(cpu);·6.35155s·(thread);·0s·(gc)33 ·--·used·9.39668s·(cpu);·7.19171s·(thread);·0s·(gc)
34 We·try·to·verify·that·$S/J$·is·regular·in·codimension·1·or·2·by·computing·the34 We·try·to·verify·that·$S/J$·is·regular·in·codimension·1·or·2·by·computing·the
35 ideal·made·up·of·a·small·number·of·minors·of·the·Jacobian·matrix.·In·this35 ideal·made·up·of·a·small·number·of·minors·of·the·Jacobian·matrix.·In·this
36 example,·instead·of·computing·all·relevant·1465128·minors·to·compute·the36 example,·instead·of·computing·all·relevant·1465128·minors·to·compute·the
37 singular·locus,·and·then·trying·to·compute·the·dimension·of·the·ideal·they37 singular·locus,·and·then·trying·to·compute·the·dimension·of·the·ideal·they
38 generate,·we·instead·compute·a·few·of·them.·regularInCodimension·returns·true38 generate,·we·instead·compute·a·few·of·them.·regularInCodimension·returns·true
39 if·it·verified·that·the·ring·is·regular·in·codim·1·or·2·(respectively)·and·null39 if·it·verified·that·the·ring·is·regular·in·codim·1·or·2·(respectively)·and·null
40 if·not.·Because·of·the·randomness·that·exists·in·terms·of·selecting·minors,·the40 if·not.·Because·of·the·randomness·that·exists·in·terms·of·selecting·minors,·the
Offset 121, 22 lines modifiedOffset 121, 22 lines modified
121 internalChooseMinor:·Choosing·LexSmallest121 internalChooseMinor:·Choosing·LexSmallest
122 internalChooseMinor:·Choosing·Random122 internalChooseMinor:·Choosing·Random
123 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices123 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices
124 considered:·49,·and·computed·=·39124 considered:·49,·and·computed·=·39
125 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast125 regularInCodimension:··singularLocus·dimension·verified·by·isCodimAtLeast
126 regularInCodimension:··partial·singular·locus·dimension·computed,·=·2126 regularInCodimension:··partial·singular·locus·dimension·computed,·=·2
127 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute127 regularInCodimension:··Loop·completed,·submatrices·considered·=·49,·and·compute
128 --·used·1.01371s·(cpu);·0.813721s·(thread);·0s·(gc)128 --·used·1.18226s·(cpu);·0.90525s·(thread);·0s·(gc)
129 d·=·39.··singular·locus·dimension·appears·to·be·=·2129 d·=·39.··singular·locus·dimension·appears·to·be·=·2
  
130 o6·=·true130 o6·=·true
131 M\x8Ma\x8ax\x8xM\x8Mi\x8in\x8no\x8or\x8rs\x8s.\x8.·The·first·output·says·that·we·will·compute·up·to·452.9·minors·before131 M\x8Ma\x8ax\x8xM\x8Mi\x8in\x8no\x8or\x8rs\x8s.\x8.·The·first·output·says·that·we·will·compute·up·to·452.9·minors·before
132 giving·up.·We·can·control·that·by·setting·the·option·MaxMinors.132 giving·up.·We·can·control·that·by·setting·the·option·MaxMinors.
133 i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)133 i7·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Verbose=>true)
134 ·--·used·0.13374s·(cpu);·0.0997883s·(thread);·0s·(gc)134 ·--·used·0.162761s·(cpu);·0.121774s·(thread);·0s·(gc)
135 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5135 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5
136 minors,·we·will·compute·up·to·10·of·them.136 minors,·we·will·compute·up·to·10·of·them.
137 regularInCodimension:·About·to·enter·loop137 regularInCodimension:·About·to·enter·loop
138 internalChooseMinor:·Choosing·Random138 internalChooseMinor:·Choosing·Random
139 internalChooseMinor:·Choosing·RandomNonZero139 internalChooseMinor:·Choosing·RandomNonZero
140 internalChooseMinor:·Choosing·GRevLexSmallestTerm140 internalChooseMinor:·Choosing·GRevLexSmallestTerm
141 internalChooseMinor:·Choosing·Random141 internalChooseMinor:·Choosing·Random
Offset 159, 15 lines modifiedOffset 159, 15 lines modified
159 There·are·other·finer·ways·to·control·the·MaxMinors·option,·but·they·will·not159 There·are·other·finer·ways·to·control·the·MaxMinors·option,·but·they·will·not
160 be·discussed·in·this·tutorial.·See·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n.160 be·discussed·in·this·tutorial.·See·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8I_\x8n_\x8C_\x8o_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n.
161 S\x8Se\x8el\x8le\x8ec\x8ct\x8ti\x8in\x8ng\x8g·s\x8su\x8ub\x8bm\x8ma\x8at\x8tr\x8ri\x8ic\x8ce\x8es\x8s·o\x8of\x8f·t\x8th\x8he\x8e·J\x8Ja\x8ac\x8co\x8ob\x8bi\x8ia\x8an\x8n.\x8.·We·also·see·output·like:·``Choosing161 S\x8Se\x8el\x8le\x8ec\x8ct\x8ti\x8in\x8ng\x8g·s\x8su\x8ub\x8bm\x8ma\x8at\x8tr\x8ri\x8ic\x8ce\x8es\x8s·o\x8of\x8f·t\x8th\x8he\x8e·J\x8Ja\x8ac\x8co\x8ob\x8bi\x8ia\x8an\x8n.\x8.·We·also·see·output·like:·``Choosing
162 LexSmallest''·or·``Choosing·Random''.·This·is·saying·how·we·are·selecting·a162 LexSmallest''·or·``Choosing·Random''.·This·is·saying·how·we·are·selecting·a
163 given·submatrix.·For·instance,·we·can·run:163 given·submatrix.·For·instance,·we·can·run:
164 i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,164 i8·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·Strategy=>StrategyRandom,
165 Verbose=>true)165 Verbose=>true)
166 ·--·used·0.120882s·(cpu);·0.0963134s·(thread);·0s·(gc)166 ·--·used·0.13672s·(cpu);·0.102678s·(thread);·0s·(gc)
167 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5167 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5
168 minors,·we·will·compute·up·to·10·of·them.168 minors,·we·will·compute·up·to·10·of·them.
169 regularInCodimension:·About·to·enter·loop169 regularInCodimension:·About·to·enter·loop
170 internalChooseMinor:·Choosing·Random170 internalChooseMinor:·Choosing·Random
171 internalChooseMinor:·Choosing·Random171 internalChooseMinor:·Choosing·Random
172 internalChooseMinor:·Choosing·Random172 internalChooseMinor:·Choosing·Random
173 internalChooseMinor:·Choosing·Random173 internalChooseMinor:·Choosing·Random
Offset 197, 15 lines modifiedOffset 197, 15 lines modified
197 C\x8Co\x8om\x8mp\x8pu\x8ut\x8ti\x8in\x8ng\x8g·m\x8mi\x8in\x8no\x8or\x8rs\x8s·v\x8vs\x8s·c\x8co\x8on\x8ns\x8si\x8id\x8de\x8er\x8ri\x8in\x8ng\x8g·t\x8th\x8he\x8e·d\x8di\x8im\x8me\x8en\x8ns\x8si\x8io\x8on\x8n·o\x8of\x8f·w\x8wh\x8ha\x8at\x8t·h\x8ha\x8as\x8s·b\x8be\x8ee\x8en\x8n·c\x8co\x8om\x8mp\x8pu\x8ut\x8te\x8ed\x8d.\x8.197 C\x8Co\x8om\x8mp\x8pu\x8ut\x8ti\x8in\x8ng\x8g·m\x8mi\x8in\x8no\x8or\x8rs\x8s·v\x8vs\x8s·c\x8co\x8on\x8ns\x8si\x8id\x8de\x8er\x8ri\x8in\x8ng\x8g·t\x8th\x8he\x8e·d\x8di\x8im\x8me\x8en\x8ns\x8si\x8io\x8on\x8n·o\x8of\x8f·w\x8wh\x8ha\x8at\x8t·h\x8ha\x8as\x8s·b\x8be\x8ee\x8en\x8n·c\x8co\x8om\x8mp\x8pu\x8ut\x8te\x8ed\x8d.\x8.
198 Periodically·we·compute·the·codimension·of·the·partial·ideal·of·minors·we·have198 Periodically·we·compute·the·codimension·of·the·partial·ideal·of·minors·we·have
199 computed·so·far.·There·are·two·options·to·control·this.·First,·we·can·tell·the199 computed·so·far.·There·are·two·options·to·control·this.·First,·we·can·tell·the
200 function·when·to·first·compute·the·dimension·of·the·working·partial·ideal·of200 function·when·to·first·compute·the·dimension·of·the·working·partial·ideal·of
201 minors.201 minors.
202 i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t-202 i9·:·time·regularInCodimension(1,·S/J,·MaxMinors=>10,·MinMinorsFunction·=>·t-
203 >3,·Verbose=>true)203 >3,·Verbose=>true)
204 ·--·used·0.473747s·(cpu);·0.383319s·(thread);·0s·(gc)204 ·--·used·0.566642s·(cpu);·0.425168s·(thread);·0s·(gc)
205 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5205 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5
206 minors,·we·will·compute·up·to·10·of·them.206 minors,·we·will·compute·up·to·10·of·them.
207 regularInCodimension:·About·to·enter·loop207 regularInCodimension:·About·to·enter·loop
208 internalChooseMinor:·Choosing·RandomNonZero208 internalChooseMinor:·Choosing·RandomNonZero
209 internalChooseMinor:·Choosing·Random209 internalChooseMinor:·Choosing·Random
210 internalChooseMinor:·Choosing·LexSmallest210 internalChooseMinor:·Choosing·LexSmallest
211 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices211 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices
Offset 243, 15 lines modifiedOffset 243, 15 lines modified
243 dimension·of·the·partial·ideal·of·minors·is·computed.·For·instance,·setting243 dimension·of·the·partial·ideal·of·minors·is·computed.·For·instance,·setting
244 CodimCheckFunction·=>·t·->·t/5·will·say·it·should·compute·dimension·after·every244 CodimCheckFunction·=>·t·->·t/5·will·say·it·should·compute·dimension·after·every
245 5·minors·are·examined.·In·general,·after·the·output·of·the·CodimCheckFunction245 5·minors·are·examined.·In·general,·after·the·output·of·the·CodimCheckFunction
246 increases·by·an·integer·we·compute·the·codimension·again.·The·default·function246 increases·by·an·integer·we·compute·the·codimension·again.·The·default·function
247 has·the·space·between·computations·grow·exponentially.247 has·the·space·between·computations·grow·exponentially.
248 i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t-248 i10·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·CodimCheckFunction·=>·t-
249 >t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)249 >t/5,·MinMinorsFunction·=>·t->2,·Verbose=>true)
250 ·--·used·0.541436s·(cpu);·0.420353s·(thread);·0s·(gc)250 ·--·used·0.561127s·(cpu);·0.428435s·(thread);·0s·(gc)
251 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5251 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5
252 minors,·we·will·compute·up·to·25·of·them.252 minors,·we·will·compute·up·to·25·of·them.
253 regularInCodimension:·About·to·enter·loop253 regularInCodimension:·About·to·enter·loop
254 internalChooseMinor:·Choosing·GRevLexSmallestTerm254 internalChooseMinor:·Choosing·GRevLexSmallestTerm
255 internalChooseMinor:·Choosing·GRevLexSmallestTerm255 internalChooseMinor:·Choosing·GRevLexSmallestTerm
256 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices256 regularInCodimension:··Loop·step,·about·to·compute·dimension.··Submatrices
257 considered:·2,·and·computed·=·2257 considered:·2,·and·computed·=·2
Offset 308, 15 lines modifiedOffset 308, 15 lines modified
308 dimension·computed''·indicates·we·did·a·complete·dimension·computation·of·the308 dimension·computed''·indicates·we·did·a·complete·dimension·computation·of·the
309 partial·ideal·defining·the·singular·locus.·How·isCodimAtLeast·is·called·can·be309 partial·ideal·defining·the·singular·locus.·How·isCodimAtLeast·is·called·can·be
310 controlled·via·the·options·SPairsFunction·and·PairLimit,·which·are·simply310 controlled·via·the·options·SPairsFunction·and·PairLimit,·which·are·simply
311 passed·to·_\x8i_\x8s_\x8C_\x8o_\x8d_\x8i_\x8m_\x8A_\x8t_\x8L_\x8e_\x8a_\x8s_\x8t.·You·can·force·the·function·to·only·use·isCodimAtLeast311 passed·to·_\x8i_\x8s_\x8C_\x8o_\x8d_\x8i_\x8m_\x8A_\x8t_\x8L_\x8e_\x8a_\x8s_\x8t.·You·can·force·the·function·to·only·use·isCodimAtLeast
312 and·not·call·dimension·by·setting·UseOnlyFastCodim·=>·true.312 and·not·call·dimension·by·setting·UseOnlyFastCodim·=>·true.
313 i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>313 i11·:·time·regularInCodimension(1,·S/J,·MaxMinors=>25,·UseOnlyFastCodim·=>
314 true,·Verbose=>true)314 true,·Verbose=>true)
315 ·--·used·0.322447s·(cpu);·0.247392s·(thread);·0s·(gc)315 ·--·used·0.413418s·(cpu);·0.288524s·(thread);·0s·(gc)
316 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5316 regularInCodimension:·ring·dimension·=4,·there·are·1465128·possible·5·by·5
317 minors,·we·will·compute·up·to·25·of·them.317 minors,·we·will·compute·up·to·25·of·them.
318 regularInCodimension:·About·to·enter·loop318 regularInCodimension:·About·to·enter·loop
319 internalChooseMinor:·Choosing·GRevLexSmallest319 internalChooseMinor:·Choosing·GRevLexSmallest
320 internalChooseMinor:·Choosing·LexSmallest320 internalChooseMinor:·Choosing·LexSmallest
321 internalChooseMinor:·Choosing·RandomNonZero321 internalChooseMinor:·Choosing·RandomNonZero
322 internalChooseMinor:·Choosing·RandomNonZero322 internalChooseMinor:·Choosing·RandomNonZero
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66 ········</ul>66 ········</ul>
67 Below·the·details·of·how·these·strategies·are·constructed·will·be·detailed·below.··But·first,·we·provide·an·example·showing·that·these·strategies·can·perform·quite·differently.··The·following·is·the·cone·over·the·product·of·two·elliptic·curves.··We·verify·that·this·ring·is·regular·in·codimension·1·using·different·strategies.··Essentially,·minors·are·computed·until·it·is·verified·that·the·ring·is·regular·in·codimension·1.········<table·class="examples">67 Below·the·details·of·how·these·strategies·are·constructed·will·be·detailed·below.··But·first,·we·provide·an·example·showing·that·these·strategies·can·perform·quite·differently.··The·following·is·the·cone·over·the·product·of·two·elliptic·curves.··We·verify·that·this·ring·is·regular·in·codimension·1·using·different·strategies.··Essentially,·minors·are·computed·until·it·is·verified·that·the·ring·is·regular·in·codimension·1.········<table·class="examples">
68 ··········<tr>68 ··········<tr>
69 <td>··············<pre><code·class="language-macaulay2">i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);</code></pre>69 <td>··············<pre><code·class="language-macaulay2">i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)72 <td>··············<pre><code·class="language-macaulay2">i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)
73 ·--·1.95279s·elapsed73 ·--·1.48846s·elapsed
  
74 o2·=·true</code></pre>74 o2·=·true</code></pre>
75 </td>··········</tr>75 </td>··········</tr>
76 ········</table>76 ········</table>
77 In·this·particular·example,·on·one·machine,·we·list·average·time·to·completion·of·each·of·the·above·strategies·after·100·runs.········<ul>77 In·this·particular·example,·on·one·machine,·we·list·average·time·to·completion·of·each·of·the·above·strategies·after·100·runs.········<ul>
78 ··········<li>78 ··········<li>
79 <span·class="tt">StrategyDefault</span>:·1.65·seconds··········</li>79 <span·class="tt">StrategyDefault</span>:·1.65·seconds··········</li>
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137 <span·class="tt">StrategyPoints</span>:·choose·all·submatrices·via·Points.··········</li>137 <span·class="tt">StrategyPoints</span>:·choose·all·submatrices·via·Points.··········</li>
138 ··········<li>138 ··········<li>
139 <span·class="tt">StrategyDefaultWithPoints</span>:·like·<span·class="tt">StrategyDefault</span>·but·replaces·the·<span·class="tt">Random</span>·and·RandomNonZero·submatrices·as·with·matrices·chosen·as·in·Points.··········</li>139 <span·class="tt">StrategyDefaultWithPoints</span>:·like·<span·class="tt">StrategyDefault</span>·but·replaces·the·<span·class="tt">Random</span>·and·RandomNonZero·submatrices·as·with·matrices·chosen·as·in·Points.··········</li>
140 ········</ul>140 ········</ul>
141 Additionally,·a·<span·class="tt">MutableHashTable</span>·named·<span·class="tt">StrategyCurrent</span>·is·also·exported.··It·begins·as·the·default·strategy,·but·the·user·can·modify·it.<br><br><b>Using·a·single·heuristic··</b>Alternatively,·if·the·user·only·wants·to·use·say·<span·class="tt">LexSmallestTerm</span>·they·can·set,·<span·class="tt">Strategy</span>·to·point·to·that·symbol,·instead·of·a·creating·a·custom·strategy·HashTable.··For·example:·········<table·class="examples">141 Additionally,·a·<span·class="tt">MutableHashTable</span>·named·<span·class="tt">StrategyCurrent</span>·is·also·exported.··It·begins·as·the·default·strategy,·but·the·user·can·modify·it.<br><br><b>Using·a·single·heuristic··</b>Alternatively,·if·the·user·only·wants·to·use·say·<span·class="tt">LexSmallestTerm</span>·they·can·set,·<span·class="tt">Strategy</span>·to·point·to·that·symbol,·instead·of·a·creating·a·custom·strategy·HashTable.··For·example:·········<table·class="examples">
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)143 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)
144 ·--·1.14197s·elapsed144 ·--·.963819s·elapsed
  
145 o4·=·true</code></pre>145 o4·=·true</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ········</table>147 ········</table>
148 ······</div>148 ······</div>
149 ······<div·class="waystouse">149 ······<div·class="waystouse">
150 ········<h2>For·the·programmer</h2>150 ········<h2>For·the·programmer</h2>
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41 i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-41 i1·:·T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-
42 b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-42 b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-
43 e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-43 e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-
44 a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-44 a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-
45 g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-45 g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-
46 h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);46 h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
47 i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)47 i2·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>StrategyDefault)
48 ·--·1.95279s·elapsed48 ·--·1.48846s·elapsed
  
49 o2·=·true49 o2·=·true
50 In·this·particular·example,·on·one·machine,·we·list·average·time·to·completion50 In·this·particular·example,·on·one·machine,·we·list·average·time·to·completion
51 of·each·of·the·above·strategies·after·100·runs.51 of·each·of·the·above·strategies·after·100·runs.
52 ····*·StrategyDefault:·1.65·seconds52 ····*·StrategyDefault:·1.65·seconds
53 ····*·StrategyRandom:·8.32·seconds53 ····*·StrategyRandom:·8.32·seconds
54 ····*·StrategyDefaultNonRandom:·0.99·seconds54 ····*·StrategyDefaultNonRandom:·0.99·seconds
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135 Additionally,·a·MutableHashTable·named·StrategyCurrent·is·also·exported.·It135 Additionally,·a·MutableHashTable·named·StrategyCurrent·is·also·exported.·It
136 begins·as·the·default·strategy,·but·the·user·can·modify·it.136 begins·as·the·default·strategy,·but·the·user·can·modify·it.
  
137 U\x8Us\x8si\x8in\x8ng\x8g·a\x8a·s\x8si\x8in\x8ng\x8gl\x8le\x8e·h\x8he\x8eu\x8ur\x8ri\x8is\x8st\x8ti\x8ic\x8c·Alternatively,·if·the·user·only·wants·to·use·say137 U\x8Us\x8si\x8in\x8ng\x8g·a\x8a·s\x8si\x8in\x8ng\x8gl\x8le\x8e·h\x8he\x8eu\x8ur\x8ri\x8is\x8st\x8ti\x8ic\x8c·Alternatively,·if·the·user·only·wants·to·use·say
138 LexSmallestTerm·they·can·set,·Strategy·to·point·to·that·symbol,·instead·of·a138 LexSmallestTerm·they·can·set,·Strategy·to·point·to·that·symbol,·instead·of·a
139 creating·a·custom·strategy·HashTable.·For·example:139 creating·a·custom·strategy·HashTable.·For·example:
140 i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)140 i4·:·elapsedTime·regularInCodimension(1,·T,·Strategy=>LexSmallestTerm)
141 ·--·1.14197s·elapsed141 ·--·.963819s·elapsed
  
142 o4·=·true142 o4·=·true
143 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*143 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
144 The·object·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y_\x8D_\x8e_\x8f_\x8a_\x8u_\x8l_\x8t·is·an·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8·_\x8t_\x8a_\x8b_\x8l_\x8e.144 The·object·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y_\x8D_\x8e_\x8f_\x8a_\x8u_\x8l_\x8t·is·an·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8·_\x8t_\x8a_\x8b_\x8l_\x8e.
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106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);107 <td>··············<pre><code·class="language-macaulay2">i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);
  
108 o6·:·Ideal·of·R</code></pre>108 o6·:·Ideal·of·R</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isCodimAtLeast(3,·J)111 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isCodimAtLeast(3,·J)
112 ·--·used·0.00399435s·(cpu);·0.00202445s·(thread);·0s·(gc)112 ·--·used·0.00400086s·(cpu);·0.00223807s·(thread);·0s·(gc)
  
113 o7·=·true</code></pre>113 o7·=·true</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ········</table>115 ········</table>
116 ········<div>116 ········<div>
117 ··········<p>The·function·works·by·computing·<span·class="tt">gb(I,·PairLimit=>f(i))</span>·for·successive·values·of·<span·class="tt">i</span>.··Here·<span·class="tt">f(i)</span>·is·a·function·that·takes·<span·class="tt">t</span>,·some·approximation·of·the·base·degree·value·of·the·polynomial·ring·(for·example,·in·a·standard·graded·polynomial·ring,·this·is·probably·expected·to·be·<span·class="tt">\{1\}</span>).··And·<span·class="tt">i</span>·is·a·counting·variable.·You·can·provide·your·own·function·by·calling·<span·class="tt">isCodimAtLeast(n,·I,·SPairsFunction=>(·(i)·->·f(i)·)</span>,·the·default·function·is·<span·class="tt">SPairsFunction=>i->ceiling(1.5^i)</span>···Perhaps·more·commonly·however,·the·user·may·want·to·instead·tell·the·function·to·compute·for·larger·values·of·<span·class="tt">i</span>.··This·is·done·via·the·option·<span·class="tt">PairLimit</span>.··This·is·the·max·value·of·<span·class="tt">i</span>·to·be·plugged·into·<span·class="tt">SPairsFunction</span>·before·the·function·gives·up.··In·other·words,·<span·class="tt">PairLimit=>5</span>·will·tell·the·function·to·check·codimension·5·times.</p>117 ··········<p>The·function·works·by·computing·<span·class="tt">gb(I,·PairLimit=>f(i))</span>·for·successive·values·of·<span·class="tt">i</span>.··Here·<span·class="tt">f(i)</span>·is·a·function·that·takes·<span·class="tt">t</span>,·some·approximation·of·the·base·degree·value·of·the·polynomial·ring·(for·example,·in·a·standard·graded·polynomial·ring,·this·is·probably·expected·to·be·<span·class="tt">\{1\}</span>).··And·<span·class="tt">i</span>·is·a·counting·variable.·You·can·provide·your·own·function·by·calling·<span·class="tt">isCodimAtLeast(n,·I,·SPairsFunction=>(·(i)·->·f(i)·)</span>,·the·default·function·is·<span·class="tt">SPairsFunction=>i->ceiling(1.5^i)</span>···Perhaps·more·commonly·however,·the·user·may·want·to·instead·tell·the·function·to·compute·for·larger·values·of·<span·class="tt">i</span>.··This·is·done·via·the·option·<span·class="tt">PairLimit</span>.··This·is·the·max·value·of·<span·class="tt">i</span>·to·be·plugged·into·<span·class="tt">SPairsFunction</span>·before·the·function·gives·up.··In·other·words,·<span·class="tt">PairLimit=>5</span>·will·tell·the·function·to·check·codimension·5·times.</p>
118 ········</div>118 ········</div>
Offset 124, 22 lines modifiedOffset 124, 22 lines modified
  
124 ···············ZZ124 ···············ZZ
125 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]125 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]
126 ··············127··11···8···1···9···12···6···5···10···2···4···3···7</code></pre>126 ··············127··11···8···1···9···12···6···5···10···2···4···3···7</code></pre>
127 </td>··········</tr>127 </td>··········</tr>
128 ··········<tr>128 ··········<tr>
129 <td>··············<pre><code·class="language-macaulay2">i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)129 <td>··············<pre><code·class="language-macaulay2">i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)
130 ·--·used·0.00134704s·(cpu);·0.00196404s·(thread);·0s·(gc)130 ·--·used·0.00172096s·(cpu);·0.00230666s·(thread);·0s·(gc)
131 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.131 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.
  
132 o9·=·true</code></pre>132 o9·=·true</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)135 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)
136 ·--·used·0.000208464s·(cpu);·0.00199842s·(thread);·0s·(gc)136 ·--·used·0.000912411s·(cpu);·0.00280648s·(thread);·0s·(gc)
  
137 o10·=·true</code></pre>137 o10·=·true</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ········</table>139 ········</table>
140 ········<div>140 ········<div>
141 ··········<p>Notice·in·the·first·case·the·function·returned·<span·class="tt">null</span>,·because·the·depth·of·search·was·not·high·enough.··It·only·computed·<span·class="tt">codim</span>·5·times.··The·second·returned·true,·but·it·did·so·as·soon·as·the·answer·was·found·(and·before·we·hit·the·<span·class="tt">PairLimit</span>·limit).</p>141 ··········<p>Notice·in·the·first·case·the·function·returned·<span·class="tt">null</span>,·because·the·depth·of·search·was·not·high·enough.··It·only·computed·<span·class="tt">codim</span>·5·times.··The·second·returned·true,·but·it·did·so·as·soon·as·the·answer·was·found·(and·before·we·hit·the·<span·class="tt">PairLimit</span>·limit).</p>
142 ········</div>142 ········</div>
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39 ·············30······1239 ·············30······12
40 o4·:·Matrix·R···<--·R40 o4·:·Matrix·R···<--·R
41 i5·:·r·=·rank·myDiff;41 i5·:·r·=·rank·myDiff;
42 i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);42 i6·:·J·=·chooseGoodMinors(15,·r,·myDiff,·Strategy=>StrategyDefaultNonRandom);
  
43 o6·:·Ideal·of·R43 o6·:·Ideal·of·R
44 i7·:·time·isCodimAtLeast(3,·J)44 i7·:·time·isCodimAtLeast(3,·J)
45 ·--·used·0.00399435s·(cpu);·0.00202445s·(thread);·0s·(gc)45 ·--·used·0.00400086s·(cpu);·0.00223807s·(thread);·0s·(gc)
  
46 o7·=·true46 o7·=·true
47 The·function·works·by·computing·gb(I,·PairLimit=>f(i))·for·successive·values·of47 The·function·works·by·computing·gb(I,·PairLimit=>f(i))·for·successive·values·of
48 i.·Here·f(i)·is·a·function·that·takes·t,·some·approximation·of·the·base·degree48 i.·Here·f(i)·is·a·function·that·takes·t,·some·approximation·of·the·base·degree
49 value·of·the·polynomial·ring·(for·example,·in·a·standard·graded·polynomial49 value·of·the·polynomial·ring·(for·example,·in·a·standard·graded·polynomial
50 ring,·this·is·probably·expected·to·be·\{1\}).·And·i·is·a·counting·variable.·You50 ring,·this·is·probably·expected·to·be·\{1\}).·And·i·is·a·counting·variable.·You
51 can·provide·your·own·function·by·calling·isCodimAtLeast(n,·I,·SPairsFunction=>51 can·provide·your·own·function·by·calling·isCodimAtLeast(n,·I,·SPairsFunction=>
Offset 73, 20 lines modifiedOffset 73, 20 lines modified
73 x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-73 x_7^3*x_8^5*x_11^3,x_2^5*x_3^3*x_11^3-
74 3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);74 3*x_2^6*x_3^2*x_11^2*x_12+3*x_2^7*x_3*x_11*x_12^2-x_2^8*x_12^3);
  
75 ···············ZZ75 ···············ZZ
76 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]76 o8·:·Ideal·of·---[x··,·x·,·x·,·x·,·x··,·x·,·x·,·x··,·x·,·x·,·x·,·x·]
77 ··············127··11···8···1···9···12···6···5···10···2···4···3···777 ··············127··11···8···1···9···12···6···5···10···2···4···3···7
78 i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)78 i9·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·5,·Verbose=>true)
79 ·--·used·0.00134704s·(cpu);·0.00196404s·(thread);·0s·(gc)79 ·--·used·0.00172096s·(cpu);·0.00230666s·(thread);·0s·(gc)
80 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.80 isCodimAtLeast:·Computing·codim·of·monomials·based·on·ideal·generators.
  
81 o9·=·true81 o9·=·true
82 i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)82 i10·:·time·isCodimAtLeast(5,·I,·PairLimit·=>·200,·Verbose=>false)
83 ·--·used·0.000208464s·(cpu);·0.00199842s·(thread);·0s·(gc)83 ·--·used·0.000912411s·(cpu);·0.00280648s·(thread);·0s·(gc)
  
84 o10·=·true84 o10·=·true
85 Notice·in·the·first·case·the·function·returned·null,·because·the·depth·of85 Notice·in·the·first·case·the·function·returned·null,·because·the·depth·of
86 search·was·not·high·enough.·It·only·computed·codim·5·times.·The·second·returned86 search·was·not·high·enough.·It·only·computed·codim·5·times.·The·second·returned
87 true,·but·it·did·so·as·soon·as·the·answer·was·found·(and·before·we·hit·the87 true,·but·it·did·so·as·soon·as·the·answer·was·found·(and·before·we·hit·the
88 PairLimit·limit).88 PairLimit·limit).
89 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sC\x8Co\x8od\x8di\x8im\x8mA\x8At\x8tL\x8Le\x8ea\x8as\x8st\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*89 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sC\x8Co\x8od\x8di\x8im\x8mA\x8At\x8tL\x8Le\x8ea\x8as\x8st\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html
    
Offset 100, 21 lines modifiedOffset 100, 21 lines modified
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·pdim(module·I)101 <td>··············<pre><code·class="language-macaulay2">i3·:·pdim(module·I)
  
102 o3·=·2</code></pre>102 o3·=·2</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)
106 ·--·used·0.190172s·(cpu);·0.134312s·(thread);·0s·(gc)106 ·--·used·0.244968s·(cpu);·0.144192s·(thread);·0s·(gc)
  
107 o4·=·1</code></pre>107 o4·=·1</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)110 <td>··············<pre><code·class="language-macaulay2">i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)
111 ·--·used·0.0109934s·(cpu);·0.0123083s·(thread);·0s·(gc)111 ·--·used·0.0126589s·(cpu);·0.0133201s·(thread);·0s·(gc)
  
112 o5·=·1</code></pre>112 o5·=·1</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ········</table>114 ········</table>
115 ········<div>115 ········<div>
116 ··········<p>The·option·<span·class="tt">MaxMinors</span>·can·be·used·to·control·how·many·minors·are·computed·at·each·step.·If·this·is·not·specified,·the·number·of·minors·is·a·function·of·the·dimension·$d$·of·the·polynomial·ring·and·the·possible·minors·$c$.·Specifically·it·is·<span·class="tt">10·*·d·+·2·*·log_1.3(c)</span>.·Otherwise·the·user·can·set·the·option·<span·class="tt">MaxMinors·=>·ZZ</span>·to·specify·that·a·fixed·integer·is·used·for·each·step.··Alternatively,·the·user·can·control·the·number·of·minors·computed·at·each·step·by·setting·the·option·<span·class="tt">MaxMinors·=>·List</span>.··In·this·case,·the·list·specifies·how·many·minors·to·be·computed·at·each·step,·(working·backwards).·Finally,·you·can·also·set·<span·class="tt">MaxMinors</span>·to·be·a·custom·function·of·the·dimension·$d$·of·the·polynomial·ring·and·the·maximum·number·of·minors.</p>116 ··········<p>The·option·<span·class="tt">MaxMinors</span>·can·be·used·to·control·how·many·minors·are·computed·at·each·step.·If·this·is·not·specified,·the·number·of·minors·is·a·function·of·the·dimension·$d$·of·the·polynomial·ring·and·the·possible·minors·$c$.·Specifically·it·is·<span·class="tt">10·*·d·+·2·*·log_1.3(c)</span>.·Otherwise·the·user·can·set·the·option·<span·class="tt">MaxMinors·=>·ZZ</span>·to·specify·that·a·fixed·integer·is·used·for·each·step.··Alternatively,·the·user·can·control·the·number·of·minors·computed·at·each·step·by·setting·the·option·<span·class="tt">MaxMinors·=>·List</span>.··In·this·case,·the·list·specifies·how·many·minors·to·be·computed·at·each·step,·(working·backwards).·Finally,·you·can·also·set·<span·class="tt">MaxMinors</span>·to·be·a·custom·function·of·the·dimension·$d$·of·the·polynomial·ring·and·the·maximum·number·of·minors.</p>
117 ········</div>117 ········</div>
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45 i2·:·I·=·ideal((x^3+y)^2,·(x^2+y^2)^2,·(x+y^3)^2,·(x*y)^2);45 i2·:·I·=·ideal((x^3+y)^2,·(x^2+y^2)^2,·(x+y^3)^2,·(x*y)^2);
  
46 o2·:·Ideal·of·R46 o2·:·Ideal·of·R
47 i3·:·pdim(module·I)47 i3·:·pdim(module·I)
  
48 o3·=·248 o3·=·2
49 i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)49 i4·:·time·projDim(module·I,·Strategy=>StrategyRandom)
50 ·--·used·0.190172s·(cpu);·0.134312s·(thread);·0s·(gc)50 ·--·used·0.244968s·(cpu);·0.144192s·(thread);·0s·(gc)
  
51 o4·=·151 o4·=·1
52 i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)52 i5·:·time·projDim(module·I,·Strategy=>StrategyRandom,·MinDimension·=>·1)
53 ·--·used·0.0109934s·(cpu);·0.0123083s·(thread);·0s·(gc)53 ·--·used·0.0126589s·(cpu);·0.0133201s·(thread);·0s·(gc)
  
54 o5·=·154 o5·=·1
55 The·option·MaxMinors·can·be·used·to·control·how·many·minors·are·computed·at55 The·option·MaxMinors·can·be·used·to·control·how·many·minors·are·computed·at
56 each·step.·If·this·is·not·specified,·the·number·of·minors·is·a·function·of·the56 each·step.·If·this·is·not·specified,·the·number·of·minors·is·a·function·of·the
57 dimension·$d$·of·the·polynomial·ring·and·the·possible·minors·$c$.·Specifically57 dimension·$d$·of·the·polynomial·ring·and·the·possible·minors·$c$.·Specifically
58 it·is·10·*·d·+·2·*·log_1.3(c).·Otherwise·the·user·can·set·the·option·MaxMinors58 it·is·10·*·d·+·2·*·log_1.3(c).·Otherwise·the·user·can·set·the·option·MaxMinors
59 =>·ZZ·to·specify·that·a·fixed·integer·is·used·for·each·step.·Alternatively,·the59 =>·ZZ·to·specify·that·a·fixed·integer·is·used·for·each·step.·Alternatively,·the
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./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html
    
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94 <td>··············<pre><code·class="language-macaulay2">i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);94 <td>··············<pre><code·class="language-macaulay2">i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);
  
95 ·············6······795 ·············6······7
96 o2·:·Matrix·R··&lt;--·R</code></pre>96 o2·:·Matrix·R··&lt;--·R</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);
100 ·--·used·0.321843s·(cpu);·0.318977s·(thread);·0s·(gc)100 ·--·used·0.434706s·(cpu);·0.435866s·(thread);·0s·(gc)
  
101 o3·:·Ideal·of·R</code></pre>101 o3·:·Ideal·of·R</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);104 <td>··············<pre><code·class="language-macaulay2">i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);
105 ·--·used·1.08533s·(cpu);·0.999458s·(thread);·0s·(gc)105 ·--·used·1.297s·(cpu);·1.1431s·(thread);·0s·(gc)
  
106 o4·:·Ideal·of·R</code></pre>106 o4·:·Ideal·of·R</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i5·:·I1·==·I2109 <td>··············<pre><code·class="language-macaulay2">i5·:·I1·==·I2
  
110 o5·=·true</code></pre>110 o5·=·true</code></pre>
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28 strategy·for·minors28 strategy·for·minors
29 i1·:·R·=·QQ[x,y];29 i1·:·R·=·QQ[x,y];
30 i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);30 i2·:·M·=·random(R^{5,5,5,5,5,5},·R^7);
  
31 ·············6······731 ·············6······7
32 o2·:·Matrix·R··<--·R32 o2·:·Matrix·R··<--·R
33 i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);33 i3·:·time·I2·=·recursiveMinors(4,·M,·Threads=>0);
34 ·--·used·0.321843s·(cpu);·0.318977s·(thread);·0s·(gc)34 ·--·used·0.434706s·(cpu);·0.435866s·(thread);·0s·(gc)
  
35 o3·:·Ideal·of·R35 o3·:·Ideal·of·R
36 i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);36 i4·:·time·I1·=·minors(4,·M,·Strategy=>Cofactor);
37 ·--·used·1.08533s·(cpu);·0.999458s·(thread);·0s·(gc)37 ·--·used·1.297s·(cpu);·1.1431s·(thread);·0s·(gc)
  
38 o4·:·Ideal·of·R38 o4·:·Ideal·of·R
39 i5·:·I1·==·I239 i5·:·I1·==·I2
  
40 o5·=·true40 o5·=·true
41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
42 ····*·_\x8m_\x8i_\x8n_\x8o_\x8r_\x8s·--·ideal·generated·by·minors42 ····*·_\x8m_\x8i_\x8n_\x8o_\x8r_\x8s·--·ideal·generated·by·minors
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./usr/share/doc/Macaulay2/FastMinors/html/_regular__In__Codimension.html
    
Offset 129, 21 lines modifiedOffset 129, 21 lines modified
129 ··········<tr>129 ··········<tr>
130 <td>··············<pre><code·class="language-macaulay2">i7·:·dim·S130 <td>··············<pre><code·class="language-macaulay2">i7·:·dim·S
  
131 o7·=·3</code></pre>131 o7·=·3</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i8·:·time·regularInCodimension(1,·S)134 <td>··············<pre><code·class="language-macaulay2">i8·:·time·regularInCodimension(1,·S)
135 ·--·used·1.18412s·(cpu);·0.96706s·(thread);·0s·(gc)135 ·--·used·1.65731s·(cpu);·1.32984s·(thread);·0s·(gc)
  
136 o8·=·true</code></pre>136 o8·=·true</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i9·:·time·regularInCodimension(2,·S)139 <td>··············<pre><code·class="language-macaulay2">i9·:·time·regularInCodimension(2,·S)
140 ·--·used·5.06537s·(cpu);·4.21399s·(thread);·0s·(gc)</code></pre>140 ·--·used·5.95441s·(cpu);·4.74918s·(thread);·0s·(gc)</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ········</table>142 ········</table>
143 ········<div>143 ········<div>
144 ··········<p>There·are·numerous·examples·where·<span·class="tt">regularInCodimension</span>·is·several·orders·of·magnitude·faster·that·calls·of·<span·class="tt">dim·singularLocus</span>.</p>144 ··········<p>There·are·numerous·examples·where·<span·class="tt">regularInCodimension</span>·is·several·orders·of·magnitude·faster·that·calls·of·<span·class="tt">dim·singularLocus</span>.</p>
145 ········</div>145 ········</div>
146 ········<div>146 ········<div>
147 ··········<p>The·following·is·a·(pruned)·affine·chart·on·an·Abelian·surface·obtained·as·a·product·of·two·elliptic·curves.··It·is·nonsingular,·as·our·function·verifies.·If·one·does·not·prune·it,·then·the·<span·class="tt">dim·singularLocus</span>·call·takes·an·enormous·amount·of·time,·otherwise·the·running·times·of·<span·class="tt">dim·singularLocus</span>·and·our·function·are·frequently·about·the·same.</p>147 ··········<p>The·following·is·a·(pruned)·affine·chart·on·an·Abelian·surface·obtained·as·a·product·of·two·elliptic·curves.··It·is·nonsingular,·as·our·function·verifies.·If·one·does·not·prune·it,·then·the·<span·class="tt">dim·singularLocus</span>·call·takes·an·enormous·amount·of·time,·otherwise·the·running·times·of·<span·class="tt">dim·singularLocus</span>·and·our·function·are·frequently·about·the·same.</p>
Offset 155, 33 lines modifiedOffset 155, 33 lines modified
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i11·:·dim(R)156 <td>··············<pre><code·class="language-macaulay2">i11·:·dim(R)
  
157 o11·=·2</code></pre>157 o11·=·2</code></pre>
158 </td>··········</tr>158 </td>··········</tr>
159 ··········<tr>159 ··········<tr>
160 <td>··············<pre><code·class="language-macaulay2">i12·:·time·(dim·singularLocus·(R))160 <td>··············<pre><code·class="language-macaulay2">i12·:·time·(dim·singularLocus·(R))
161 ·--·used·0.0157777s·(cpu);·0.0148119s·(thread);·0s·(gc)161 ·--·used·0.0279973s·(cpu);·0.0254638s·(thread);·0s·(gc)
  
162 o12·=·-1</code></pre>162 o12·=·-1</code></pre>
163 </td>··········</tr>163 </td>··········</tr>
164 ··········<tr>164 ··········<tr>
165 <td>··············<pre><code·class="language-macaulay2">i13·:·time·regularInCodimension(2,·R)165 <td>··············<pre><code·class="language-macaulay2">i13·:·time·regularInCodimension(2,·R)
166 ·--·used·0.369174s·(cpu);·0.287095s·(thread);·0s·(gc)166 ·--·used·0.516182s·(cpu);·0.383168s·(thread);·0s·(gc)
  
167 o13·=·true</code></pre>167 o13·=·true</code></pre>
168 </td>··········</tr>168 </td>··········</tr>
169 ··········<tr>169 ··········<tr>
170 <td>··············<pre><code·class="language-macaulay2">i14·:·time·regularInCodimension(2,·R)170 <td>··············<pre><code·class="language-macaulay2">i14·:·time·regularInCodimension(2,·R)
171 ·--·used·0.649844s·(cpu);·0.490454s·(thread);·0s·(gc)171 ·--·used·0.812796s·(cpu);·0.626848s·(thread);·0s·(gc)
  
172 o14·=·true</code></pre>172 o14·=·true</code></pre>
173 </td>··········</tr>173 </td>··········</tr>
174 ··········<tr>174 ··········<tr>
175 <td>··············<pre><code·class="language-macaulay2">i15·:·time·regularInCodimension(2,·R)175 <td>··············<pre><code·class="language-macaulay2">i15·:·time·regularInCodimension(2,·R)
176 ·--·used·0.260796s·(cpu);·0.201669s·(thread);·0s·(gc)176 ·--·used·0.353608s·(cpu);·0.256535s·(thread);·0s·(gc)
  
177 o15·=·true</code></pre>177 o15·=·true</code></pre>
178 </td>··········</tr>178 </td>··········</tr>
179 ········</table>179 ········</table>
180 ········<div>180 ········<div>
181 ··········<p>The·function·works·by·choosing·interesting·looking·submatrices,·computing·their·determinants,·and·periodically·(based·on·a·logarithmic·growth·setting),·computing·the·dimension·of·a·subideal·of·the·Jacobian.·The·option·<span·class="tt">Verbose</span>·can·be·used·to·see·this·in·action.</p>181 ··········<p>The·function·works·by·choosing·interesting·looking·submatrices,·computing·their·determinants,·and·periodically·(based·on·a·logarithmic·growth·setting),·computing·the·dimension·of·a·subideal·of·the·Jacobian.·The·option·<span·class="tt">Verbose</span>·can·be·used·to·see·this·in·action.</p>
182 ········</div>182 ········</div>
Offset 519, 15 lines modifiedOffset 519, 15 lines modified
519 internalChooseMinor:·Choosing·GRevLexSmallestTerm519 internalChooseMinor:·Choosing·GRevLexSmallestTerm
520 internalChooseMinor:·Choosing·RandomNonZero520 internalChooseMinor:·Choosing·RandomNonZero
521 internalChooseMinor:·Choosing·LexSmallest521 internalChooseMinor:·Choosing·LexSmallest
522 internalChooseMinor:·Choosing·Random522 internalChooseMinor:·Choosing·Random
523 internalChooseMinor:·Choosing·Random523 internalChooseMinor:·Choosing·Random
524 internalChooseMinor:·Choosing·LexSmallestTerm524 internalChooseMinor:·Choosing·LexSmallestTerm
525 internalChooseMinor:·Choosing·GRevLexSmallestTerm525 internalChooseMinor:·Choosing·GRevLexSmallestTerm
526 internalChooseMinor:·Choosing·--·used·5.05989s·(cpu);·4.23214s·(thread);·0s·(gc)526 internalChooseMinor:·Choosing·--·used·6.09346s·(cpu);·4.79091s·(thread);·0s·(gc)
527 ·LexSmallestTerm527 ·LexSmallestTerm
528 internalChooseMinor:·Choosing·Random528 internalChooseMinor:·Choosing·Random
529 internalChooseMinor:·Choosing·Random529 internalChooseMinor:·Choosing·Random
530 internalChooseMinor:·Choosing·GRevLexSmallest530 internalChooseMinor:·Choosing·GRevLexSmallest
531 internalChooseMinor:·Choosing·RandomNonZero531 internalChooseMinor:·Choosing·RandomNonZero
532 internalChooseMinor:·Choosing·GRevLexSmallest532 internalChooseMinor:·Choosing·GRevLexSmallest
533 internalChooseMinor:·Choosing·GRevLexSmallestTerm533 internalChooseMinor:·Choosing·GRevLexSmallestTerm
Offset 569, 15 lines modifiedOffset 569, 15 lines modified
569 ········</table>569 ········</table>
570 ········<div>570 ········<div>
571 ··········<p>The·maximum·number·of·minors·considered·can·be·controlled·by·the·option·<span·class="tt">MaxMinors</span>.··Alternatively,·it·can·be·controlled·in·a·more·precise·way·by·passing·a·function·to·the·option·<span·class="tt">MaxMinors</span>.··This·function·should·have·two·inputs;·the·first·is·minimum·number·of·minors·needed·to·determine·whether·the·ring·is·regular·in·codimension·n,·and·the·second·is·the·total·number·of·minors·available·in·the·Jacobian.·The·function·<span·class="tt">regularInCodimension</span>·does·not·recompute·determinants,·so·<span·class="tt">MaxMinors</span>·or·is·only·an·upper·bound·on·the·number·of·minors·computed.</p>571 ··········<p>The·maximum·number·of·minors·considered·can·be·controlled·by·the·option·<span·class="tt">MaxMinors</span>.··Alternatively,·it·can·be·controlled·in·a·more·precise·way·by·passing·a·function·to·the·option·<span·class="tt">MaxMinors</span>.··This·function·should·have·two·inputs;·the·first·is·minimum·number·of·minors·needed·to·determine·whether·the·ring·is·regular·in·codimension·n,·and·the·second·is·the·total·number·of·minors·available·in·the·Jacobian.·The·function·<span·class="tt">regularInCodimension</span>·does·not·recompute·determinants,·so·<span·class="tt">MaxMinors</span>·or·is·only·an·upper·bound·on·the·number·of·minors·computed.</p>
572 ········</div>572 ········</div>
573 ········<table·class="examples">573 ········<table·class="examples">
574 ··········<tr>574 ··········<tr>
575 <td>··············<pre><code·class="language-macaulay2">i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)575 <td>··············<pre><code·class="language-macaulay2">i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)
576 ·--·used·1.04176s·(cpu);·0.888601s·(thread);·0s·(gc)576 ·--·used·1.44101s·(cpu);·1.19387s·(thread);·0s·(gc)
577 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·30·of·them.577 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4·minors,·we·will·compute·up·to·30·of·them.
578 regularInCodimension:·About·to·enter·loop578 regularInCodimension:·About·to·enter·loop
579 internalChooseMinor:·Choosing·Random579 internalChooseMinor:·Choosing·Random
580 internalChooseMinor:·Choosing·Random580 internalChooseMinor:·Choosing·Random
581 internalChooseMinor:·Choosing·RandomNonZero581 internalChooseMinor:·Choosing·RandomNonZero
582 internalChooseMinor:·Choosing·LexSmallestTerm582 internalChooseMinor:·Choosing·LexSmallestTerm
583 internalChooseMinor:·Choosing·RandomNonZero583 internalChooseMinor:·Choosing·RandomNonZero
Offset 638, 73 lines modifiedOffset 638, 73 lines modified
638 <td>··············<pre><code·class="language-macaulay2">i19·:·StrategyCurrent#LexSmallest·=·100;</code></pre>638 <td>··············<pre><code·class="language-macaulay2">i19·:·StrategyCurrent#LexSmallest·=·100;</code></pre>
639 </td>··········</tr>639 </td>··········</tr>
640 ··········<tr>640 ··········<tr>
641 <td>··············<pre><code·class="language-macaulay2">i20·:·StrategyCurrent#LexSmallestTerm·=·0;</code></pre>641 <td>··············<pre><code·class="language-macaulay2">i20·:·StrategyCurrent#LexSmallestTerm·=·0;</code></pre>
642 </td>··········</tr>642 </td>··········</tr>
643 ··········<tr>643 ··········<tr>
644 <td>··············<pre><code·class="language-macaulay2">i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)644 <td>··············<pre><code·class="language-macaulay2">i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
645 ·--·used·0.101049s·(cpu);·0.073258s·(thread);·0s·(gc)645 ·--·used·0.160646s·(cpu);·0.107785s·(thread);·0s·(gc)
  
646 o21·=·true</code></pre>646 o21·=·true</code></pre>
647 </td>··········</tr>647 </td>··········</tr>
648 ··········<tr>648 ··········<tr>
649 <td>··············<pre><code·class="language-macaulay2">i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)649 <td>··············<pre><code·class="language-macaulay2">i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
650 ·--·used·0.14821s·(cpu);·0.12367s·(thread);·0s·(gc)650 ·--·used·0.253229s·(cpu);·0.168654s·(thread);·0s·(gc)
  
651 o22·=·true</code></pre>651 o22·=·true</code></pre>
652 </td>··········</tr>652 </td>··········</tr>
653 ··········<tr>653 ··········<tr>
654 <td>··············<pre><code·class="language-macaulay2">i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)654 <td>··············<pre><code·class="language-macaulay2">i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
655 ·--·used·1.15046s·(cpu);·0.964828s·(thread);·0s·(gc)655 ·--·used·1.34427s·(cpu);·1.03428s·(thread);·0s·(gc)
  
656 o23·=·true</code></pre>656 o23·=·true</code></pre>
657 </td>··········</tr>657 </td>··········</tr>
658 ··········<tr>658 ··········<tr>
659 <td>··············<pre><code·class="language-macaulay2">i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)659 <td>··············<pre><code·class="language-macaulay2">i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
660 ·--·used·0.964293s·(cpu);·0.782207s·(thread);·0s·(gc)660 ·--·used·1.10973s·(cpu);·0.865713s·(thread);·0s·(gc)
  
661 o24·=·true</code></pre>661 o24·=·true</code></pre>
662 </td>··········</tr>662 </td>··········</tr>
663 ··········<tr>663 ··········<tr>
664 <td>··············<pre><code·class="language-macaulay2">i25·:·StrategyCurrent#LexSmallest·=·0;</code></pre>664 <td>··············<pre><code·class="language-macaulay2">i25·:·StrategyCurrent#LexSmallest·=·0;</code></pre>
665 </td>··········</tr>665 </td>··········</tr>
666 ··········<tr>666 ··········<tr>
667 <td>··············<pre><code·class="language-macaulay2">i26·:·StrategyCurrent#LexSmallestTerm·=·100;</code></pre>667 <td>··············<pre><code·class="language-macaulay2">i26·:·StrategyCurrent#LexSmallestTerm·=·100;</code></pre>
668 </td>··········</tr>668 </td>··········</tr>
669 ··········<tr>669 ··········<tr>
670 <td>··············<pre><code·class="language-macaulay2">i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)670 <td>··············<pre><code·class="language-macaulay2">i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
671 ·--·used·0.30488s·(cpu);·0.264081s·(thread);·0s·(gc)671 ·--·used·0.445197s·(cpu);·0.282256s·(thread);·0s·(gc)
  
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Offset 73, 19 lines modifiedOffset 73, 19 lines modified
  
73 o5·:·Ideal·of·T73 o5·:·Ideal·of·T
74 i6·:·S·=·T/I;74 i6·:·S·=·T/I;
75 i7·:·dim·S75 i7·:·dim·S
  
76 o7·=·376 o7·=·3
77 i8·:·time·regularInCodimension(1,·S)77 i8·:·time·regularInCodimension(1,·S)
78 ·--·used·1.18412s·(cpu);·0.96706s·(thread);·0s·(gc)78 ·--·used·1.65731s·(cpu);·1.32984s·(thread);·0s·(gc)
  
79 o8·=·true79 o8·=·true
80 i9·:·time·regularInCodimension(2,·S)80 i9·:·time·regularInCodimension(2,·S)
81 ·--·used·5.06537s·(cpu);·4.21399s·(thread);·0s·(gc)81 ·--·used·5.95441s·(cpu);·4.74918s·(thread);·0s·(gc)
82 There·are·numerous·examples·where·regularInCodimension·is·several·orders·of82 There·are·numerous·examples·where·regularInCodimension·is·several·orders·of
83 magnitude·faster·that·calls·of·dim·singularLocus.83 magnitude·faster·that·calls·of·dim·singularLocus.
84 The·following·is·a·(pruned)·affine·chart·on·an·Abelian·surface·obtained·as·a84 The·following·is·a·(pruned)·affine·chart·on·an·Abelian·surface·obtained·as·a
85 product·of·two·elliptic·curves.·It·is·nonsingular,·as·our·function·verifies.·If85 product·of·two·elliptic·curves.·It·is·nonsingular,·as·our·function·verifies.·If
86 one·does·not·prune·it,·then·the·dim·singularLocus·call·takes·an·enormous·amount86 one·does·not·prune·it,·then·the·dim·singularLocus·call·takes·an·enormous·amount
87 of·time,·otherwise·the·running·times·of·dim·singularLocus·and·our·function·are87 of·time,·otherwise·the·running·times·of·dim·singularLocus·and·our·function·are
88 frequently·about·the·same.88 frequently·about·the·same.
Offset 93, 27 lines modifiedOffset 93, 27 lines modified
93 (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-93 (g^3+h^3+1,f*g^3+f*h^3+f,c*g^3+c*h^3+c,f^2*g^3+f^2*h^3+f^2,c*f*g^3+c*f*h^3+c*f,c^2*g^3+c^2*h^3+c^2,f^3*g^3+f^3*h^3+f^3,c*f^2*g^3+c*f^2*h^3+c*f^2,c^2*f*g^3+c^2*f*h^3+c^2*f,c^3-
94 f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-94 f^2-c,c^3*h-f^2*h-c*h,c^3*g-f^2*g-c*g,c^3*h^2-f^2*h^2-c*h^2,c^3*g*h-f^2*g*h-c*g*h,c^3*g^2-f^2*g^2-c*g^2,c^3*h^3-f^2*h^3-c*h^3,c^3*g*h^2-f^2*g*h^2-c*g*h^2,c^3*g^2*h-f^2*g^2*h-
95 c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);95 c*g^2*h,c^3*g^3+f^2*h^3+c*h^3+f^2+c);
96 i11·:·dim(R)96 i11·:·dim(R)
  
97 o11·=·297 o11·=·2
98 i12·:·time·(dim·singularLocus·(R))98 i12·:·time·(dim·singularLocus·(R))
99 ·--·used·0.0157777s·(cpu);·0.0148119s·(thread);·0s·(gc)99 ·--·used·0.0279973s·(cpu);·0.0254638s·(thread);·0s·(gc)
  
100 o12·=·-1100 o12·=·-1
101 i13·:·time·regularInCodimension(2,·R)101 i13·:·time·regularInCodimension(2,·R)
102 ·--·used·0.369174s·(cpu);·0.287095s·(thread);·0s·(gc)102 ·--·used·0.516182s·(cpu);·0.383168s·(thread);·0s·(gc)
  
103 o13·=·true103 o13·=·true
104 i14·:·time·regularInCodimension(2,·R)104 i14·:·time·regularInCodimension(2,·R)
105 ·--·used·0.649844s·(cpu);·0.490454s·(thread);·0s·(gc)105 ·--·used·0.812796s·(cpu);·0.626848s·(thread);·0s·(gc)
  
106 o14·=·true106 o14·=·true
107 i15·:·time·regularInCodimension(2,·R)107 i15·:·time·regularInCodimension(2,·R)
108 ·--·used·0.260796s·(cpu);·0.201669s·(thread);·0s·(gc)108 ·--·used·0.353608s·(cpu);·0.256535s·(thread);·0s·(gc)
  
109 o15·=·true109 o15·=·true
110 The·function·works·by·choosing·interesting·looking·submatrices,·computing·their110 The·function·works·by·choosing·interesting·looking·submatrices,·computing·their
111 determinants,·and·periodically·(based·on·a·logarithmic·growth·setting),111 determinants,·and·periodically·(based·on·a·logarithmic·growth·setting),
112 computing·the·dimension·of·a·subideal·of·the·Jacobian.·The·option·Verbose·can112 computing·the·dimension·of·a·subideal·of·the·Jacobian.·The·option·Verbose·can
113 be·used·to·see·this·in·action.113 be·used·to·see·this·in·action.
114 i16·:·time·regularInCodimension(2,·S,·Verbose=>true)114 i16·:·time·regularInCodimension(2,·S,·Verbose=>true)
Offset 462, 15 lines modifiedOffset 462, 15 lines modified
462 internalChooseMinor:·Choosing·GRevLexSmallestTerm462 internalChooseMinor:·Choosing·GRevLexSmallestTerm
463 internalChooseMinor:·Choosing·RandomNonZero463 internalChooseMinor:·Choosing·RandomNonZero
464 internalChooseMinor:·Choosing·LexSmallest464 internalChooseMinor:·Choosing·LexSmallest
465 internalChooseMinor:·Choosing·Random465 internalChooseMinor:·Choosing·Random
466 internalChooseMinor:·Choosing·Random466 internalChooseMinor:·Choosing·Random
467 internalChooseMinor:·Choosing·LexSmallestTerm467 internalChooseMinor:·Choosing·LexSmallestTerm
468 internalChooseMinor:·Choosing·GRevLexSmallestTerm468 internalChooseMinor:·Choosing·GRevLexSmallestTerm
469 internalChooseMinor:·Choosing·--·used·5.05989s·(cpu);·4.23214s·(thread);·0s469 internalChooseMinor:·Choosing·--·used·6.09346s·(cpu);·4.79091s·(thread);·0s
470 (gc)470 (gc)
471 ·LexSmallestTerm471 ·LexSmallestTerm
472 internalChooseMinor:·Choosing·Random472 internalChooseMinor:·Choosing·Random
473 internalChooseMinor:·Choosing·Random473 internalChooseMinor:·Choosing·Random
474 internalChooseMinor:·Choosing·GRevLexSmallest474 internalChooseMinor:·Choosing·GRevLexSmallest
475 internalChooseMinor:·Choosing·RandomNonZero475 internalChooseMinor:·Choosing·RandomNonZero
476 internalChooseMinor:·Choosing·GRevLexSmallest476 internalChooseMinor:·Choosing·GRevLexSmallest
Offset 517, 15 lines modifiedOffset 517, 15 lines modified
517 a·function·to·the·option·MaxMinors.·This·function·should·have·two·inputs;·the517 a·function·to·the·option·MaxMinors.·This·function·should·have·two·inputs;·the
518 first·is·minimum·number·of·minors·needed·to·determine·whether·the·ring·is518 first·is·minimum·number·of·minors·needed·to·determine·whether·the·ring·is
519 regular·in·codimension·n,·and·the·second·is·the·total·number·of·minors519 regular·in·codimension·n,·and·the·second·is·the·total·number·of·minors
520 available·in·the·Jacobian.·The·function·regularInCodimension·does·not·recompute520 available·in·the·Jacobian.·The·function·regularInCodimension·does·not·recompute
521 determinants,·so·MaxMinors·or·is·only·an·upper·bound·on·the·number·of·minors521 determinants,·so·MaxMinors·or·is·only·an·upper·bound·on·the·number·of·minors
522 computed.522 computed.
523 i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)523 i17·:·time·regularInCodimension(2,·S,·Verbose=>true,·MaxMinors=>30)
524 ·--·used·1.04176s·(cpu);·0.888601s·(thread);·0s·(gc)524 ·--·used·1.44101s·(cpu);·1.19387s·(thread);·0s·(gc)
525 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4525 regularInCodimension:·ring·dimension·=3,·there·are·17325·possible·4·by·4
526 minors,·we·will·compute·up·to·30·of·them.526 minors,·we·will·compute·up·to·30·of·them.
527 regularInCodimension:·About·to·enter·loop527 regularInCodimension:·About·to·enter·loop
528 internalChooseMinor:·Choosing·Random528 internalChooseMinor:·Choosing·Random
529 internalChooseMinor:·Choosing·Random529 internalChooseMinor:·Choosing·Random
530 internalChooseMinor:·Choosing·RandomNonZero530 internalChooseMinor:·Choosing·RandomNonZero
531 internalChooseMinor:·Choosing·LexSmallestTerm531 internalChooseMinor:·Choosing·LexSmallestTerm
Offset 592, 51 lines modifiedOffset 592, 51 lines modified
592 because·there·are·a·small·number·of·entries·with·nonzero·constant·terms,·which592 because·there·are·a·small·number·of·entries·with·nonzero·constant·terms,·which
593 are·selected·repeatedly).·However,·in·our·first·example,·the·LexSmallestTerm·is593 are·selected·repeatedly).·However,·in·our·first·example,·the·LexSmallestTerm·is
594 much·faster,·and·Random·does·not·perform·well·at·all.594 much·faster,·and·Random·does·not·perform·well·at·all.
595 i18·:·StrategyCurrent#Random·=·0;595 i18·:·StrategyCurrent#Random·=·0;
596 i19·:·StrategyCurrent#LexSmallest·=·100;596 i19·:·StrategyCurrent#LexSmallest·=·100;
597 i20·:·StrategyCurrent#LexSmallestTerm·=·0;597 i20·:·StrategyCurrent#LexSmallestTerm·=·0;
598 i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)598 i21·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
599 ·--·used·0.101049s·(cpu);·0.073258s·(thread);·0s·(gc)599 ·--·used·0.160646s·(cpu);·0.107785s·(thread);·0s·(gc)
  
600 o21·=·true600 o21·=·true
601 i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)601 i22·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
602 ·--·used·0.14821s·(cpu);·0.12367s·(thread);·0s·(gc)602 ·--·used·0.253229s·(cpu);·0.168654s·(thread);·0s·(gc)
  
603 o22·=·true603 o22·=·true
604 i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)604 i23·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
605 ·--·used·1.15046s·(cpu);·0.964828s·(thread);·0s·(gc)605 ·--·used·1.34427s·(cpu);·1.03428s·(thread);·0s·(gc)
  
606 o23·=·true606 o23·=·true
607 i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)607 i24·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
608 ·--·used·0.964293s·(cpu);·0.782207s·(thread);·0s·(gc)608 ·--·used·1.10973s·(cpu);·0.865713s·(thread);·0s·(gc)
  
609 o24·=·true609 o24·=·true
610 i25·:·StrategyCurrent#LexSmallest·=·0;610 i25·:·StrategyCurrent#LexSmallest·=·0;
611 i26·:·StrategyCurrent#LexSmallestTerm·=·100;611 i26·:·StrategyCurrent#LexSmallestTerm·=·100;
612 i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)612 i27·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
613 ·--·used·0.30488s·(cpu);·0.264081s·(thread);·0s·(gc)613 ·--·used·0.445197s·(cpu);·0.282256s·(thread);·0s·(gc)
  
614 o27·=·true614 o27·=·true
615 i28·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)615 i28·:·time·regularInCodimension(2,·R,·Strategy=>StrategyCurrent)
616 ·--·used·1.70445s·(cpu);·1.34525s·(thread);·0s·(gc)616 ·--·used·2.08877s·(cpu);·1.52933s·(thread);·0s·(gc)
617 i29·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)617 i29·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
618 ·--·used·0.129454s·(cpu);·0.102854s·(thread);·0s·(gc)618 ·--·used·0.149107s·(cpu);·0.12124s·(thread);·0s·(gc)
  
619 o29·=·true619 o29·=·true
620 i30·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)620 i30·:·time·regularInCodimension(1,·S,·Strategy=>StrategyCurrent)
621 ·--·used·0.16017s·(cpu);·0.135024s·(thread);·0s·(gc)621 ·--·used·0.202604s·(cpu);·0.144515s·(thread);·0s·(gc)
  
622 o30·=·true622 o30·=·true
623 i31·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)623 i31·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)
624 ·--·used·1.17756s·(cpu);·0.999373s·(thread);·0s·(gc)624 ·--·used·1.55939s·(cpu);·1.30429s·(thread);·0s·(gc)
  
625 o31·=·true625 o31·=·true
626 i32·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)626 i32·:·time·regularInCodimension(1,·S,·Strategy=>StrategyRandom)
627 ·--·used·1.18958s·(cpu);·1.06471s·(thread);·0s·(gc)627 ·--·used·1.47327s·(cpu);·1.20417s·(thread);·0s·(gc)
  
628 o32·=·true628 o32·=·true
629 The·minimum·number·of·minors·computed·before·checking·the·codimension·can·also629 The·minimum·number·of·minors·computed·before·checking·the·codimension·can·also
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9 #:len=2889 #:len=288
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV411 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV4
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./usr/share/doc/Macaulay2/FiniteFittingIdeals/example-output/___Fitting_spideals_spof_spfinite_spmodules.out
    
Offset 81, 23 lines modifiedOffset 81, 23 lines modified
  
81 i14·:·K3=nextDegree(gens·ker·Q2,2,S);81 i14·:·K3=nextDegree(gens·ker·Q2,2,S);
  
82 ··············8······882 ··············8······8
83 o14·:·Matrix·R··<--·R83 o14·:·Matrix·R··<--·R
  
84 i15·:·time·I=co1Fitting(K3)84 i15·:·time·I=co1Fitting(K3)
85 ·--·used·0.00399531s·(cpu);·0.00219235s·(thread);·0s·(gc)85 ·--·used·0.00375382s·(cpu);·0.00231872s·(thread);·0s·(gc)
  
86 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)86 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)
87 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····587 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····5
  
88 o15·:·Ideal·of·R88 o15·:·Ideal·of·R
  
89 i16·:·time·J=fittingIdeal(2-1,coker·K3);89 i16·:·time·J=fittingIdeal(2-1,coker·K3);
90 ·--·used·0.00480311s·(cpu);·0.00474183s·(thread);·0s·(gc)90 ·--·used·0.0040188s·(cpu);·0.00489896s·(thread);·0s·(gc)
  
91 o16·:·Ideal·of·R91 o16·:·Ideal·of·R
  
92 i17·:·I==J92 i17·:·I==J
  
93 o17·=·true93 o17·=·true
  
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./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html
    
Offset 166, 24 lines modifiedOffset 166, 24 lines modified
166 <td>··············<pre><code·class="language-macaulay2">i14·:·K3=nextDegree(gens·ker·Q2,2,S);166 <td>··············<pre><code·class="language-macaulay2">i14·:·K3=nextDegree(gens·ker·Q2,2,S);
  
167 ··············8······8167 ··············8······8
168 o14·:·Matrix·R··&lt;--·R</code></pre>168 o14·:·Matrix·R··&lt;--·R</code></pre>
169 </td>··········</tr>169 </td>··········</tr>
170 ··········<tr>170 ··········<tr>
171 <td>··············<pre><code·class="language-macaulay2">i15·:·time·I=co1Fitting(K3)171 <td>··············<pre><code·class="language-macaulay2">i15·:·time·I=co1Fitting(K3)
172 ·--·used·0.00399531s·(cpu);·0.00219235s·(thread);·0s·(gc)172 ·--·used·0.00375382s·(cpu);·0.00231872s·(thread);·0s·(gc)
  
173 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)173 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)
174 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····5174 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····5
  
175 o15·:·Ideal·of·R</code></pre>175 o15·:·Ideal·of·R</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ··········<tr>177 ··········<tr>
178 <td>··············<pre><code·class="language-macaulay2">i16·:·time·J=fittingIdeal(2-1,coker·K3);178 <td>··············<pre><code·class="language-macaulay2">i16·:·time·J=fittingIdeal(2-1,coker·K3);
179 ·--·used·0.00480311s·(cpu);·0.00474183s·(thread);·0s·(gc)179 ·--·used·0.0040188s·(cpu);·0.00489896s·(thread);·0s·(gc)
  
180 o16·:·Ideal·of·R</code></pre>180 o16·:·Ideal·of·R</code></pre>
181 </td>··········</tr>181 </td>··········</tr>
182 ··········<tr>182 ··········<tr>
183 <td>··············<pre><code·class="language-macaulay2">i17·:·I==J183 <td>··············<pre><code·class="language-macaulay2">i17·:·I==J
  
184 o17·=·true</code></pre>184 o17·=·true</code></pre>
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95 ··············2······695 ··············2······6
96 o13·:·Matrix·R··<--·R96 o13·:·Matrix·R··<--·R
97 i14·:·K3=nextDegree(gens·ker·Q2,2,S);97 i14·:·K3=nextDegree(gens·ker·Q2,2,S);
  
98 ··············8······898 ··············8······8
99 o14·:·Matrix·R··<--·R99 o14·:·Matrix·R··<--·R
100 i15·:·time·I=co1Fitting(K3)100 i15·:·time·I=co1Fitting(K3)
101 ·--·used·0.00399531s·(cpu);·0.00219235s·(thread);·0s·(gc)101 ·--·used·0.00375382s·(cpu);·0.00231872s·(thread);·0s·(gc)
  
102 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)102 o15·=·ideal·(a·a···+·a··-·a··,·a·a···-·a·,·a·a···+·a··-·a··,·a·a···-·a·)
103 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····5103 ··············9·11····5····12···3·11····6···9·10····4····11···3·10····5
  
104 o15·:·Ideal·of·R104 o15·:·Ideal·of·R
105 i16·:·time·J=fittingIdeal(2-1,coker·K3);105 i16·:·time·J=fittingIdeal(2-1,coker·K3);
106 ·--·used·0.00480311s·(cpu);·0.00474183s·(thread);·0s·(gc)106 ·--·used·0.0040188s·(cpu);·0.00489896s·(thread);·0s·(gc)
  
107 o16·:·Ideal·of·R107 o16·:·Ideal·of·R
108 i17·:·I==J108 i17·:·I==J
  
109 o17·=·true109 o17·=·true
110 Note·that·our·method·is·a·bit·faster·for·this·small·example,·and·for·rank·2110 Note·that·our·method·is·a·bit·faster·for·this·small·example,·and·for·rank·2
111 quotients·of·S^3=\mathbb{Z}[x,y]^3·the·time·difference·is·massive.111 quotients·of·S^3=\mathbb{Z}[x,y]^3·the·time·difference·is·massive.
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./usr/share/doc/Macaulay2/ForeignFunctions/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Object.out
    
Offset 4, 19 lines modifiedOffset 4, 19 lines modified
  
4 o1·=·54 o1·=·5
  
5 o1·:·ForeignObject·of·type·int325 o1·:·ForeignObject·of·type·int32
  
6 i2·:·peek·x6 i2·:·peek·x
  
7 o2·=·int32{Address·=>·0x7f4d8d6d4a30}7 o2·=·int32{Address·=>·0x7f40c93a7320}
  
8 i3·:·address·x8 i3·:·address·x
  
9 o3·=·0x7f4d8d6d4a309 o3·=·0x7f40c93a7320
  
10 o3·:·Pointer10 o3·:·Pointer
  
11 i4·:·class·x11 i4·:·class·x
  
12 o4·=·int3212 o4·=·int32
  
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./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type.out
    
Offset 11, 15 lines modifiedOffset 11, 15 lines modified
  
11 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}11 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}
  
12 o2·:·ForeignObject·of·type·char**12 o2·:·ForeignObject·of·type·char**
  
13 i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}13 i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
14 o3·=·{0x7f8c945e37c0,·0x7f8c945e3780,·0x7f8c945e3750}14 o3·=·{0x7f0df9fb49b0,·0x7f0df9fb4990,·0x7f0df9fb4980}
  
15 o3·:·ForeignObject·of·type·void**15 o3·:·ForeignObject·of·type·void**
  
16 i4·:·x·=·charstarstar·{"foo",·"bar",·"baz"}16 i4·:·x·=·charstarstar·{"foo",·"bar",·"baz"}
  
17 o4·=·{foo,·bar,·baz}17 o4·=·{foo,·bar,·baz}
  
⊟
586 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Array__Type_sp__Visible__List.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·{foo,·bar}4 o1·=·{foo,·bar}
  
5 o1·:·ForeignObject·of·type·char**5 o1·:·ForeignObject·of·type·char**
  
6 i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}6 i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
7 o2·=·{0x7f2dc6976030,·0x7f2dc6976020,·0x7f2dc6993ff0}7 o2·=·{0x7f2e672a6350,·0x7f2e672a6340,·0x7f2e672a6330}
  
8 o2·:·ForeignObject·of·type·void**8 o2·:·ForeignObject·of·type·void**
  
9 i3·:·int2star·=·foreignPointerArrayType(2·*·int)9 i3·:·int2star·=·foreignPointerArrayType(2·*·int)
  
10 o3·=·int32[2]*10 o3·=·int32[2]*
  
⊟
462 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Pointer__Type_sp__Pointer.out
    
Offset 1, 15 lines modifiedOffset 1, 15 lines modified
1 --·-*-·M2-comint·-*-·hash:·17308351698883994501 --·-*-·M2-comint·-*-·hash:·1730835169888399450
  
2 i1·:·ptr·=·address·int·02 i1·:·ptr·=·address·int·0
  
3 o1·=·0x7f2be575edc03 o1·=·0x7f51635cf350
  
4 o1·:·Pointer4 o1·:·Pointer
  
5 i2·:·voidstar·ptr5 i2·:·voidstar·ptr
  
6 o2·=·0x7f2be575edc06 o2·=·0x7f51635cf350
  
7 o2·:·ForeignObject·of·type·void*7 o2·:·ForeignObject·of·type·void*
  
8 i3·:·8 i3·:·
⊟
356 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp__Pointer.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·54 o1·=·5
  
5 o1·:·ForeignObject·of·type·int325 o1·:·ForeignObject·of·type·int32
  
6 i2·:·ptr·=·address·x6 i2·:·ptr·=·address·x
  
7 o2·=·0x7ff49a9006f07 o2·=·0x7f231ffc59f0
  
8 o2·:·Pointer8 o2·:·Pointer
  
9 i3·:·int·ptr9 i3·:·int·ptr
  
10 o3·=·510 o3·=·5
  
⊟
403 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Type_sp_st_spvoidstar.out
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
1 --·-*-·M2-comint·-*-·hash:·17312308291836839301 --·-*-·M2-comint·-*-·hash:·1731230829183683930
  
2 i1·:·ptr·=·voidstar·address·int·52 i1·:·ptr·=·voidstar·address·int·5
  
3 o1·=·0x7fd42edebd703 o1·=·0x7febfd633780
  
4 o1·:·ForeignObject·of·type·void*4 o1·:·ForeignObject·of·type·void*
  
5 i2·:·int·*·ptr5 i2·:·int·*·ptr
  
6 o2·=·56 o2·=·5
  
⊟
471 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Foreign__Union__Type_sp__Thing.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·myunion4 o1·=·myunion
  
5 o1·:·ForeignUnionType5 o1·:·ForeignUnionType
  
6 i2·:·myunion·276 i2·:·myunion·27
  
7 o2·=·HashTable{"bar"·=>·6.9455e-310}7 o2·=·HashTable{"bar"·=>·1.33398e-322}
8 ···············"foo"·=>·278 ···············"foo"·=>·27
  
9 o2·:·ForeignObject·of·type·myunion9 o2·:·ForeignObject·of·type·myunion
  
10 i3·:·myunion·pi10 i3·:·myunion·pi
  
11 o3·=·HashTable{"bar"·=>·3.14159···}11 o3·=·HashTable{"bar"·=>·3.14159···}
⊟
549 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Pointer.out
    
Offset 4, 28 lines modifiedOffset 4, 28 lines modified
  
4 o1·=·204 o1·=·20
  
5 o1·:·ForeignObject·of·type·int325 o1·:·ForeignObject·of·type·int32
  
6 i2·:·peek·x6 i2·:·peek·x
  
7 o2·=·int32{Address·=>·0x7fce2dddbd40}7 o2·=·int32{Address·=>·0x7f1d176a6990}
  
8 i3·:·ptr·=·address·x8 i3·:·ptr·=·address·x
  
9 o3·=·0x7fce2dddbd409 o3·=·0x7f1d176a6990
  
10 o3·:·Pointer10 o3·:·Pointer
  
11 i4·:·ptr·+·511 i4·:·ptr·+·5
  
12 o4·=·0x7fce2dddbd4512 o4·=·0x7f1d176a6995
  
13 o4·:·Pointer13 o4·:·Pointer
  
14 i5·:·ptr·-·314 i5·:·ptr·-·3
  
15 o5·=·0x7fce2dddbd3d15 o5·=·0x7f1d176a698d
  
16 o5·:·Pointer16 o5·:·Pointer
  
17 i6·:·17 i6·:·
⊟
326 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/___Shared__Library.out
    
Offset 4, 10 lines modifiedOffset 4, 10 lines modified
  
4 o1·=·mpfr4 o1·=·mpfr
  
5 o1·:·SharedLibrary5 o1·:·SharedLibrary
  
6 i2·:·peek·mpfr6 i2·:·peek·mpfr
  
7 o2·=·SharedLibrary{0x7f343de40550,·mpfr}7 o2·=·SharedLibrary{0x7fa8fea43550,·mpfr}
  
8 i3·:·8 i3·:·
⊟
367 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/__st_spvoidstar_sp_eq_sp__Thing.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·54 o1·=·5
  
5 o1·:·ForeignObject·of·type·int325 o1·:·ForeignObject·of·type·int32
  
6 i2·:·ptr·=·address·x6 i2·:·ptr·=·address·x
  
7 o2·=·0x7f996db8d7207 o2·=·0x7fd606bff850
  
8 o2·:·Pointer8 o2·:·Pointer
  
9 i3·:·*ptr·=·int·69 i3·:·*ptr·=·int·6
  
10 o3·=·610 o3·=·6
  
⊟
377 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_address.out
    
Offset 1, 15 lines modifiedOffset 1, 15 lines modified
1 --·-*-·M2-comint·-*-·hash:·17301818843773735951 --·-*-·M2-comint·-*-·hash:·1730181884377373595
  
2 i1·:·address·int2 i1·:·address·int
  
3 o1·=·0x55afb6af0bc03 o1·=·0x562f1b446bc0
  
4 o1·:·Pointer4 o1·:·Pointer
  
5 i2·:·address·int·55 i2·:·address·int·5
  
6 o2·=·0x7fc99b3ad3b06 o2·=·0x7f3da59f04f0
  
7 o2·:·Pointer7 o2·:·Pointer
  
8 i3·:·8 i3·:·
⊟
370 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_foreign__Function.out
    
Offset 78, 14 lines modifiedOffset 78, 14 lines modified
  
78 o16·=·free78 o16·=·free
  
79 o16·:·ForeignFunction79 o16·:·ForeignFunction
  
80 i17·:·x·=·malloc·880 i17·:·x·=·malloc·8
  
81 o17·=·0x7f56a00639a081 o17·=·0x7f56400639a0
  
82 o17·:·ForeignObject·of·type·void*82 o17·:·ForeignObject·of·type·void*
  
83 i18·:·registerFinalizer(x,·free)83 i18·:·registerFinalizer(x,·free)
  
84 i19·:·84 i19·:·
⊟
561 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_get__Memory.out
    
Offset 1, 21 lines modifiedOffset 1, 21 lines modified
1 --·-*-·M2-comint·-*-·hash:·106479884127672803101 --·-*-·M2-comint·-*-·hash:·10647988412767280310
  
2 i1·:·ptr·=·getMemory·82 i1·:·ptr·=·getMemory·8
  
3 o1·=·0x7f4ae4d639a03 o1·=·0x7f91a0c849a0
  
4 o1·:·ForeignObject·of·type·void*4 o1·:·ForeignObject·of·type·void*
  
5 i2·:·ptr·=·getMemory(8,·Atomic·=>·true)5 i2·:·ptr·=·getMemory(8,·Atomic·=>·true)
  
6 o2·=·0x7f4ae52700206 o2·=·0x7f919e2a64b0
  
7 o2·:·ForeignObject·of·type·void*7 o2·:·ForeignObject·of·type·void*
  
8 i3·:·ptr·=·getMemory·int8 i3·:·ptr·=·getMemory·int
  
9 o3·=·0x7f4ae528edb09 o3·=·0x7f919e2a63c0
  
10 o3·:·ForeignObject·of·type·void*10 o3·:·ForeignObject·of·type·void*
  
11 i4·:·11 i4·:·
⊟
1010 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.out
    
Offset 17, 18 lines modifiedOffset 17, 18 lines modified
17 o3·=·finalizer17 o3·=·finalizer
  
18 o3·:·FunctionClosure18 o3·:·FunctionClosure
  
19 i4·:·for·i·to·9·do·(x·:=·malloc·8;·registerFinalizer(x,·finalizer))19 i4·:·for·i·to·9·do·(x·:=·malloc·8;·registerFinalizer(x,·finalizer))
  
20 i5·:·collectGarbage()20 i5·:·collectGarbage()
21 freeing·memory·at·0x7f38e406a1e0 
22 freeing·memory·at·0x7f38e40639c0 
23 freeing·memory·at·0x7f38e4063b90 
24 freeing·memory·at·0x7f38e4066c2021 freeing·memory·at·0x7f3208066c20
25 freeing·memory·at·0x7f38e4069150 
26 freeing·memory·at·0x7f38e4063bc0 
27 freeing·memory·at·0x7f38e40639a022 freeing·memory·at·0x7f32080639a0
28 freeing·memory·at·0x7f38e405f430 
29 freeing·memory·at·0x7f38e406a20023 freeing·memory·at·0x7f320806a200
 24 freeing·memory·at·0x7f320806a1e0
 25 freeing·memory·at·0x7f32080639c0
 26 freeing·memory·at·0x7f320805f430
 27 freeing·memory·at·0x7f3208063b90
 28 freeing·memory·at·0x7f3208063bc0
 29 freeing·memory·at·0x7f3208069150
  
30 i6·:·30 i6·:·
⊟
493 B
./usr/share/doc/Macaulay2/ForeignFunctions/example-output/_value_lp__Foreign__Object_rp.out
    
Offset 20, 21 lines modifiedOffset 20, 21 lines modified
  
20 o4·=·520 o4·=·5
  
21 o4·:·RR·(of·precision·53)21 o4·:·RR·(of·precision·53)
  
22 i5·:·x·=·voidstar·address·int·522 i5·:·x·=·voidstar·address·int·5
  
23 o5·=·0x7fcba2525fe023 o5·=·0x7f36caaa62a0
  
24 o5·:·ForeignObject·of·type·void*24 o5·:·ForeignObject·of·type·void*
  
25 i6·:·value·x25 i6·:·value·x
  
26 o6·=·0x7fcba2525fe026 o6·=·0x7f36caaa62a0
  
27 o6·:·Pointer27 o6·:·Pointer
  
28 i7·:·x·=·charstar·"Hello,·world!"28 i7·:·x·=·charstar·"Hello,·world!"
  
29 o7·=·Hello,·world!29 o7·=·Hello,·world!
  
⊟
1.39 KB
./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
    
Offset 54, 25 lines modifiedOffset 54, 25 lines modified
54 o1·=·554 o1·=·5
  
55 o1·:·ForeignObject·of·type·int32</code></pre>55 o1·:·ForeignObject·of·type·int32</code></pre>
56 </td>··········</tr>56 </td>··········</tr>
57 ··········<tr>57 ··········<tr>
58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·x58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·x
  
59 o2·=·int32{Address·=>·0x7f4d8d6d4a30}</code></pre>59 o2·=·int32{Address·=>·0x7f40c93a7320}</code></pre>
60 </td>··········</tr>60 </td>··········</tr>
61 ········</table>61 ········</table>
62 ········<div>62 ········<div>
63 ··········<p>To·get·this,·use·<a·title="pointer·to·type·or·object"·href="_address.html">address</a>.</p>63 ··········<p>To·get·this,·use·<a·title="pointer·to·type·or·object"·href="_address.html">address</a>.</p>
64 ········</div>64 ········</div>
65 ········<table·class="examples">65 ········<table·class="examples">
66 ··········<tr>66 ··········<tr>
67 <td>··············<pre><code·class="language-macaulay2">i3·:·address·x67 <td>··············<pre><code·class="language-macaulay2">i3·:·address·x
  
68 o3·=·0x7f4d8d6d4a3068 o3·=·0x7f40c93a7320
  
69 o3·:·Pointer</code></pre>69 o3·:·Pointer</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ········</table>71 ········</table>
72 ········<div>72 ········<div>
73 ··········<p>Use·<a·title="class·of·an·object"·href="../../Macaulay2Doc/html/_class.html">class</a>·to·determine·the·type·of·the·object.</p>73 ··········<p>Use·<a·title="class·of·an·object"·href="../../Macaulay2Doc/html/_class.html">class</a>·to·determine·the·type·of·the·object.</p>
74 ········</div>74 ········</div>
⊟
404 B
html2text {}
    
Offset 10, 19 lines modifiedOffset 10, 19 lines modified
10 i1·:·x·=·int·510 i1·:·x·=·int·5
  
11 o1·=·511 o1·=·5
  
12 o1·:·ForeignObject·of·type·int3212 o1·:·ForeignObject·of·type·int32
13 i2·:·peek·x13 i2·:·peek·x
  
14 o2·=·int32{Address·=>·0x7f4d8d6d4a30}14 o2·=·int32{Address·=>·0x7f40c93a7320}
15 To·get·this,·use·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s.15 To·get·this,·use·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s.
16 i3·:·address·x16 i3·:·address·x
  
17 o3·=·0x7f4d8d6d4a3017 o3·=·0x7f40c93a7320
  
18 o3·:·Pointer18 o3·:·Pointer
19 Use·_\x8c_\x8l_\x8a_\x8s_\x8s·to·determine·the·type·of·the·object.19 Use·_\x8c_\x8l_\x8a_\x8s_\x8s·to·determine·the·type·of·the·object.
20 i4·:·class·x20 i4·:·class·x
  
21 o4·=·int3221 o4·=·int32
  
⊟
1.36 KB
./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html
    
Offset 62, 15 lines modifiedOffset 62, 15 lines modified
62 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}62 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}
  
63 o2·:·ForeignObject·of·type·char**</code></pre>63 o2·:·ForeignObject·of·type·char**</code></pre>
64 </td>··········</tr>64 </td>··········</tr>
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}66 <td>··············<pre><code·class="language-macaulay2">i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
67 o3·=·{0x7f8c945e37c0,·0x7f8c945e3780,·0x7f8c945e3750}67 o3·=·{0x7f0df9fb49b0,·0x7f0df9fb4990,·0x7f0df9fb4980}
  
68 o3·:·ForeignObject·of·type·void**</code></pre>68 o3·:·ForeignObject·of·type·void**</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ········</table>70 ········</table>
71 ········<div>71 ········<div>
72 ··········<p>Foreign·pointer·arrays·may·be·subscripted·using·<a·title="a·binary·operator,·used·for·subscripting·and·access·to·elements"·href="../../Macaulay2Doc/html/__us.html">_</a>.</p>72 ··········<p>Foreign·pointer·arrays·may·be·subscripted·using·<a·title="a·binary·operator,·used·for·subscripting·and·access·to·elements"·href="../../Macaulay2Doc/html/__us.html">_</a>.</p>
73 ········</div>73 ········</div>
⊟
507 B
html2text {}
    
Offset 20, 15 lines modifiedOffset 20, 15 lines modified
20 ·········"lazy",·"dog"}20 ·········"lazy",·"dog"}
  
21 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}21 o2·=·{the,·quick,·brown,·fox,·jumps,·over,·the,·lazy,·dog}
  
22 o2·:·ForeignObject·of·type·char**22 o2·:·ForeignObject·of·type·char**
23 i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}23 i3·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
24 o3·=·{0x7f8c945e37c0,·0x7f8c945e3780,·0x7f8c945e3750}24 o3·=·{0x7f0df9fb49b0,·0x7f0df9fb4990,·0x7f0df9fb4980}
  
25 o3·:·ForeignObject·of·type·void**25 o3·:·ForeignObject·of·type·void**
26 Foreign·pointer·arrays·may·be·subscripted·using·_\x8_.26 Foreign·pointer·arrays·may·be·subscripted·using·_\x8_.
27 i4·:·x·=·charstarstar·{"foo",·"bar",·"baz"}27 i4·:·x·=·charstarstar·{"foo",·"bar",·"baz"}
  
28 o4·=·{foo,·bar,·baz}28 o4·=·{foo,·bar,·baz}
  
⊟
1.2 KB
./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html
    
Offset 81, 15 lines modifiedOffset 81, 15 lines modified
81 o1·=·{foo,·bar}81 o1·=·{foo,·bar}
  
82 o1·:·ForeignObject·of·type·char**</code></pre>82 o1·:·ForeignObject·of·type·char**</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}85 <td>··············<pre><code·class="language-macaulay2">i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
86 o2·=·{0x7f2dc6976030,·0x7f2dc6976020,·0x7f2dc6993ff0}86 o2·=·{0x7f2e672a6350,·0x7f2e672a6340,·0x7f2e672a6330}
  
87 o2·:·ForeignObject·of·type·void**</code></pre>87 o2·:·ForeignObject·of·type·void**</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i3·:·int2star·=·foreignPointerArrayType(2·*·int)90 <td>··············<pre><code·class="language-macaulay2">i3·:·int2star·=·foreignPointerArrayType(2·*·int)
  
91 o3·=·int32[2]*91 o3·=·int32[2]*
⊟
448 B
html2text {}
    
Offset 21, 15 lines modifiedOffset 21, 15 lines modified
21 i1·:·charstarstar·{"foo",·"bar"}21 i1·:·charstarstar·{"foo",·"bar"}
  
22 o1·=·{foo,·bar}22 o1·=·{foo,·bar}
  
23 o1·:·ForeignObject·of·type·char**23 o1·:·ForeignObject·of·type·char**
24 i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}24 i2·:·voidstarstar·{address·int·0,·address·int·1,·address·int·2}
  
25 o2·=·{0x7f2dc6976030,·0x7f2dc6976020,·0x7f2dc6993ff0}25 o2·=·{0x7f2e672a6350,·0x7f2e672a6340,·0x7f2e672a6330}
  
26 o2·:·ForeignObject·of·type·void**26 o2·:·ForeignObject·of·type·void**
27 i3·:·int2star·=·foreignPointerArrayType(2·*·int)27 i3·:·int2star·=·foreignPointerArrayType(2·*·int)
  
28 o3·=·int32[2]*28 o3·=·int32[2]*
  
29 o3·:·ForeignPointerArrayType29 o3·:·ForeignPointerArrayType
⊟
1.58 KB
./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
    
Offset 74, 22 lines modifiedOffset 74, 22 lines modified
74 ········<div>74 ········<div>
75 ··········<p>To·cast·a·Macaulay2·pointer·to·a·foreign·object·with·a·pointer·type,·give·the·type·followed·by·the·pointer.</p>75 ··········<p>To·cast·a·Macaulay2·pointer·to·a·foreign·object·with·a·pointer·type,·give·the·type·followed·by·the·pointer.</p>
76 ········</div>76 ········</div>
77 ········<table·class="examples">77 ········<table·class="examples">
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·address·int·079 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·address·int·0
  
80 o1·=·0x7f2be575edc080 o1·=·0x7f51635cf350
  
81 o1·:·Pointer</code></pre>81 o1·:·Pointer</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i2·:·voidstar·ptr84 <td>··············<pre><code·class="language-macaulay2">i2·:·voidstar·ptr
  
85 o2·=·0x7f2be575edc085 o2·=·0x7f51635cf350
  
86 o2·:·ForeignObject·of·type·void*</code></pre>86 o2·:·ForeignObject·of·type·void*</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ········</table>88 ········</table>
89 ······</div>89 ······</div>
90 ······<div·class="waystouse">90 ······<div·class="waystouse">
91 ········<h2>Ways·to·use·this·method:</h2>91 ········<h2>Ways·to·use·this·method:</h2>
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16 ····*·Outputs:16 ····*·Outputs:
17 ··········o·a·_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8·_\x8o_\x8b_\x8j_\x8e_\x8c_\x8t,17 ··········o·a·_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8·_\x8o_\x8b_\x8j_\x8e_\x8c_\x8t,
18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
19 To·cast·a·Macaulay2·pointer·to·a·foreign·object·with·a·pointer·type,·give·the19 To·cast·a·Macaulay2·pointer·to·a·foreign·object·with·a·pointer·type,·give·the
20 type·followed·by·the·pointer.20 type·followed·by·the·pointer.
21 i1·:·ptr·=·address·int·021 i1·:·ptr·=·address·int·0
  
22 o1·=·0x7f2be575edc022 o1·=·0x7f51635cf350
  
23 o1·:·Pointer23 o1·:·Pointer
24 i2·:·voidstar·ptr24 i2·:·voidstar·ptr
  
25 o2·=·0x7f2be575edc025 o2·=·0x7f51635cf350
  
26 o2·:·ForeignObject·of·type·void*26 o2·:·ForeignObject·of·type·void*
27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
28 ····*·_\x8F_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r_\x8T_\x8y_\x8p_\x8e_\x8·_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r·--·cast·a·Macaulay2·pointer·to·a·foreign28 ····*·_\x8F_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r_\x8T_\x8y_\x8p_\x8e_\x8·_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r·--·cast·a·Macaulay2·pointer·to·a·foreign
29 ······pointer29 ······pointer
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Offset 81, 15 lines modifiedOffset 81, 15 lines modified
81 o1·=·581 o1·=·5
  
82 o1·:·ForeignObject·of·type·int32</code></pre>82 o1·:·ForeignObject·of·type·int32</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·address·x85 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·address·x
  
86 o2·=·0x7ff49a9006f086 o2·=·0x7f231ffc59f0
  
87 o2·:·Pointer</code></pre>87 o2·:·Pointer</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i3·:·int·ptr90 <td>··············<pre><code·class="language-macaulay2">i3·:·int·ptr
  
91 o3·=·591 o3·=·5
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19 i1·:·x·=·int·519 i1·:·x·=·int·5
  
20 o1·=·520 o1·=·5
  
21 o1·:·ForeignObject·of·type·int3221 o1·:·ForeignObject·of·type·int32
22 i2·:·ptr·=·address·x22 i2·:·ptr·=·address·x
  
23 o2·=·0x7ff49a9006f023 o2·=·0x7f231ffc59f0
  
24 o2·:·Pointer24 o2·:·Pointer
25 i3·:·int·ptr25 i3·:·int·ptr
  
26 o3·=·526 o3·=·5
  
27 o3·:·ForeignObject·of·type·int3227 o3·:·ForeignObject·of·type·int32
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74 ········<div>74 ········<div>
75 ··········<p>This·is·syntactic·sugar·for·<code·class="language-macaulay2">T·value·ptr</code>·(see·<a·title="dereference·a·pointer"·href="___Foreign__Type_sp__Pointer.html">ForeignType·Pointer</a>)·for·dereferencing·pointers.</p>75 ··········<p>This·is·syntactic·sugar·for·<code·class="language-macaulay2">T·value·ptr</code>·(see·<a·title="dereference·a·pointer"·href="___Foreign__Type_sp__Pointer.html">ForeignType·Pointer</a>)·for·dereferencing·pointers.</p>
76 ········</div>76 ········</div>
77 ········<table·class="examples">77 ········<table·class="examples">
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·voidstar·address·int·579 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·voidstar·address·int·5
  
80 o1·=·0x7fd42edebd7080 o1·=·0x7febfd633780
  
81 o1·:·ForeignObject·of·type·void*</code></pre>81 o1·:·ForeignObject·of·type·void*</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i2·:·int·*·ptr84 <td>··············<pre><code·class="language-macaulay2">i2·:·int·*·ptr
  
85 o2·=·585 o2·=·5
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15 ····*·Outputs:15 ····*·Outputs:
16 ··········o·a·_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8·_\x8o_\x8b_\x8j_\x8e_\x8c_\x8t,·of·type·T;16 ··········o·a·_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8·_\x8o_\x8b_\x8j_\x8e_\x8c_\x8t,·of·type·T;
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 This·is·syntactic·sugar·for·T·value·ptr·(see·_\x8F_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8T_\x8y_\x8p_\x8e_\x8·_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r)·for18 This·is·syntactic·sugar·for·T·value·ptr·(see·_\x8F_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8T_\x8y_\x8p_\x8e_\x8·_\x8P_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r)·for
19 dereferencing·pointers.19 dereferencing·pointers.
20 i1·:·ptr·=·voidstar·address·int·520 i1·:·ptr·=·voidstar·address·int·5
  
21 o1·=·0x7fd42edebd7021 o1·=·0x7febfd633780
  
22 o1·:·ForeignObject·of·type·void*22 o1·:·ForeignObject·of·type·void*
23 i2·:·int·*·ptr23 i2·:·int·*·ptr
  
24 o2·=·524 o2·=·5
  
25 o2·:·ForeignObject·of·type·int3225 o2·:·ForeignObject·of·type·int32
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81 o1·=·myunion81 o1·=·myunion
  
82 o1·:·ForeignUnionType</code></pre>82 o1·:·ForeignUnionType</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·myunion·2785 <td>··············<pre><code·class="language-macaulay2">i2·:·myunion·27
  
86 o2·=·HashTable{&quot;bar&quot;·=>·6.9455e-310}86 o2·=·HashTable{&quot;bar&quot;·=>·1.33398e-322}
87 ···············&quot;foo&quot;·=>·2787 ···············&quot;foo&quot;·=>·27
  
88 o2·:·ForeignObject·of·type·myunion</code></pre>88 o2·:·ForeignObject·of·type·myunion</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·myunion·pi91 <td>··············<pre><code·class="language-macaulay2">i3·:·myunion·pi
  
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21 i1·:·myunion·=·foreignUnionType("myunion",·{"foo"·=>·int,·"bar"·=>·double})21 i1·:·myunion·=·foreignUnionType("myunion",·{"foo"·=>·int,·"bar"·=>·double})
  
22 o1·=·myunion22 o1·=·myunion
  
23 o1·:·ForeignUnionType23 o1·:·ForeignUnionType
24 i2·:·myunion·2724 i2·:·myunion·27
  
25 o2·=·HashTable{"bar"·=>·6.9455e-310}25 o2·=·HashTable{"bar"·=>·1.33398e-322}
26 ···············"foo"·=>·2726 ···············"foo"·=>·27
  
27 o2·:·ForeignObject·of·type·myunion27 o2·:·ForeignObject·of·type·myunion
28 i3·:·myunion·pi28 i3·:·myunion·pi
  
29 o3·=·HashTable{"bar"·=>·3.14159···}29 o3·=·HashTable{"bar"·=>·3.14159···}
30 ···············"foo"·=>·141375413630 ···············"foo"·=>·1413754136
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Offset 54, 44 lines modifiedOffset 54, 44 lines modified
54 o1·=·2054 o1·=·20
  
55 o1·:·ForeignObject·of·type·int32</code></pre>55 o1·:·ForeignObject·of·type·int32</code></pre>
56 </td>··········</tr>56 </td>··········</tr>
57 ··········<tr>57 ··········<tr>
58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·x58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·x
  
59 o2·=·int32{Address·=>·0x7fce2dddbd40}</code></pre>59 o2·=·int32{Address·=>·0x7f1d176a6990}</code></pre>
60 </td>··········</tr>60 </td>··········</tr>
61 ········</table>61 ········</table>
62 ········<div>62 ········<div>
63 ··········<p>These·pointers·can·be·accessed·using·<a·title="pointer·to·type·or·object"·href="_address.html">address</a>.</p>63 ··········<p>These·pointers·can·be·accessed·using·<a·title="pointer·to·type·or·object"·href="_address.html">address</a>.</p>
64 ········</div>64 ········</div>
65 ········<table·class="examples">65 ········<table·class="examples">
66 ··········<tr>66 ··········<tr>
67 <td>··············<pre><code·class="language-macaulay2">i3·:·ptr·=·address·x67 <td>··············<pre><code·class="language-macaulay2">i3·:·ptr·=·address·x
  
68 o3·=·0x7fce2dddbd4068 o3·=·0x7f1d176a6990
  
69 o3·:·Pointer</code></pre>69 o3·:·Pointer</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ········</table>71 ········</table>
72 ········<div>72 ········<div>
73 ··········<p>Simple·arithmetic·can·be·performed·on·pointers.</p>73 ··········<p>Simple·arithmetic·can·be·performed·on·pointers.</p>
74 ········</div>74 ········</div>
75 ········<table·class="examples">75 ········<table·class="examples">
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i4·:·ptr·+·577 <td>··············<pre><code·class="language-macaulay2">i4·:·ptr·+·5
  
78 o4·=·0x7fce2dddbd4578 o4·=·0x7f1d176a6995
  
79 o4·:·Pointer</code></pre>79 o4·:·Pointer</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i5·:·ptr·-·382 <td>··············<pre><code·class="language-macaulay2">i5·:·ptr·-·3
  
83 o5·=·0x7fce2dddbd3d83 o5·=·0x7f1d176a698d
  
84 o5·:·Pointer</code></pre>84 o5·:·Pointer</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ········</table>86 ········</table>
87 ······</div>87 ······</div>
88 ······<div·class="waystouse">88 ······<div·class="waystouse">
89 ········<h2>Functions·and·methods·returning·a·pointer:</h2>89 ········<h2>Functions·and·methods·returning·a·pointer:</h2>
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10 i1·:·x·=·int·2010 i1·:·x·=·int·20
  
11 o1·=·2011 o1·=·20
  
12 o1·:·ForeignObject·of·type·int3212 o1·:·ForeignObject·of·type·int32
13 i2·:·peek·x13 i2·:·peek·x
  
14 o2·=·int32{Address·=>·0x7fce2dddbd40}14 o2·=·int32{Address·=>·0x7f1d176a6990}
15 These·pointers·can·be·accessed·using·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s.15 These·pointers·can·be·accessed·using·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s.
16 i3·:·ptr·=·address·x16 i3·:·ptr·=·address·x
  
17 o3·=·0x7fce2dddbd4017 o3·=·0x7f1d176a6990
  
18 o3·:·Pointer18 o3·:·Pointer
19 Simple·arithmetic·can·be·performed·on·pointers.19 Simple·arithmetic·can·be·performed·on·pointers.
20 i4·:·ptr·+·520 i4·:·ptr·+·5
  
21 o4·=·0x7fce2dddbd4521 o4·=·0x7f1d176a6995
  
22 o4·:·Pointer22 o4·:·Pointer
23 i5·:·ptr·-·323 i5·:·ptr·-·3
  
24 o5·=·0x7fce2dddbd3d24 o5·=·0x7f1d176a698d
  
25 o5·:·Pointer25 o5·:·Pointer
26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8a·p\x8po\x8oi\x8in\x8nt\x8te\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8a·p\x8po\x8oi\x8in\x8nt\x8te\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
27 ····*·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s·--·pointer·to·type·or·object27 ····*·_\x8a_\x8d_\x8d_\x8r_\x8e_\x8s_\x8s·--·pointer·to·type·or·object
28 *\x8**\x8**\x8**\x8**\x8*·M\x8Me\x8et\x8th\x8ho\x8od\x8ds\x8s·t\x8th\x8ha\x8at\x8t·u\x8us\x8se\x8e·a\x8a·p\x8po\x8oi\x8in\x8nt\x8te\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*28 *\x8**\x8**\x8**\x8**\x8*·M\x8Me\x8et\x8th\x8ho\x8od\x8ds\x8s·t\x8th\x8ha\x8at\x8t·u\x8us\x8se\x8e·a\x8a·p\x8po\x8oi\x8in\x8nt\x8te\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
29 ····*·*·Pointer·=·Thing·--·see·_\x8*_\x8·_\x8v_\x8o_\x8i_\x8d_\x8s_\x8t_\x8a_\x8r_\x8·_\x8=_\x8·_\x8T_\x8h_\x8i_\x8n_\x8g·--·assign·value·to·object·at29 ····*·*·Pointer·=·Thing·--·see·_\x8*_\x8·_\x8v_\x8o_\x8i_\x8d_\x8s_\x8t_\x8a_\x8r_\x8·_\x8=_\x8·_\x8T_\x8h_\x8i_\x8n_\x8g·--·assign·value·to·object·at
30 ······address30 ······address
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Offset 54, 15 lines modifiedOffset 54, 15 lines modified
54 o1·=·mpfr54 o1·=·mpfr
  
55 o1·:·SharedLibrary</code></pre>55 o1·:·SharedLibrary</code></pre>
56 </td>··········</tr>56 </td>··········</tr>
57 ··········<tr>57 ··········<tr>
58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·mpfr58 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·mpfr
  
59 o2·=·SharedLibrary{0x7f343de40550,·mpfr}</code></pre>59 o2·=·SharedLibrary{0x7fa8fea43550,·mpfr}</code></pre>
60 </td>··········</tr>60 </td>··········</tr>
61 ········</table>61 ········</table>
62 ······</div>62 ······</div>
63 ······<div·class="waystouse">63 ······<div·class="waystouse">
64 ········<h2>Functions·and·methods·returning·a·shared·library:</h2>64 ········<h2>Functions·and·methods·returning·a·shared·library:</h2>
65 ········<ul>65 ········<ul>
66 ··········<li>66 ··········<li>
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12 i1·:·mpfr·=·openSharedLibrary·"mpfr"12 i1·:·mpfr·=·openSharedLibrary·"mpfr"
  
13 o1·=·mpfr13 o1·=·mpfr
  
14 o1·:·SharedLibrary14 o1·:·SharedLibrary
15 i2·:·peek·mpfr15 i2·:·peek·mpfr
  
16 o2·=·SharedLibrary{0x7f343de40550,·mpfr}16 o2·=·SharedLibrary{0x7fa8fea43550,·mpfr}
17 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8a·s\x8sh\x8ha\x8ar\x8re\x8ed\x8d·l\x8li\x8ib\x8br\x8ra\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·a\x8an\x8nd\x8d·m\x8me\x8et\x8th\x8ho\x8od\x8ds\x8s·r\x8re\x8et\x8tu\x8ur\x8rn\x8ni\x8in\x8ng\x8g·a\x8a·s\x8sh\x8ha\x8ar\x8re\x8ed\x8d·l\x8li\x8ib\x8br\x8ra\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*
18 ····*·_\x8o_\x8p_\x8e_\x8n_\x8S_\x8h_\x8a_\x8r_\x8e_\x8d_\x8L_\x8i_\x8b_\x8r_\x8a_\x8r_\x8y·--·open·a·shared·library18 ····*·_\x8o_\x8p_\x8e_\x8n_\x8S_\x8h_\x8a_\x8r_\x8e_\x8d_\x8L_\x8i_\x8b_\x8r_\x8a_\x8r_\x8y·--·open·a·shared·library
19 *\x8**\x8**\x8**\x8**\x8*·M\x8Me\x8et\x8th\x8ho\x8od\x8ds\x8s·t\x8th\x8ha\x8at\x8t·u\x8us\x8se\x8e·a\x8a·s\x8sh\x8ha\x8ar\x8re\x8ed\x8d·l\x8li\x8ib\x8br\x8ra\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·M\x8Me\x8et\x8th\x8ho\x8od\x8ds\x8s·t\x8th\x8ha\x8at\x8t·u\x8us\x8se\x8e·a\x8a·s\x8sh\x8ha\x8ar\x8re\x8ed\x8d·l\x8li\x8ib\x8br\x8ra\x8ar\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*
20 ····*·foreignFunction(SharedLibrary,String,ForeignType,ForeignType)·--·see20 ····*·foreignFunction(SharedLibrary,String,ForeignType,ForeignType)·--·see
21 ······_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8F_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n·--·construct·a·foreign·function21 ······_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8F_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n·--·construct·a·foreign·function
22 ····*·foreignFunction(SharedLibrary,String,ForeignType,VisibleList)·--·see22 ····*·foreignFunction(SharedLibrary,String,ForeignType,VisibleList)·--·see
23 ······_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8F_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n·--·construct·a·foreign·function23 ······_\x8f_\x8o_\x8r_\x8e_\x8i_\x8g_\x8n_\x8F_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n·--·construct·a·foreign·function
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75 o1·=·575 o1·=·5
  
76 o1·:·ForeignObject·of·type·int32</code></pre>76 o1·:·ForeignObject·of·type·int32</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·address·x79 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·address·x
  
80 o2·=·0x7f996db8d72080 o2·=·0x7fd606bff850
  
81 o2·:·Pointer</code></pre>81 o2·:·Pointer</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i3·:·*ptr·=·int·684 <td>··············<pre><code·class="language-macaulay2">i3·:·*ptr·=·int·6
  
85 o3·=·685 o3·=·6
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17 i1·:·x·=·int·517 i1·:·x·=·int·5
  
18 o1·=·518 o1·=·5
  
19 o1·:·ForeignObject·of·type·int3219 o1·:·ForeignObject·of·type·int32
20 i2·:·ptr·=·address·x20 i2·:·ptr·=·address·x
  
21 o2·=·0x7f996db8d72021 o2·=·0x7fd606bff850
  
22 o2·:·Pointer22 o2·:·Pointer
23 i3·:·*ptr·=·int·623 i3·:·*ptr·=·int·6
  
24 o3·=·624 o3·=·6
  
25 o3·:·ForeignObject·of·type·int3225 o3·:·ForeignObject·of·type·int32
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70 ········<div>70 ········<div>
71 ··········<p>If·<span·class="tt">x</span>·is·a·foreign·type,·then·this·returns·the·address·to·the·<span·class="tt">ffi_type</span>·struct·used·by·<span·class="tt">libffi</span>·to·identify·the·type.</p>71 ··········<p>If·<span·class="tt">x</span>·is·a·foreign·type,·then·this·returns·the·address·to·the·<span·class="tt">ffi_type</span>·struct·used·by·<span·class="tt">libffi</span>·to·identify·the·type.</p>
72 ········</div>72 ········</div>
73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·address·int75 <td>··············<pre><code·class="language-macaulay2">i1·:·address·int
  
76 o1·=·0x55afb6af0bc076 o1·=·0x562f1b446bc0
  
77 o1·:·Pointer</code></pre>77 o1·:·Pointer</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ········</table>79 ········</table>
80 ········<div>80 ········<div>
81 ··········<p>If·<span·class="tt">x</span>·is·a·foreign·object,·then·this·returns·the·address·to·the·object.··It·behaves·like·the·<span·class="tt">&amp;</span>·&quot;address-of&quot;·operator·in·C.</p>81 ··········<p>If·<span·class="tt">x</span>·is·a·foreign·object,·then·this·returns·the·address·to·the·object.··It·behaves·like·the·<span·class="tt">&amp;</span>·&quot;address-of&quot;·operator·in·C.</p>
82 ········</div>82 ········</div>
83 ········<table·class="examples">83 ········<table·class="examples">
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·address·int·585 <td>··············<pre><code·class="language-macaulay2">i2·:·address·int·5
  
86 o2·=·0x7fc99b3ad3b086 o2·=·0x7f3da59f04f0
  
87 o2·:·Pointer</code></pre>87 o2·:·Pointer</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
90 ······</div>90 ······</div>
91 ······<div·class="waystouse">91 ······<div·class="waystouse">
92 ········<h2>Ways·to·use·<span·class="tt">address</span>:</h2>92 ········<h2>Ways·to·use·<span·class="tt">address</span>:</h2>
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12 ····*·Outputs:12 ····*·Outputs:
13 ··········o·a·_\x8p_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r,13 ··········o·a·_\x8p_\x8o_\x8i_\x8n_\x8t_\x8e_\x8r,
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 If·x·is·a·foreign·type,·then·this·returns·the·address·to·the·ffi_type·struct15 If·x·is·a·foreign·type,·then·this·returns·the·address·to·the·ffi_type·struct
16 used·by·libffi·to·identify·the·type.16 used·by·libffi·to·identify·the·type.
17 i1·:·address·int17 i1·:·address·int
  
18 o1·=·0x55afb6af0bc018 o1·=·0x562f1b446bc0
  
19 o1·:·Pointer19 o1·:·Pointer
20 If·x·is·a·foreign·object,·then·this·returns·the·address·to·the·object.·It20 If·x·is·a·foreign·object,·then·this·returns·the·address·to·the·object.·It
21 behaves·like·the·&·"address-of"·operator·in·C.21 behaves·like·the·&·"address-of"·operator·in·C.
22 i2·:·address·int·522 i2·:·address·int·5
  
23 o2·=·0x7fc99b3ad3b023 o2·=·0x7f3da59f04f0
  
24 o2·:·Pointer24 o2·:·Pointer
25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·a\x8ad\x8dd\x8dr\x8re\x8es\x8ss\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·a\x8ad\x8dd\x8dr\x8re\x8es\x8ss\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
26 ····*·address(ForeignObject)26 ····*·address(ForeignObject)
27 ····*·address(ForeignType)27 ····*·address(ForeignType)
28 ····*·address(Nothing)·(missing·documentation)28 ····*·address(Nothing)·(missing·documentation)
29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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204 o16·=·free204 o16·=·free
  
205 o16·:·ForeignFunction</code></pre>205 o16·:·ForeignFunction</code></pre>
206 </td>··········</tr>206 </td>··········</tr>
207 ··········<tr>207 ··········<tr>
208 <td>··············<pre><code·class="language-macaulay2">i17·:·x·=·malloc·8208 <td>··············<pre><code·class="language-macaulay2">i17·:·x·=·malloc·8
  
209 o17·=·0x7f56a00639a0209 o17·=·0x7f56400639a0
  
210 o17·:·ForeignObject·of·type·void*</code></pre>210 o17·:·ForeignObject·of·type·void*</code></pre>
211 </td>··········</tr>211 </td>··········</tr>
212 ··········<tr>212 ··········<tr>
213 <td>··············<pre><code·class="language-macaulay2">i18·:·registerFinalizer(x,·free)</code></pre>213 <td>··············<pre><code·class="language-macaulay2">i18·:·registerFinalizer(x,·free)</code></pre>
214 </td>··········</tr>214 </td>··········</tr>
215 ········</table>215 ········</table>
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96 i16·:·free·=·foreignFunction("free",·void,·voidstar)96 i16·:·free·=·foreignFunction("free",·void,·voidstar)
  
97 o16·=·free97 o16·=·free
  
98 o16·:·ForeignFunction98 o16·:·ForeignFunction
99 i17·:·x·=·malloc·899 i17·:·x·=·malloc·8
  
100 o17·=·0x7f56a00639a0100 o17·=·0x7f56400639a0
  
101 o17·:·ForeignObject·of·type·void*101 o17·:·ForeignObject·of·type·void*
102 i18·:·registerFinalizer(x,·free)102 i18·:·registerFinalizer(x,·free)
103 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·f\x8fo\x8or\x8re\x8ei\x8ig\x8gn\x8nF\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8n:\x8:·*\x8**\x8**\x8**\x8**\x8*103 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·f\x8fo\x8or\x8re\x8ei\x8ig\x8gn\x8nF\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8n:\x8:·*\x8**\x8**\x8**\x8**\x8*
104 ····*·foreignFunction(Pointer,String,ForeignType,VisibleList)104 ····*·foreignFunction(Pointer,String,ForeignType,VisibleList)
105 ····*·foreignFunction(SharedLibrary,String,ForeignType,ForeignType)105 ····*·foreignFunction(SharedLibrary,String,ForeignType,ForeignType)
106 ····*·foreignFunction(SharedLibrary,String,ForeignType,VisibleList)106 ····*·foreignFunction(SharedLibrary,String,ForeignType,VisibleList)
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79 ········<div>79 ········<div>
80 ··········<p>Allocate·<span·class="tt">n</span>·bytes·of·memory·using·the·<a·href="../../Macaulay2Doc/html/___G__C_spgarbage_spcollector.html">GC·garbage·collector</a>.</p>80 ··········<p>Allocate·<span·class="tt">n</span>·bytes·of·memory·using·the·<a·href="../../Macaulay2Doc/html/___G__C_spgarbage_spcollector.html">GC·garbage·collector</a>.</p>
81 ········</div>81 ········</div>
82 ········<table·class="examples">82 ········<table·class="examples">
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·getMemory·884 <td>··············<pre><code·class="language-macaulay2">i1·:·ptr·=·getMemory·8
  
85 o1·=·0x7f4ae4d639a085 o1·=·0x7f91a0c849a0
  
86 o1·:·ForeignObject·of·type·void*</code></pre>86 o1·:·ForeignObject·of·type·void*</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ········</table>88 ········</table>
89 ········<div>89 ········<div>
90 ··········<p>If·the·memory·will·not·contain·any·pointers,·then·set·the·<span·class="tt">Atomic</span>·option·to·<a·href="../../Macaulay2Doc/html/_true.html">true</a>.</p>90 ··········<p>If·the·memory·will·not·contain·any·pointers,·then·set·the·<span·class="tt">Atomic</span>·option·to·<a·href="../../Macaulay2Doc/html/_true.html">true</a>.</p>
91 ········</div>91 ········</div>
92 ········<table·class="examples">92 ········<table·class="examples">
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·getMemory(8,·Atomic·=>·true)94 <td>··············<pre><code·class="language-macaulay2">i2·:·ptr·=·getMemory(8,·Atomic·=>·true)
  
95 o2·=·0x7f4ae527002095 o2·=·0x7f919e2a64b0
  
96 o2·:·ForeignObject·of·type·void*</code></pre>96 o2·:·ForeignObject·of·type·void*</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ········</table>98 ········</table>
99 ········<div>99 ········<div>
100 ··········<p>Alternatively,·a·foreign·object·type·<span·class="tt">T</span>·may·be·specified.··In·this·case,·the·number·of·bytes·and·whether·the·<span·class="tt">Atomic</span>·option·should·be·set·will·be·determined·automatically.</p>100 ··········<p>Alternatively,·a·foreign·object·type·<span·class="tt">T</span>·may·be·specified.··In·this·case,·the·number·of·bytes·and·whether·the·<span·class="tt">Atomic</span>·option·should·be·set·will·be·determined·automatically.</p>
101 ········</div>101 ········</div>
102 ········<table·class="examples">102 ········<table·class="examples">
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i3·:·ptr·=·getMemory·int104 <td>··············<pre><code·class="language-macaulay2">i3·:·ptr·=·getMemory·int
  
105 o3·=·0x7f4ae528edb0105 o3·=·0x7f919e2a63c0
  
106 o3·:·ForeignObject·of·type·void*</code></pre>106 o3·:·ForeignObject·of·type·void*</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ········</table>108 ········</table>
109 ······</div>109 ······</div>
110 ······<div>110 ······<div>
111 ········<h2>See·also</h2>111 ········<h2>See·also</h2>
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15 ··········o·Atomic·=>·...,·default·value·false15 ··········o·Atomic·=>·...,·default·value·false
16 ····*·Outputs:16 ····*·Outputs:
17 ··········o·an·instance·of·the·type·_\x8v_\x8o_\x8i_\x8d_\x8s_\x8t_\x8a_\x8r,17 ··········o·an·instance·of·the·type·_\x8v_\x8o_\x8i_\x8d_\x8s_\x8t_\x8a_\x8r,
18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
19 Allocate·n·bytes·of·memory·using·the·_\x8G_\x8C_\x8·_\x8g_\x8a_\x8r_\x8b_\x8a_\x8g_\x8e_\x8·_\x8c_\x8o_\x8l_\x8l_\x8e_\x8c_\x8t_\x8o_\x8r.19 Allocate·n·bytes·of·memory·using·the·_\x8G_\x8C_\x8·_\x8g_\x8a_\x8r_\x8b_\x8a_\x8g_\x8e_\x8·_\x8c_\x8o_\x8l_\x8l_\x8e_\x8c_\x8t_\x8o_\x8r.
20 i1·:·ptr·=·getMemory·820 i1·:·ptr·=·getMemory·8
  
21 o1·=·0x7f4ae4d639a021 o1·=·0x7f91a0c849a0
  
22 o1·:·ForeignObject·of·type·void*22 o1·:·ForeignObject·of·type·void*
23 If·the·memory·will·not·contain·any·pointers,·then·set·the·Atomic·option·to23 If·the·memory·will·not·contain·any·pointers,·then·set·the·Atomic·option·to
24 _\x8t_\x8r_\x8u_\x8e.24 _\x8t_\x8r_\x8u_\x8e.
25 i2·:·ptr·=·getMemory(8,·Atomic·=>·true)25 i2·:·ptr·=·getMemory(8,·Atomic·=>·true)
  
26 o2·=·0x7f4ae527002026 o2·=·0x7f919e2a64b0
  
27 o2·:·ForeignObject·of·type·void*27 o2·:·ForeignObject·of·type·void*
28 Alternatively,·a·foreign·object·type·T·may·be·specified.·In·this·case,·the28 Alternatively,·a·foreign·object·type·T·may·be·specified.·In·this·case,·the
29 number·of·bytes·and·whether·the·Atomic·option·should·be·set·will·be·determined29 number·of·bytes·and·whether·the·Atomic·option·should·be·set·will·be·determined
30 automatically.30 automatically.
31 i3·:·ptr·=·getMemory·int31 i3·:·ptr·=·getMemory·int
  
32 o3·=·0x7f4ae528edb032 o3·=·0x7f919e2a63c0
  
33 o3·:·ForeignObject·of·type·void*33 o3·:·ForeignObject·of·type·void*
34 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*34 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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36 ······foreign·object36 ······foreign·object
37 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8tM\x8Me\x8em\x8mo\x8or\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8tM\x8Me\x8em\x8mo\x8or\x8ry\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*
38 ····*·getMemory(ForeignType)38 ····*·getMemory(ForeignType)
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50 ······foreign·object50 ······foreign·object
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./usr/share/doc/Macaulay2/ForeignFunctions/html/_value_lp__Foreign__Object_rp.html
    
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108 ········<div>108 ········<div>
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124 ··········<p>Foreign·string·objects·are·converted·to·strings.</p>124 ··········<p>Foreign·string·objects·are·converted·to·strings.</p>
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=327 #:len=32
8 aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=8 aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=
9 #:len=2989 #:len=298
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo
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923 B
./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_fpt.out
    
Offset 155, 31 lines modifiedOffset 155, 31 lines modified
155 i26·:·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)·--·FinalAttempt·improves·the·estimate·slightly155 i26·:·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)·--·FinalAttempt·improves·the·estimate·slightly
  
156 o26·=·{.142067,·.144}156 o26·=·{.142067,·.144}
  
157 o26·:·List157 o26·:·List
  
158 i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)158 i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)
159 ·--·used·1.69619s·(cpu);·1.15727s·(thread);·0s·(gc)159 ·--·used·1.82989s·(cpu);·1.1147s·(thread);·0s·(gc)
  
160 o27·=·{.142067,·.144}160 o27·=·{.142067,·.144}
  
161 o27·:·List161 o27·:·List
  
162 i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)162 i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)
163 ·--·used·1.04947s·(cpu);·0.705317s·(thread);·0s·(gc)163 ·--·used·1.08603s·(cpu);·0.680628s·(thread);·0s·(gc)
  
164 ······1164 ······1
165 o28·=·-165 o28·=·-
166 ······7166 ······7
  
167 o28·:·QQ167 o28·:·QQ
  
168 i29·:·time·fpt(f,·DepthOfSearch·=>·4)168 i29·:·time·fpt(f,·DepthOfSearch·=>·4)
169 ·--·used·0.798542s·(cpu);·0.538397s·(thread);·0s·(gc)169 ·--·used·1.03659s·(cpu);·0.626257s·(thread);·0s·(gc)
  
170 ······1170 ······1
171 o29·=·-171 o29·=·-
172 ······7172 ······7
  
173 o29·:·QQ173 o29·:·QQ
  
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2.05 KB
./usr/share/doc/Macaulay2/FrobeniusThresholds/example-output/_frobenius__Nu.out
    
Offset 43, 34 lines modifiedOffset 43, 34 lines modified
43 o12·=·22043 o12·=·220
  
44 i13·:·R·=·ZZ/17[x,y,z];44 i13·:·R·=·ZZ/17[x,y,z];
  
45 i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial45 i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial
  
46 i15·:·time·frobeniusNu(3,·f)46 i15·:·time·frobeniusNu(3,·f)
47 ·--·used·0.00397514s·(cpu);·0.00371911s·(thread);·0s·(gc)47 ·--·used·0.00497191s·(cpu);·0.00570084s·(thread);·0s·(gc)
  
48 o15·=·375648 o15·=·3756
  
49 i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)49 i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)
50 ·--·used·0.270534s·(cpu);·0.21608s·(thread);·0s·(gc)50 ·--·used·0.399843s·(cpu);·0.281503s·(thread);·0s·(gc)
  
51 o16·=·375651 o16·=·3756
  
52 i17·:·R·=·ZZ/5[x,y,z];52 i17·:·R·=·ZZ/5[x,y,z];
  
53 i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;53 i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;
  
54 i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by·default54 i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by·default
55 ·--·used·0.265207s·(cpu);·0.186318s·(thread);·0s·(gc)55 ·--·used·0.301189s·(cpu);·0.192267s·(thread);·0s·(gc)
  
56 o19·=·49956 o19·=·499
  
57 i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)57 i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)
58 ·--·used·0.959131s·(cpu);·0.834749s·(thread);·0s·(gc)58 ·--·used·1.20809s·(cpu);·0.983564s·(thread);·0s·(gc)
  
59 o20·=·49959 o20·=·499
  
60 i21·:·R·=·ZZ/3[x,y];60 i21·:·R·=·ZZ/3[x,y];
  
61 i22·:·M·=·ideal(x,·y);61 i22·:·M·=·ideal(x,·y);
  
Offset 85, 34 lines modifiedOffset 85, 34 lines modified
85 o24·=·885 o24·=·8
  
86 i25·:·R·=·ZZ/5[x,y,z];86 i25·:·R·=·ZZ/5[x,y,z];
  
87 i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;87 i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;
  
88 i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)88 i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)
89 ·--·used·0.728426s·(cpu);·0.51059s·(thread);·0s·(gc)89 ·--·used·0.882056s·(cpu);·0.589107s·(thread);·0s·(gc)
  
90 o27·=·112490 o27·=·1124
  
91 i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)91 i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)
92 ·--·used·1.06851s·(cpu);·0.77176s·(thread);·0s·(gc)92 ·--·used·1.38206s·(cpu);·1.00392s·(thread);·0s·(gc)
  
93 o28·=·112493 o28·=·1124
  
94 i29·:·M·=·ideal(x,·y,·z);94 i29·:·M·=·ideal(x,·y,·z);
  
95 o29·:·Ideal·of·R95 o29·:·Ideal·of·R
  
96 i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)96 i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)
97 ·--·used·1.18947s·(cpu);·1.05905s·(thread);·0s·(gc)97 ·--·used·1.41601s·(cpu);·1.25292s·(thread);·0s·(gc)
  
98 o30·=·9798 o30·=·97
  
99 i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets·luckier99 i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets·luckier
100 ·--·used·0.444604s·(cpu);·0.388727s·(thread);·0s·(gc)100 ·--·used·0.521833s·(cpu);·0.443712s·(thread);·0s·(gc)
  
101 o31·=·97101 o31·=·97
  
102 i32·:·R·=·ZZ/7[x,y];102 i32·:·R·=·ZZ/7[x,y];
  
103 i33·:·f·=·(x·-·1)^3·-·(y·-·2)^2;103 i33·:·f·=·(x·-·1)^3·-·(y·-·2)^2;
  
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2.48 KB
./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
    
Offset 324, 33 lines modifiedOffset 324, 33 lines modified
324 ········</table>324 ········</table>
325 ········<div>325 ········<div>
326 ··········<p>The·computations·performed·when·<span·class="tt">FinalAttempt</span>·is·set·to·<span·class="tt">true</span>·are·often·slow,·and·often·fail·to·improve·the·estimate,·and·for·this·reason,·this·option·should·be·used·sparingly.·It·is·often·more·effective·to·increase·the·values·of·<span·class="tt">Attempts</span>·or·<span·class="tt">DepthOfSearch</span>,·instead.</p>326 ··········<p>The·computations·performed·when·<span·class="tt">FinalAttempt</span>·is·set·to·<span·class="tt">true</span>·are·often·slow,·and·often·fail·to·improve·the·estimate,·and·for·this·reason,·this·option·should·be·used·sparingly.·It·is·often·more·effective·to·increase·the·values·of·<span·class="tt">Attempts</span>·or·<span·class="tt">DepthOfSearch</span>,·instead.</p>
327 ········</div>327 ········</div>
328 ········<table·class="examples">328 ········<table·class="examples">
329 ··········<tr>329 ··········<tr>
330 <td>··············<pre><code·class="language-macaulay2">i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)330 <td>··············<pre><code·class="language-macaulay2">i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)
331 ·--·used·1.69619s·(cpu);·1.15727s·(thread);·0s·(gc)331 ·--·used·1.82989s·(cpu);·1.1147s·(thread);·0s·(gc)
  
332 o27·=·{.142067,·.144}332 o27·=·{.142067,·.144}
  
333 o27·:·List</code></pre>333 o27·:·List</code></pre>
334 </td>··········</tr>334 </td>··········</tr>
335 ··········<tr>335 ··········<tr>
336 <td>··············<pre><code·class="language-macaulay2">i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)336 <td>··············<pre><code·class="language-macaulay2">i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)
337 ·--·used·1.04947s·(cpu);·0.705317s·(thread);·0s·(gc)337 ·--·used·1.08603s·(cpu);·0.680628s·(thread);·0s·(gc)
  
338 ······1338 ······1
339 o28·=·-339 o28·=·-
340 ······7340 ······7
  
341 o28·:·QQ</code></pre>341 o28·:·QQ</code></pre>
342 </td>··········</tr>342 </td>··········</tr>
343 ··········<tr>343 ··········<tr>
344 <td>··············<pre><code·class="language-macaulay2">i29·:·time·fpt(f,·DepthOfSearch·=>·4)344 <td>··············<pre><code·class="language-macaulay2">i29·:·time·fpt(f,·DepthOfSearch·=>·4)
345 ·--·used·0.798542s·(cpu);·0.538397s·(thread);·0s·(gc)345 ·--·used·1.03659s·(cpu);·0.626257s·(thread);·0s·(gc)
  
346 ······1346 ······1
347 o29·=·-347 o29·=·-
348 ······7348 ······7
  
349 o29·:·QQ</code></pre>349 o29·:·QQ</code></pre>
350 </td>··········</tr>350 </td>··········</tr>
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1010 B
html2text {}
    
Offset 229, 29 lines modifiedOffset 229, 29 lines modified
  
229 o26·:·List229 o26·:·List
230 The·computations·performed·when·FinalAttempt·is·set·to·true·are·often·slow,·and230 The·computations·performed·when·FinalAttempt·is·set·to·true·are·often·slow,·and
231 often·fail·to·improve·the·estimate,·and·for·this·reason,·this·option·should·be231 often·fail·to·improve·the·estimate,·and·for·this·reason,·this·option·should·be
232 used·sparingly.·It·is·often·more·effective·to·increase·the·values·of·Attempts232 used·sparingly.·It·is·often·more·effective·to·increase·the·values·of·Attempts
233 or·DepthOfSearch,·instead.233 or·DepthOfSearch,·instead.
234 i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)234 i27·:·time·numeric·fpt(f,·DepthOfSearch·=>·3,·FinalAttempt·=>·true)
235 ·--·used·1.69619s·(cpu);·1.15727s·(thread);·0s·(gc)235 ·--·used·1.82989s·(cpu);·1.1147s·(thread);·0s·(gc)
  
236 o27·=·{.142067,·.144}236 o27·=·{.142067,·.144}
  
237 o27·:·List237 o27·:·List
238 i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)238 i28·:·time·fpt(f,·DepthOfSearch·=>·3,·Attempts·=>·7)
239 ·--·used·1.04947s·(cpu);·0.705317s·(thread);·0s·(gc)239 ·--·used·1.08603s·(cpu);·0.680628s·(thread);·0s·(gc)
  
240 ······1240 ······1
241 o28·=·-241 o28·=·-
242 ······7242 ······7
  
243 o28·:·QQ243 o28·:·QQ
244 i29·:·time·fpt(f,·DepthOfSearch·=>·4)244 i29·:·time·fpt(f,·DepthOfSearch·=>·4)
245 ·--·used·0.798542s·(cpu);·0.538397s·(thread);·0s·(gc)245 ·--·used·1.03659s·(cpu);·0.626257s·(thread);·0s·(gc)
  
246 ······1246 ······1
247 o29·=·-247 o29·=·-
248 ······7248 ······7
  
249 o29·:·QQ249 o29·:·QQ
250 As·seen·in·several·examples·above,·when·the·exact·answer·is·not·found,·a·list250 As·seen·in·several·examples·above,·when·the·exact·answer·is·not·found,·a·list
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9.38 KB
./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
    
Offset 175, 21 lines modifiedOffset 175, 21 lines modified
175 <td>··············<pre><code·class="language-macaulay2">i13·:·R·=·ZZ/17[x,y,z];</code></pre>175 <td>··············<pre><code·class="language-macaulay2">i13·:·R·=·ZZ/17[x,y,z];</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ··········<tr>177 ··········<tr>
178 <td>··············<pre><code·class="language-macaulay2">i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial</code></pre>178 <td>··············<pre><code·class="language-macaulay2">i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial</code></pre>
179 </td>··········</tr>179 </td>··········</tr>
180 ··········<tr>180 ··········<tr>
181 <td>··············<pre><code·class="language-macaulay2">i15·:·time·frobeniusNu(3,·f)181 <td>··············<pre><code·class="language-macaulay2">i15·:·time·frobeniusNu(3,·f)
182 ·--·used·0.00397514s·(cpu);·0.00371911s·(thread);·0s·(gc)182 ·--·used·0.00497191s·(cpu);·0.00570084s·(thread);·0s·(gc)
  
183 o15·=·3756</code></pre>183 o15·=·3756</code></pre>
184 </td>··········</tr>184 </td>··········</tr>
185 ··········<tr>185 ··········<tr>
186 <td>··············<pre><code·class="language-macaulay2">i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)186 <td>··············<pre><code·class="language-macaulay2">i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)
187 ·--·used·0.270534s·(cpu);·0.21608s·(thread);·0s·(gc)187 ·--·used·0.399843s·(cpu);·0.281503s·(thread);·0s·(gc)
  
188 o16·=·3756</code></pre>188 o16·=·3756</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ········</table>190 ········</table>
191 ········<div>191 ········<div>
192 ··········<p>The·valid·values·for·the·option·<span·class="tt">ContainmentTest</span>·are·<span·class="tt">FrobeniusPower</span>,·<span·class="tt">FrobeniusRoot</span>,·and·<span·class="tt">StandardPower</span>.·The·default·value·of·this·option·depends·on·what·is·passed·to·<span·class="tt">frobeniusNu</span>.·Indeed,·by·default,·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">FrobeniusRoot</span>·if·<span·class="tt">frobeniusNu</span>·is·passed·a·ring·element·$f$,·and·is·set·to·<span·class="tt">StandardPower</span>·if·<span·class="tt">frobeniusNu</span>·is·passed·an·ideal·$I$.·We·describe·the·consequences·of·setting·<span·class="tt">ContainmentTest</span>·to·each·of·these·values·below.</p>192 ··········<p>The·valid·values·for·the·option·<span·class="tt">ContainmentTest</span>·are·<span·class="tt">FrobeniusPower</span>,·<span·class="tt">FrobeniusRoot</span>,·and·<span·class="tt">StandardPower</span>.·The·default·value·of·this·option·depends·on·what·is·passed·to·<span·class="tt">frobeniusNu</span>.·Indeed,·by·default,·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">FrobeniusRoot</span>·if·<span·class="tt">frobeniusNu</span>·is·passed·a·ring·element·$f$,·and·is·set·to·<span·class="tt">StandardPower</span>·if·<span·class="tt">frobeniusNu</span>·is·passed·an·ideal·$I$.·We·describe·the·consequences·of·setting·<span·class="tt">ContainmentTest</span>·to·each·of·these·values·below.</p>
193 ··········<p>First,·if·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">StandardPower</span>,·then·the·ideal·containments·checked·when·computing·<span·class="tt">frobeniusNu(e,I,J)</span>·are·verified·directly.·That·is,·the·standard·power·$I^n$·is·first·computed,·and·a·check·is·then·run·to·see·if·it·is·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.</p>193 ··········<p>First,·if·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">StandardPower</span>,·then·the·ideal·containments·checked·when·computing·<span·class="tt">frobeniusNu(e,I,J)</span>·are·verified·directly.·That·is,·the·standard·power·$I^n$·is·first·computed,·and·a·check·is·then·run·to·see·if·it·is·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.</p>
Offset 200, 21 lines modifiedOffset 200, 21 lines modified
200 <td>··············<pre><code·class="language-macaulay2">i17·:·R·=·ZZ/5[x,y,z];</code></pre>200 <td>··············<pre><code·class="language-macaulay2">i17·:·R·=·ZZ/5[x,y,z];</code></pre>
201 </td>··········</tr>201 </td>··········</tr>
202 ··········<tr>202 ··········<tr>
203 <td>··············<pre><code·class="language-macaulay2">i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;</code></pre>203 <td>··············<pre><code·class="language-macaulay2">i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;</code></pre>
204 </td>··········</tr>204 </td>··········</tr>
205 ··········<tr>205 ··········<tr>
206 <td>··············<pre><code·class="language-macaulay2">i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by·default206 <td>··············<pre><code·class="language-macaulay2">i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by·default
207 ·--·used·0.265207s·(cpu);·0.186318s·(thread);·0s·(gc)207 ·--·used·0.301189s·(cpu);·0.192267s·(thread);·0s·(gc)
  
208 o19·=·499</code></pre>208 o19·=·499</code></pre>
209 </td>··········</tr>209 </td>··········</tr>
210 ··········<tr>210 ··········<tr>
211 <td>··············<pre><code·class="language-macaulay2">i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)211 <td>··············<pre><code·class="language-macaulay2">i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)
212 ·--·used·0.959131s·(cpu);·0.834749s·(thread);·0s·(gc)212 ·--·used·1.20809s·(cpu);·0.983564s·(thread);·0s·(gc)
  
213 o20·=·499</code></pre>213 o20·=·499</code></pre>
214 </td>··········</tr>214 </td>··········</tr>
215 ········</table>215 ········</table>
216 ········<div>216 ········<div>
217 ··········<p>Finally,·when·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">FrobeniusPower</span>,·then·instead·of·producing·the·invariant·$\nu_I^J(p^e)$·as·defined·above,·<span·class="tt">frobeniusNu</span>·instead·outputs·the·maximal·integer·$n$·such·that·the·$n$^{th}·(generalized)·Frobenius·power·of·$I$·is·not·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.·Here,·the·$n$^{th}·Frobenius·power·of·$I$,·when·$n$·is·a·nonnegative·integer,·is·as·defined·in·the·paper·<i>Frobenius·Powers</i>·by·Hernández,·Teixeira,·and·Witt,·which·can·be·computed·with·the·function·<a·title="compute·a·(generalized)·Frobenius·power·of·an·ideal"·href="../../TestIdeals/html/_frobenius__Power.html">frobeniusPower</a>,·from·the·<a·title="a·package·for·calculations·of·singularities·in·positive·characteristic·"·href="../../TestIdeals/html/index.html">TestIdeals</a>·package.·In·particular,·<span·class="tt">frobeniusNu(e,I,J)</span>·and·<span·class="tt">frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower)</span>·need·not·agree.·However,·they·will·agree·when·$I$·is·a·principal·ideal.</p>217 ··········<p>Finally,·when·<span·class="tt">ContainmentTest</span>·is·set·to·<span·class="tt">FrobeniusPower</span>,·then·instead·of·producing·the·invariant·$\nu_I^J(p^e)$·as·defined·above,·<span·class="tt">frobeniusNu</span>·instead·outputs·the·maximal·integer·$n$·such·that·the·$n$^{th}·(generalized)·Frobenius·power·of·$I$·is·not·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.·Here,·the·$n$^{th}·Frobenius·power·of·$I$,·when·$n$·is·a·nonnegative·integer,·is·as·defined·in·the·paper·<i>Frobenius·Powers</i>·by·Hernández,·Teixeira,·and·Witt,·which·can·be·computed·with·the·function·<a·title="compute·a·(generalized)·Frobenius·power·of·an·ideal"·href="../../TestIdeals/html/_frobenius__Power.html">frobeniusPower</a>,·from·the·<a·title="a·package·for·calculations·of·singularities·in·positive·characteristic·"·href="../../TestIdeals/html/index.html">TestIdeals</a>·package.·In·particular,·<span·class="tt">frobeniusNu(e,I,J)</span>·and·<span·class="tt">frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower)</span>·need·not·agree.·However,·they·will·agree·when·$I$·is·a·principal·ideal.</p>
218 ········</div>218 ········</div>
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246 <td>··············<pre><code·class="language-macaulay2">i25·:·R·=·ZZ/5[x,y,z];</code></pre>246 <td>··············<pre><code·class="language-macaulay2">i25·:·R·=·ZZ/5[x,y,z];</code></pre>
247 </td>··········</tr>247 </td>··········</tr>
248 ··········<tr>248 ··········<tr>
249 <td>··············<pre><code·class="language-macaulay2">i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;</code></pre>249 <td>··············<pre><code·class="language-macaulay2">i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;</code></pre>
250 </td>··········</tr>250 </td>··········</tr>
251 ··········<tr>251 ··········<tr>
252 <td>··············<pre><code·class="language-macaulay2">i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)252 <td>··············<pre><code·class="language-macaulay2">i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)
253 ·--·used·0.728426s·(cpu);·0.51059s·(thread);·0s·(gc)253 ·--·used·0.882056s·(cpu);·0.589107s·(thread);·0s·(gc)
  
254 o27·=·1124</code></pre>254 o27·=·1124</code></pre>
255 </td>··········</tr>255 </td>··········</tr>
256 ··········<tr>256 ··········<tr>
257 <td>··············<pre><code·class="language-macaulay2">i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)257 <td>··············<pre><code·class="language-macaulay2">i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)
258 ·--·used·1.06851s·(cpu);·0.77176s·(thread);·0s·(gc)258 ·--·used·1.38206s·(cpu);·1.00392s·(thread);·0s·(gc)
  
259 o28·=·1124</code></pre>259 o28·=·1124</code></pre>
260 </td>··········</tr>260 </td>··········</tr>
261 ··········<tr>261 ··········<tr>
262 <td>··············<pre><code·class="language-macaulay2">i29·:·M·=·ideal(x,·y,·z);262 <td>··············<pre><code·class="language-macaulay2">i29·:·M·=·ideal(x,·y,·z);
  
263 o29·:·Ideal·of·R</code></pre>263 o29·:·Ideal·of·R</code></pre>
264 </td>··········</tr>264 </td>··········</tr>
265 ··········<tr>265 ··········<tr>
266 <td>··············<pre><code·class="language-macaulay2">i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)266 <td>··············<pre><code·class="language-macaulay2">i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)
267 ·--·used·1.18947s·(cpu);·1.05905s·(thread);·0s·(gc)267 ·--·used·1.41601s·(cpu);·1.25292s·(thread);·0s·(gc)
  
268 o30·=·97</code></pre>268 o30·=·97</code></pre>
269 </td>··········</tr>269 </td>··········</tr>
270 ··········<tr>270 ··········<tr>
271 <td>··············<pre><code·class="language-macaulay2">i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets·luckier271 <td>··············<pre><code·class="language-macaulay2">i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets·luckier
272 ·--·used·0.444604s·(cpu);·0.388727s·(thread);·0s·(gc)272 ·--·used·0.521833s·(cpu);·0.443712s·(thread);·0s·(gc)
  
273 o31·=·97</code></pre>273 o31·=·97</code></pre>
274 </td>··········</tr>274 </td>··········</tr>
275 ········</table>275 ········</table>
276 ········<div>276 ········<div>
277 ··········<p>The·option·<span·class="tt">AtOrigin</span>·(default·value·<span·class="tt">true</span>)·can·be·turned·off·to·tell·<span·class="tt">frobeniusNu</span>·to·effectively·do·the·computation·over·all·possible·maximal·ideals·$J$·and·take·the·minimum.</p>277 ··········<p>The·option·<span·class="tt">AtOrigin</span>·(default·value·<span·class="tt">true</span>)·can·be·turned·off·to·tell·<span·class="tt">frobeniusNu</span>·to·effectively·do·the·computation·over·all·possible·maximal·ideals·$J$·and·take·the·minimum.</p>
278 ········</div>278 ········</div>
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107 algorithms,·namely·diagonal·polynomials,·binomials,·forms·in·two·variables,·and107 algorithms,·namely·diagonal·polynomials,·binomials,·forms·in·two·variables,·and
108 polynomials·whose·factors·are·in·simple·normal·crossing.·This·feature·can·be108 polynomials·whose·factors·are·in·simple·normal·crossing.·This·feature·can·be
109 disabled·by·setting·the·option·UseSpecialAlgorithms·(default·value·true)·to109 disabled·by·setting·the·option·UseSpecialAlgorithms·(default·value·true)·to
110 false.110 false.
111 i13·:·R·=·ZZ/17[x,y,z];111 i13·:·R·=·ZZ/17[x,y,z];
112 i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial112 i14·:·f·=·x^3·+·y^4·+·z^5;·--·a·diagonal·polynomial
113 i15·:·time·frobeniusNu(3,·f)113 i15·:·time·frobeniusNu(3,·f)
114 ·--·used·0.00397514s·(cpu);·0.00371911s·(thread);·0s·(gc)114 ·--·used·0.00497191s·(cpu);·0.00570084s·(thread);·0s·(gc)
  
115 o15·=·3756115 o15·=·3756
116 i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)116 i16·:·time·frobeniusNu(3,·f,·UseSpecialAlgorithms·=>·false)
117 ·--·used·0.270534s·(cpu);·0.21608s·(thread);·0s·(gc)117 ·--·used·0.399843s·(cpu);·0.281503s·(thread);·0s·(gc)
  
118 o16·=·3756118 o16·=·3756
119 The·valid·values·for·the·option·ContainmentTest·are·FrobeniusPower,119 The·valid·values·for·the·option·ContainmentTest·are·FrobeniusPower,
120 FrobeniusRoot,·and·StandardPower.·The·default·value·of·this·option·depends·on120 FrobeniusRoot,·and·StandardPower.·The·default·value·of·this·option·depends·on
121 what·is·passed·to·frobeniusNu.·Indeed,·by·default,·ContainmentTest·is·set·to121 what·is·passed·to·frobeniusNu.·Indeed,·by·default,·ContainmentTest·is·set·to
122 FrobeniusRoot·if·frobeniusNu·is·passed·a·ring·element·$f$,·and·is·set·to122 FrobeniusRoot·if·frobeniusNu·is·passed·a·ring·element·$f$,·and·is·set·to
123 StandardPower·if·frobeniusNu·is·passed·an·ideal·$I$.·We·describe·the123 StandardPower·if·frobeniusNu·is·passed·an·ideal·$I$.·We·describe·the
Offset 134, 19 lines modifiedOffset 134, 19 lines modified
134 is·contained·in·$J$.·The·output·is·unaffected,·but·this·option·often·speeds·up134 is·contained·in·$J$.·The·output·is·unaffected,·but·this·option·often·speeds·up
135 computations,·specially·when·a·polynomial·or·principal·ideal·is·passed·as·the135 computations,·specially·when·a·polynomial·or·principal·ideal·is·passed·as·the
136 second·argument.136 second·argument.
137 i17·:·R·=·ZZ/5[x,y,z];137 i17·:·R·=·ZZ/5[x,y,z];
138 i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;138 i18·:·f·=·x^3·+·y^3·+·z^3·+·x*y*z;
139 i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by139 i19·:·time·frobeniusNu(4,·f)·--·ContainmentTest·is·set·to·FrobeniusRoot,·by
140 default140 default
141 ·--·used·0.265207s·(cpu);·0.186318s·(thread);·0s·(gc)141 ·--·used·0.301189s·(cpu);·0.192267s·(thread);·0s·(gc)
  
142 o19·=·499142 o19·=·499
143 i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)143 i20·:·time·frobeniusNu(4,·f,·ContainmentTest·=>·StandardPower)
144 ·--·used·0.959131s·(cpu);·0.834749s·(thread);·0s·(gc)144 ·--·used·1.20809s·(cpu);·0.983564s·(thread);·0s·(gc)
  
145 o20·=·499145 o20·=·499
146 Finally,·when·ContainmentTest·is·set·to·FrobeniusPower,·then·instead·of146 Finally,·when·ContainmentTest·is·set·to·FrobeniusPower,·then·instead·of
147 producing·the·invariant·$\nu_I^J(p^e)$·as·defined·above,·frobeniusNu·instead147 producing·the·invariant·$\nu_I^J(p^e)$·as·defined·above,·frobeniusNu·instead
148 outputs·the·maximal·integer·$n$·such·that·the·$n$^{th}·(generalized)·Frobenius148 outputs·the·maximal·integer·$n$·such·that·the·$n$^{th}·(generalized)·Frobenius
149 power·of·$I$·is·not·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.·Here,·the149 power·of·$I$·is·not·contained·in·the·$p^e$-th·Frobenius·power·of·$J$.·Here,·the
150 $n$^{th}·Frobenius·power·of·$I$,·when·$n$·is·a·nonnegative·integer,·is·as150 $n$^{th}·Frobenius·power·of·$I$,·when·$n$·is·a·nonnegative·integer,·is·as
Offset 168, 31 lines modifiedOffset 168, 31 lines modified
168 The·function·frobeniusNu·works·by·searching·through·the·list·of·potential168 The·function·frobeniusNu·works·by·searching·through·the·list·of·potential
169 integers·$n$·and·checking·containments·of·$I^n$·in·the·specified·Frobenius169 integers·$n$·and·checking·containments·of·$I^n$·in·the·specified·Frobenius
170 power·of·$J$.·The·way·this·search·is·approached·is·specified·by·the·option170 power·of·$J$.·The·way·this·search·is·approached·is·specified·by·the·option
171 Search,·which·can·be·set·to·Binary·(the·default·value)·or·Linear.171 Search,·which·can·be·set·to·Binary·(the·default·value)·or·Linear.
172 i25·:·R·=·ZZ/5[x,y,z];172 i25·:·R·=·ZZ/5[x,y,z];
173 i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;173 i26·:·f·=·x^2*y^4·+·y^2*z^7·+·z^2*x^8;
174 i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)174 i27·:·time·frobeniusNu(5,·f)·--·uses·binary·search·(default)
175 ·--·used·0.728426s·(cpu);·0.51059s·(thread);·0s·(gc)175 ·--·used·0.882056s·(cpu);·0.589107s·(thread);·0s·(gc)
  
176 o27·=·1124176 o27·=·1124
177 i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)177 i28·:·time·frobeniusNu(5,·f,·Search·=>·Linear)
178 ·--·used·1.06851s·(cpu);·0.77176s·(thread);·0s·(gc)178 ·--·used·1.38206s·(cpu);·1.00392s·(thread);·0s·(gc)
  
179 o28·=·1124179 o28·=·1124
180 i29·:·M·=·ideal(x,·y,·z);180 i29·:·M·=·ideal(x,·y,·z);
  
181 o29·:·Ideal·of·R181 o29·:·Ideal·of·R
182 i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)182 i30·:·time·frobeniusNu(2,·M,·M^2)·--·uses·binary·search·(default)
183 ·--·used·1.18947s·(cpu);·1.05905s·(thread);·0s·(gc)183 ·--·used·1.41601s·(cpu);·1.25292s·(thread);·0s·(gc)
  
184 o30·=·97184 o30·=·97
185 i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets185 i31·:·time·frobeniusNu(2,·M,·M^2,·Search·=>·Linear)·--·but·linear·search·gets
186 luckier186 luckier
187 ·--·used·0.444604s·(cpu);·0.388727s·(thread);·0s·(gc)187 ·--·used·0.521833s·(cpu);·0.443712s·(thread);·0s·(gc)
  
188 o31·=·97188 o31·=·97
189 The·option·AtOrigin·(default·value·true)·can·be·turned·off·to·tell·frobeniusNu189 The·option·AtOrigin·(default·value·true)·can·be·turned·off·to·tell·frobeniusNu
190 to·effectively·do·the·computation·over·all·possible·maximal·ideals·$J$·and·take190 to·effectively·do·the·computation·over·all·possible·maximal·ideals·$J$·and·take
191 the·minimum.191 the·minimum.
192 i32·:·R·=·ZZ/7[x,y];192 i32·:·R·=·ZZ/7[x,y];
193 i33·:·f·=·(x·-·1)^3·-·(y·-·2)^2;193 i33·:·f·=·(x·-·1)^3·-·(y·-·2)^2;
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7 #:len=47 #:len=4
8 YXJjcw==8 YXJjcw==
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./usr/share/doc/Macaulay2/GKMVarieties/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out
    
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208 ······|·7/9··7/10·3/7·2/3·|208 ······|·7/9··7/10·3/7·2/3·|
209 ······|·7/10·7/3··5/2·1···|209 ······|·7/10·7/3··5/2·1···|
  
210 ···············3·······4210 ···············3·······4
211 o26·:·Matrix·QQ··<--·QQ211 o26·:·Matrix·QQ··<--·QQ
  
212 i27·:·time·C·=·orbitClosure(X,Mat)212 i27·:·time·C·=·orbitClosure(X,Mat)
213 ·--·used·30.5655s·(cpu);·3.94247s·(thread);·0s·(gc)213 ·--·used·27.2797s·(cpu);·3.44032s·(thread);·0s·(gc)
  
214 o27·=·an·"equivariant·K-class"·on·a·GKM·variety·214 o27·=·an·"equivariant·K-class"·on·a·GKM·variety·
  
215 o27·:·KClass215 o27·:·KClass
  
216 i28·:·time·C·=·orbitClosure(X,Mat,·RREFMethod·=>·true)216 i28·:·time·C·=·orbitClosure(X,Mat,·RREFMethod·=>·true)
217 ·--·used·33.3878s·(cpu);·4.26022s·(thread);·0s·(gc)217 ·--·used·28.3139s·(cpu);·3.85187s·(thread);·0s·(gc)
  
218 o28·=·an·"equivariant·K-class"·on·a·GKM·variety·218 o28·=·an·"equivariant·K-class"·on·a·GKM·variety·
  
219 o28·:·KClass219 o28·:·KClass
  
220 i29·:·220 i29·:·
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336 ······|·7/10·7/3··5/2·1···|336 ······|·7/10·7/3··5/2·1···|
  
337 ···············3·······4337 ···············3·······4
338 o26·:·Matrix·QQ··&lt;--·QQ</code></pre>338 o26·:·Matrix·QQ··&lt;--·QQ</code></pre>
339 </td>··········</tr>339 </td>··········</tr>
340 ··········<tr>340 ··········<tr>
341 <td>··············<pre><code·class="language-macaulay2">i27·:·time·C·=·orbitClosure(X,Mat)341 <td>··············<pre><code·class="language-macaulay2">i27·:·time·C·=·orbitClosure(X,Mat)
342 ·--·used·30.5655s·(cpu);·3.94247s·(thread);·0s·(gc)342 ·--·used·27.2797s·(cpu);·3.44032s·(thread);·0s·(gc)
  
343 o27·=·an·&quot;equivariant·K-class&quot;·on·a·GKM·variety·343 o27·=·an·&quot;equivariant·K-class&quot;·on·a·GKM·variety·
  
344 o27·:·KClass</code></pre>344 o27·:·KClass</code></pre>
345 </td>··········</tr>345 </td>··········</tr>
346 ··········<tr>346 ··········<tr>
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243 ······|······2····················2··············································2··2·····································································2·················2·········2··2···············2·······2··2··2·|243 ······|······2····················2··············································2··2·····································································2·················2·········2··2···············2·······2··2··2·|
244 ······|t··-·t···+·t··t··t···-·t··t···+·t··t··t···+·t··t··t···-·t··t··t··t···+·25t··t···+·4t··t··t···-·2t··t··t··t···-·2t··t··t··t···-·2t··t··t··t···+·t··t··t··t···+·t··t··t··t···-·3t··t···+·3t··t··t··t···-·26t··t··t··|244 ······|t··-·t···+·t··t··t···-·t··t···+·t··t··t···+·t··t··t···-·t··t··t··t···+·25t··t···+·4t··t··t···-·2t··t··t··t···-·2t··t··t··t···-·2t··t··t··t···+·t··t··t··t···+·t··t··t··t···-·3t··t···+·3t··t··t··t···-·26t··t··t··|
245 ······|·6····23····16·20·22····14·22····23·22·13····23·16·19····16·22·13·19······16·19·····23·20·21·····20·22·13·21·····16·20·19·21·····23·13·19·21····22·13·19·21····16·13·19·21·····20·21·····20·13·19·21······13·19·21|245 ······|·6····23····16·20·22····14·22····23·22·13····23·16·19····16·22·13·19······16·19·····23·20·21·····20·22·13·21·····16·20·19·21·····23·13·19·21····22·13·19·21····16·13·19·21·····20·21·····20·13·19·21······13·19·21|
246 ······+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+246 ······+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
  
247 i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0247 i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
248 o12·=·|·48·33·19·-49·36·14·-34·2·-27·-30·-11·21·19·19·42·-10·24·-36·-29·33·-8248 o12·=·|·-36·-30·42·21·-22·23·23·16·-7·-30·-11·-40·19·19·-41·-22·24·-36·-8·33
249 ······-----------------------------------------------------------------------249 ······-----------------------------------------------------------------------
250 ······-22·-29·-29·|250 ······-29·-10·-29·-29·|
  
251 ···············1·······24251 ···············1·······24
252 o12·:·Matrix·kk··<--·kk252 o12·:·Matrix·kk··<--·kk
  
253 i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1253 i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
254 o13·=·|·20·-50·18·-20·-48·-11·3·50·-19·38·20·49·26·39·22·-27·-24·-38·21·34·19254 o13·=·|·24·-7·-25·32·-28·13·-49·11·-17·-17·-2·-50·30·19·-41·-21·-38·-24·21·39
255 ······-----------------------------------------------------------------------255 ······-----------------------------------------------------------------------
256 ······0·-16·-47·|256 ······34·0·-16·-47·|
  
257 ···············1·······24257 ···············1·······24
258 o13·:·Matrix·kk··<--·kk258 o13·:·Matrix·kk··<--·kk
  
259 i14·:·F1·=·sub(F,·(vars·S)|pt1)259 i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
260 ··············2··············2······························2···············260 ··············2·············2······························2···············
261 o14·=·ideal·(a··-·27b*c·-·49c··-·30a*d·+·36b*d·+·33c*d·+·48d·,·a*b·+·42b*c·-261 o14·=·ideal·(a··-·7b*c·+·21c··-·30a*d·-·22b*d·-·30c*d·-·36d·,·a*b·-·41b*c·-
262 ······-----------------------------------------------------------------------262 ······-----------------------------------------------------------------------
263 ·········2······························2···2·············2··················263 ·········2······························2···2··············2················
264 ······11c··-·10a*d·+·21b*d·+·14c*d·+·19d·,·b··-·8b*c·+·24c··-·22a*d·-·36b*d·+264 ······11c··-·22a*d·-·40b*d·+·23c*d·+·42d·,·b··-·29b*c·+·24c··-·10a*d·-·36b*d
265 ······-----------------------------------------------------------------------265 ······-----------------------------------------------------------------------
266 ·················2···················2·····························2266 ···················2··················2······························2
267 ······19c*d·-·34d·,·a*c·-·29b*c·-·29c··-·29a*d·+·33b*d·+·19c*d·+·2d·)267 ······+·19c*d·+·23d·,·a*c·-·29b*c·-·8c··-·29a*d·+·33b*d·+·19c*d·+·16d·)
  
268 o14·:·Ideal·of·S268 o14·:·Ideal·of·S
  
269 i15·:·F2·=·sub(F,·(vars·S)|pt2)269 i15·:·F2·=·sub(F,·(vars·S)|pt2)
  
270 ··············2··············2······························2···············270 ··············2··············2·····························2···············
271 o15·=·ideal·(a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·,·a*b·+·22b*c·+271 o15·=·ideal·(a··-·17b*c·+·32c··-·17a*d·-·28b*d·-·7c*d·+·24d·,·a*b·-·41b*c·-
272 ······-----------------------------------------------------------------------272 ······-----------------------------------------------------------------------
273 ·········2······························2···2··············2················273 ········2······························2···2··············2··················
274 ······20c··-·27a*d·+·49b*d·-·11c*d·+·18d·,·b··+·19b*c·-·24c··-·38b*d·+·26c*d274 ······2c··-·21a*d·-·50b*d·+·13c*d·-·25d·,·b··+·34b*c·-·38c··-·24b*d·+·30c*d·-
275 ······-----------------------------------------------------------------------275 ······-----------------------------------------------------------------------
276 ··········2···················2······························2276 ·········2···················2······························2
277 ······+·3d·,·a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·)277 ······49d·,·a*c·-·16b*c·+·21c··-·47a*d·+·39b*d·+·19c*d·+·11d·)
  
278 o15·:·Ideal·of·S278 o15·:·Ideal·of·S
  
279 i16·:·decompose·F1279 i16·:·decompose·F1
  
280 ····································2·············2······················2280 ···································2··············2······················2
281 o16·=·{ideal·(a·-·29b·-·29c·-·14d,·b··-·8b*c·+·24c··+·33b*d·-·13c*d·-·39d·),281 o16·=·{ideal·(a·-·29b·-·8c·-·11d,·b··-·29b*c·+·24c··-·23b*d·+·40c*d·+·14d·),
282 ······-----------------------------------------------------------------------282 ······-----------------------------------------------------------------------
283 ······ideal·(c·-·29d,·b·+·22d,·a·+·4d)}283 ······ideal·(c·-·29d,·b·-·35d,·a·+·43d)}
  
284 o16·:·List284 o16·:·List
  
285 i17·:·decompose·F2285 i17·:·decompose·F2
  
 286 o17·=·{ideal·(b·-·6c·-·42d,·a·+·26c·-·5d),·ideal·(b·+·40c·+·18d,·a·-·46c·+
286 ·······························2······························2···2·········· 
287 o17·=·{ideal·(a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·,·b··+·19b*c·- 
288 ······----------------------------------------------------------------------- 
289 ·········2·····················2···················2························ 
290 ······24c··-·38b*d·+·26c*d·+·3d·,·a*b·+·22b*c·+·20c··-·27a*d·+·49b*d·-·11c*d 
291 ······-----------------------------------------------------------------------287 ······-----------------------------------------------------------------------
 288 ······19d)}
292 ···········2···2··············2······························2 
293 ······+·18d·,·a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·)} 
  
294 o17·:·List289 o17·:·List
  
295 i18·:·290 i18·:·
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Offset 12, 15 lines modifiedOffset 12, 15 lines modified
12 ·············································212 ·············································2
13 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)13 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)
  
14 o3·:·Ideal·of·S14 o3·:·Ideal·of·S
  
15 i4·:·pt·=·randomPointOnRationalVariety·I15 i4·:·pt·=·randomPointOnRationalVariety·I
  
16 o4·=·|·-25·20·-30·-16·24·-36·|16 o4·=·|·1·49·24·-23·-36·-30·|
  
17 ··············1·······617 ··············1·······6
18 o4·:·Matrix·kk··<--·kk18 o4·:·Matrix·kk··<--·kk
  
19 i5·:·sub(I,·pt)·==·019 i5·:·sub(I,·pt)·==·0
  
20 o5·=·true20 o5·=·true
Offset 200, 67 lines modifiedOffset 200, 67 lines modified
  
200 o12·=·{11,·8}200 o12·=·{11,·8}
  
201 o12·:·List201 o12·:·List
  
202 i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0202 i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
203 o13·=·|·2·-18·6·4·20·-11·44·-13·0·19·11·-47·-29·-8·-19·-22·19·-20·-29·-38·-24203 o13·=·|·50·15·46·-33·2·-43·-46·8·33·19·-2·-18·-8·-22·43·-29·19·3·-16·-29·-38
204 ······-----------------------------------------------------------------------204 ······-----------------------------------------------------------------------
205 ······-16·-10·-29·|205 ······-24·-10·-29·|
  
206 ···············1·······24206 ···············1·······24
207 o13·:·Matrix·kk··<--·kk207 o13·:·Matrix·kk··<--·kk
  
208 i14·:·F1·=·sub(F,·(vars·S)|pt1)208 i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
209 ··············2·····2·····························2···················2··209 ··············2··············2·····························2···············
210 o14·=·ideal·(a··+·4c··+·19a*d·+·20b*d·-·18c*d·+·2d·,·a*b·-·19b*c·+·11c··-210 o14·=·ideal·(a··+·33b*c·-·33c··+·19a*d·+·2b*d·+·15c*d·+·50d·,·a*b·+·43b*c·-
211 ······-----------------------------------------------------------------------211 ······-----------------------------------------------------------------------
212 ································2···················2························212 ········2······························2···················2·················
213 ······22a*d·-·47b*d·-·11c*d·+·6d·,·a*c·-·24b*c·+·19c··-·16a*d·-·20b*d·-·29c*d213 ······2c··-·29a*d·-·18b*d·-·43c*d·+·46d·,·a*c·-·38b*c·+·19c··-·24a*d·+·3b*d·-
214 ······-----------------------------------------------------------------------214 ······-----------------------------------------------------------------------
215 ···········2···2··············2·····························2215 ················2···2··············2·····························2
216 ······+·44d·,·b··-·10b*c·-·29c··-·29a*d·-·38b*d·-·8c*d·-·13d·)216 ······8c*d·-·46d·,·b··-·10b*c·-·16c··-·29a*d·-·29b*d·-·22c*d·+·8d·)
  
217 o14·:·Ideal·of·S217 o14·:·Ideal·of·S
  
218 i15·:·decompose·F1218 i15·:·decompose·F1
  
219 ····································2··············2······················2219 ····································2··············2······················2
220 o15·=·{ideal·(a·-·24b·+·19c·-·28d,·b··-·10b*c·-·29c··-·27b*d·+·38c*d·-·17d·),220 o15·=·{ideal·(a·-·38b·+·19c·+·44d,·b··-·10b*c·-·16c··-·20b*d·+·24c*d·-·29d·),
221 ······-----------------------------------------------------------------------221 ······-----------------------------------------------------------------------
222 ······ideal·(c·-·16d,·b·-·33d,·a·-·32d)}222 ······ideal·(c·-·24d,·b·-·38d,·a·+·15d)}
  
223 o15·:·List223 o15·:·List
  
224 i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1224 i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
225 o16·=·|·-8·28·25·30·-12·-42·-20·-35·-7·-37·-5·45·19·20·-24·34·39·21·-47·-39225 o16·=·|·-14·40·-5·26·-48·-26·-35·41·-8·-15·-38·31·-13·29·21·16·39·21·-18·19
226 ······-----------------------------------------------------------------------226 ······-----------------------------------------------------------------------
227 ······-13·-18·34·0·|227 ······-47·-39·34·0·|
  
228 ···············1·······24228 ···············1·······24
229 o16·:·Matrix·kk··<--·kk229 o16·:·Matrix·kk··<--·kk
  
230 i17·:·F2·=·sub(F,·(vars·S)|pt2)230 i17·:·F2·=·sub(F,·(vars·S)|pt2)
  
231 ··············2·············2·····························2···············231 ··············2·············2······························2···············
232 o17·=·ideal·(a··-·7b*c·+·30c··-·37a*d·-·12b*d·+·28c*d·-·8d·,·a*b·-·24b*c·-232 o17·=·ideal·(a··-·8b*c·+·26c··-·15a*d·-·48b*d·+·40c*d·-·14d·,·a*b·+·21b*c·-
233 ······-----------------------------------------------------------------------233 ······-----------------------------------------------------------------------
234 ········2······························2···················2················234 ·········2·····························2···················2················
235 ······5c··+·34a*d·+·45b*d·-·42c*d·+·25d·,·a*c·-·13b*c·+·39c··-·18a*d·+·21b*d235 ······38c··+·16a*d·+·31b*d·-·26c*d·-·5d·,·a*c·-·47b*c·+·39c··-·39a*d·+·21b*d
236 ······-----------------------------------------------------------------------236 ······-----------------------------------------------------------------------
237 ···················2···2··············2······················2237 ···················2···2··············2······················2
238 ······+·19c*d·-·20d·,·b··+·34b*c·-·47c··-·39b*d·+·20c*d·-·35d·)238 ······-·13c*d·-·35d·,·b··+·34b*c·-·18c··+·19b*d·+·29c*d·+·41d·)
  
239 o17·:·Ideal·of·S239 o17·:·Ideal·of·S
  
240 i18·:·decompose·F2240 i18·:·decompose·F2
  
241 o18·=·{ideal·(b·-·12c·+·48d,·a·-·16c·+·d),·ideal·(b·+·46c·+·14d,·a·+·31c·-241 o18·=·{ideal·(b·+·19c·-·18d,·a·+·23c·+·43d),·ideal·(b·+·15c·+·37d,·a·+·37c·+
242 ······-----------------------------------------------------------------------242 ······-----------------------------------------------------------------------
243 ······5d)}243 ······26d)}
  
244 o18·:·List244 o18·:·List
  
245 i19·:·245 i19·:·
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90 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)90 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)
  
91 o3·:·Ideal·of·S</code></pre>91 o3·:·Ideal·of·S</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·pt·=·randomPointOnRationalVariety·I94 <td>··············<pre><code·class="language-macaulay2">i4·:·pt·=·randomPointOnRationalVariety·I
  
95 o4·=·|·-25·20·-30·-16·24·-36·|95 o4·=·|·1·49·24·-23·-36·-30·|
  
96 ··············1·······696 ··············1·······6
97 o4·:·Matrix·kk··&lt;--·kk</code></pre>97 o4·:·Matrix·kk··&lt;--·kk</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i5·:·sub(I,·pt)·==·0100 <td>··············<pre><code·class="language-macaulay2">i5·:·sub(I,·pt)·==·0
  
Offset 294, 80 lines modifiedOffset 294, 80 lines modified
294 ········<div>294 ········<div>
295 ··········<p>There·are·2·components.··We·attempt·to·find·a·point·on·the·first·component</p>295 ··········<p>There·are·2·components.··We·attempt·to·find·a·point·on·the·first·component</p>
296 ········</div>296 ········</div>
297 ········<table·class="examples">297 ········<table·class="examples">
298 ··········<tr>298 ··········<tr>
299 <td>··············<pre><code·class="language-macaulay2">i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0299 <td>··············<pre><code·class="language-macaulay2">i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
300 o13·=·|·2·-18·6·4·20·-11·44·-13·0·19·11·-47·-29·-8·-19·-22·19·-20·-29·-38·-24300 o13·=·|·50·15·46·-33·2·-43·-46·8·33·19·-2·-18·-8·-22·43·-29·19·3·-16·-29·-38
301 ······-----------------------------------------------------------------------301 ······-----------------------------------------------------------------------
302 ······-16·-10·-29·|302 ······-24·-10·-29·|
  
303 ···············1·······24303 ···············1·······24
304 o13·:·Matrix·kk··&lt;--·kk</code></pre>304 o13·:·Matrix·kk··&lt;--·kk</code></pre>
305 </td>··········</tr>305 </td>··········</tr>
306 ··········<tr>306 ··········<tr>
307 <td>··············<pre><code·class="language-macaulay2">i14·:·F1·=·sub(F,·(vars·S)|pt1)307 <td>··············<pre><code·class="language-macaulay2">i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
308 ··············2·····2·····························2···················2··308 ··············2··············2·····························2···············
309 o14·=·ideal·(a··+·4c··+·19a*d·+·20b*d·-·18c*d·+·2d·,·a*b·-·19b*c·+·11c··-309 o14·=·ideal·(a··+·33b*c·-·33c··+·19a*d·+·2b*d·+·15c*d·+·50d·,·a*b·+·43b*c·-
310 ······-----------------------------------------------------------------------310 ······-----------------------------------------------------------------------
311 ································2···················2························311 ········2······························2···················2·················
312 ······22a*d·-·47b*d·-·11c*d·+·6d·,·a*c·-·24b*c·+·19c··-·16a*d·-·20b*d·-·29c*d312 ······2c··-·29a*d·-·18b*d·-·43c*d·+·46d·,·a*c·-·38b*c·+·19c··-·24a*d·+·3b*d·-
313 ······-----------------------------------------------------------------------313 ······-----------------------------------------------------------------------
314 ···········2···2··············2·····························2314 ················2···2··············2·····························2
315 ······+·44d·,·b··-·10b*c·-·29c··-·29a*d·-·38b*d·-·8c*d·-·13d·)315 ······8c*d·-·46d·,·b··-·10b*c·-·16c··-·29a*d·-·29b*d·-·22c*d·+·8d·)
  
316 o14·:·Ideal·of·S</code></pre>316 o14·:·Ideal·of·S</code></pre>
317 </td>··········</tr>317 </td>··········</tr>
318 ··········<tr>318 ··········<tr>
319 <td>··············<pre><code·class="language-macaulay2">i15·:·decompose·F1319 <td>··············<pre><code·class="language-macaulay2">i15·:·decompose·F1
  
320 ····································2··············2······················2320 ····································2··············2······················2
321 o15·=·{ideal·(a·-·24b·+·19c·-·28d,·b··-·10b*c·-·29c··-·27b*d·+·38c*d·-·17d·),321 o15·=·{ideal·(a·-·38b·+·19c·+·44d,·b··-·10b*c·-·16c··-·20b*d·+·24c*d·-·29d·),
322 ······-----------------------------------------------------------------------322 ······-----------------------------------------------------------------------
323 ······ideal·(c·-·16d,·b·-·33d,·a·-·32d)}323 ······ideal·(c·-·24d,·b·-·38d,·a·+·15d)}
  
324 o15·:·List</code></pre>324 o15·:·List</code></pre>
325 </td>··········</tr>325 </td>··········</tr>
326 ········</table>326 ········</table>
327 ········<div>327 ········<div>
328 ··········<p>We·attempt·to·find·a·point·on·the·second·component·in·parameter·space,·and·its·corresponding·ideal.</p>328 ··········<p>We·attempt·to·find·a·point·on·the·second·component·in·parameter·space,·and·its·corresponding·ideal.</p>
329 ········</div>329 ········</div>
330 ········<table·class="examples">330 ········<table·class="examples">
331 ··········<tr>331 ··········<tr>
332 <td>··············<pre><code·class="language-macaulay2">i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1332 <td>··············<pre><code·class="language-macaulay2">i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
333 o16·=·|·-8·28·25·30·-12·-42·-20·-35·-7·-37·-5·45·19·20·-24·34·39·21·-47·-39333 o16·=·|·-14·40·-5·26·-48·-26·-35·41·-8·-15·-38·31·-13·29·21·16·39·21·-18·19
334 ······-----------------------------------------------------------------------334 ······-----------------------------------------------------------------------
335 ······-13·-18·34·0·|335 ······-47·-39·34·0·|
  
336 ···············1·······24336 ···············1·······24
337 o16·:·Matrix·kk··&lt;--·kk</code></pre>337 o16·:·Matrix·kk··&lt;--·kk</code></pre>
338 </td>··········</tr>338 </td>··········</tr>
339 ··········<tr>339 ··········<tr>
340 <td>··············<pre><code·class="language-macaulay2">i17·:·F2·=·sub(F,·(vars·S)|pt2)340 <td>··············<pre><code·class="language-macaulay2">i17·:·F2·=·sub(F,·(vars·S)|pt2)
  
341 ··············2·············2·····························2···············341 ··············2·············2······························2···············
342 o17·=·ideal·(a··-·7b*c·+·30c··-·37a*d·-·12b*d·+·28c*d·-·8d·,·a*b·-·24b*c·-342 o17·=·ideal·(a··-·8b*c·+·26c··-·15a*d·-·48b*d·+·40c*d·-·14d·,·a*b·+·21b*c·-
343 ······-----------------------------------------------------------------------343 ······-----------------------------------------------------------------------
344 ········2······························2···················2················344 ·········2·····························2···················2················
345 ······5c··+·34a*d·+·45b*d·-·42c*d·+·25d·,·a*c·-·13b*c·+·39c··-·18a*d·+·21b*d345 ······38c··+·16a*d·+·31b*d·-·26c*d·-·5d·,·a*c·-·47b*c·+·39c··-·39a*d·+·21b*d
346 ······-----------------------------------------------------------------------346 ······-----------------------------------------------------------------------
347 ···················2···2··············2······················2347 ···················2···2··············2······················2
348 ······+·19c*d·-·20d·,·b··+·34b*c·-·47c··-·39b*d·+·20c*d·-·35d·)348 ······-·13c*d·-·35d·,·b··+·34b*c·-·18c··+·19b*d·+·29c*d·+·41d·)
  
349 o17·:·Ideal·of·S</code></pre>349 o17·:·Ideal·of·S</code></pre>
350 </td>··········</tr>350 </td>··········</tr>
351 ··········<tr>351 ··········<tr>
352 <td>··············<pre><code·class="language-macaulay2">i18·:·decompose·F2352 <td>··············<pre><code·class="language-macaulay2">i18·:·decompose·F2
  
353 o18·=·{ideal·(b·-·12c·+·48d,·a·-·16c·+·d),·ideal·(b·+·46c·+·14d,·a·+·31c·-353 o18·=·{ideal·(b·+·19c·-·18d,·a·+·23c·+·43d),·ideal·(b·+·15c·+·37d,·a·+·37c·+
354 ······-----------------------------------------------------------------------354 ······-----------------------------------------------------------------------
355 ······5d)}355 ······26d)}
  
356 o18·:·List</code></pre>356 o18·:·List</code></pre>
357 </td>··········</tr>357 </td>··········</tr>
358 ········</table>358 ········</table>
359 ········<div>359 ········<div>
360 ··········<p>It·turns·out·that·this·is·the·ideal·of·2·skew·lines,·just·not·defined·over·this·field.</p>360 ··········<p>It·turns·out·that·this·is·the·ideal·of·2·skew·lines,·just·not·defined·over·this·field.</p>
361 ········</div>361 ········</div>
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29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
30 ·············································230 ·············································2
31 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)31 ·····c*e·+·b*f,·-·c*d·+·b*e,·-·c*e·+·b*f,·-·e··+·d*f)
  
32 o3·:·Ideal·of·S32 o3·:·Ideal·of·S
33 i4·:·pt·=·randomPointOnRationalVariety·I33 i4·:·pt·=·randomPointOnRationalVariety·I
  
34 o4·=·|·-25·20·-30·-16·24·-36·|34 o4·=·|·1·49·24·-23·-36·-30·|
  
35 ··············1·······635 ··············1·······6
36 o4·:·Matrix·kk··<--·kk36 o4·:·Matrix·kk··<--·kk
37 i5·:·sub(I,·pt)·==·037 i5·:·sub(I,·pt)·==·0
  
38 o5·=·true38 o5·=·true
39 i6·:·S·=·kk[a..d];39 i6·:·S·=·kk[a..d];
Offset 213, 67 lines modifiedOffset 213, 67 lines modified
  
213 o12·=·{11,·8}213 o12·=·{11,·8}
  
214 o12·:·List214 o12·:·List
215 There·are·2·components.·We·attempt·to·find·a·point·on·the·first·component215 There·are·2·components.·We·attempt·to·find·a·point·on·the·first·component
216 i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0216 i13·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
217 o13·=·|·2·-18·6·4·20·-11·44·-13·0·19·11·-47·-29·-8·-19·-22·19·-20·-29·-38·-24217 o13·=·|·50·15·46·-33·2·-43·-46·8·33·19·-2·-18·-8·-22·43·-29·19·3·-16·-29·-38
218 ······-----------------------------------------------------------------------218 ······-----------------------------------------------------------------------
219 ······-16·-10·-29·|219 ······-24·-10·-29·|
  
220 ···············1·······24220 ···············1·······24
221 o13·:·Matrix·kk··<--·kk221 o13·:·Matrix·kk··<--·kk
222 i14·:·F1·=·sub(F,·(vars·S)|pt1)222 i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
223 ··············2·····2·····························2···················2223 ··············2··············2·····························2
224 o14·=·ideal·(a··+·4c··+·19a*d·+·20b*d·-·18c*d·+·2d·,·a*b·-·19b*c·+·11c··-224 o14·=·ideal·(a··+·33b*c·-·33c··+·19a*d·+·2b*d·+·15c*d·+·50d·,·a*b·+·43b*c·-
225 ······-----------------------------------------------------------------------225 ······-----------------------------------------------------------------------
226 ································2···················2226 ········2······························2···················2
227 ······22a*d·-·47b*d·-·11c*d·+·6d·,·a*c·-·24b*c·+·19c··-·16a*d·-·20b*d·-·29c*d227 ······2c··-·29a*d·-·18b*d·-·43c*d·+·46d·,·a*c·-·38b*c·+·19c··-·24a*d·+·3b*d·-
228 ······-----------------------------------------------------------------------228 ······-----------------------------------------------------------------------
229 ···········2···2··············2·····························2229 ················2···2··············2·····························2
230 ······+·44d·,·b··-·10b*c·-·29c··-·29a*d·-·38b*d·-·8c*d·-·13d·)230 ······8c*d·-·46d·,·b··-·10b*c·-·16c··-·29a*d·-·29b*d·-·22c*d·+·8d·)
  
231 o14·:·Ideal·of·S231 o14·:·Ideal·of·S
232 i15·:·decompose·F1232 i15·:·decompose·F1
  
233 ····································2··············2······················2233 ····································2··············2······················2
234 o15·=·{ideal·(a·-·24b·+·19c·-·28d,·b··-·10b*c·-·29c··-·27b*d·+·38c*d·-·17d·),234 o15·=·{ideal·(a·-·38b·+·19c·+·44d,·b··-·10b*c·-·16c··-·20b*d·+·24c*d·-·29d·),
235 ······-----------------------------------------------------------------------235 ······-----------------------------------------------------------------------
236 ······ideal·(c·-·16d,·b·-·33d,·a·-·32d)}236 ······ideal·(c·-·24d,·b·-·38d,·a·+·15d)}
  
237 o15·:·List237 o15·:·List
238 We·attempt·to·find·a·point·on·the·second·component·in·parameter·space,·and·its238 We·attempt·to·find·a·point·on·the·second·component·in·parameter·space,·and·its
239 corresponding·ideal.239 corresponding·ideal.
240 i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1240 i16·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
241 o16·=·|·-8·28·25·30·-12·-42·-20·-35·-7·-37·-5·45·19·20·-24·34·39·21·-47·-39241 o16·=·|·-14·40·-5·26·-48·-26·-35·41·-8·-15·-38·31·-13·29·21·16·39·21·-18·19
242 ······-----------------------------------------------------------------------242 ······-----------------------------------------------------------------------
243 ······-13·-18·34·0·|243 ······-47·-39·34·0·|
  
244 ···············1·······24244 ···············1·······24
245 o16·:·Matrix·kk··<--·kk245 o16·:·Matrix·kk··<--·kk
246 i17·:·F2·=·sub(F,·(vars·S)|pt2)246 i17·:·F2·=·sub(F,·(vars·S)|pt2)
  
247 ··············2·············2·····························2247 ··············2·············2······························2
248 o17·=·ideal·(a··-·7b*c·+·30c··-·37a*d·-·12b*d·+·28c*d·-·8d·,·a*b·-·24b*c·-248 o17·=·ideal·(a··-·8b*c·+·26c··-·15a*d·-·48b*d·+·40c*d·-·14d·,·a*b·+·21b*c·-
249 ······-----------------------------------------------------------------------249 ······-----------------------------------------------------------------------
250 ········2······························2···················2250 ·········2·····························2···················2
251 ······5c··+·34a*d·+·45b*d·-·42c*d·+·25d·,·a*c·-·13b*c·+·39c··-·18a*d·+·21b*d251 ······38c··+·16a*d·+·31b*d·-·26c*d·-·5d·,·a*c·-·47b*c·+·39c··-·39a*d·+·21b*d
252 ······-----------------------------------------------------------------------252 ······-----------------------------------------------------------------------
253 ···················2···2··············2······················2253 ···················2···2··············2······················2
254 ······+·19c*d·-·20d·,·b··+·34b*c·-·47c··-·39b*d·+·20c*d·-·35d·)254 ······-·13c*d·-·35d·,·b··+·34b*c·-·18c··+·19b*d·+·29c*d·+·41d·)
  
255 o17·:·Ideal·of·S255 o17·:·Ideal·of·S
256 i18·:·decompose·F2256 i18·:·decompose·F2
  
257 o18·=·{ideal·(b·-·12c·+·48d,·a·-·16c·+·d),·ideal·(b·+·46c·+·14d,·a·+·31c·-257 o18·=·{ideal·(b·+·19c·-·18d,·a·+·23c·+·43d),·ideal·(b·+·15c·+·37d,·a·+·37c·+
258 ······-----------------------------------------------------------------------258 ······-----------------------------------------------------------------------
259 ······5d)}259 ······26d)}
  
260 o18·:·List260 o18·:·List
261 It·turns·out·that·this·is·the·ideal·of·2·skew·lines,·just·not·defined·over·this261 It·turns·out·that·this·is·the·ideal·of·2·skew·lines,·just·not·defined·over·this
262 field.262 field.
263 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*263 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
264 This·routine·expects·the·input·to·represent·an·irreducible·variety264 This·routine·expects·the·input·to·represent·an·irreducible·variety
265 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*265 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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320 ········<div>320 ········<div>
321 ··········<p>We·can·find·random·points·on·each·component,·since·these·components·are·rational.</p>321 ··········<p>We·can·find·random·points·on·each·component,·since·these·components·are·rational.</p>
322 ········</div>322 ········</div>
323 ········<table·class="examples">323 ········<table·class="examples">
324 ··········<tr>324 ··········<tr>
325 <td>··············<pre><code·class="language-macaulay2">i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0325 <td>··············<pre><code·class="language-macaulay2">i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
326 o12·=·|·48·33·19·-49·36·14·-34·2·-27·-30·-11·21·19·19·42·-10·24·-36·-29·33·-8326 o12·=·|·-36·-30·42·21·-22·23·23·16·-7·-30·-11·-40·19·19·-41·-22·24·-36·-8·33
327 ······-----------------------------------------------------------------------327 ······-----------------------------------------------------------------------
328 ······-22·-29·-29·|328 ······-29·-10·-29·-29·|
  
329 ···············1·······24329 ···············1·······24
330 o12·:·Matrix·kk··&lt;--·kk</code></pre>330 o12·:·Matrix·kk··&lt;--·kk</code></pre>
331 </td>··········</tr>331 </td>··········</tr>
332 ··········<tr>332 ··········<tr>
333 <td>··············<pre><code·class="language-macaulay2">i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1333 <td>··············<pre><code·class="language-macaulay2">i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
334 o13·=·|·20·-50·18·-20·-48·-11·3·50·-19·38·20·49·26·39·22·-27·-24·-38·21·34·19334 o13·=·|·24·-7·-25·32·-28·13·-49·11·-17·-17·-2·-50·30·19·-41·-21·-38·-24·21·39
335 ······-----------------------------------------------------------------------335 ······-----------------------------------------------------------------------
336 ······0·-16·-47·|336 ······34·0·-16·-47·|
  
337 ···············1·······24337 ···············1·······24
338 o13·:·Matrix·kk··&lt;--·kk</code></pre>338 o13·:·Matrix·kk··&lt;--·kk</code></pre>
339 </td>··········</tr>339 </td>··········</tr>
340 ··········<tr>340 ··········<tr>
341 <td>··············<pre><code·class="language-macaulay2">i14·:·F1·=·sub(F,·(vars·S)|pt1)341 <td>··············<pre><code·class="language-macaulay2">i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
342 ··············2··············2······························2···············342 ··············2·············2······························2···············
343 o14·=·ideal·(a··-·27b*c·-·49c··-·30a*d·+·36b*d·+·33c*d·+·48d·,·a*b·+·42b*c·-343 o14·=·ideal·(a··-·7b*c·+·21c··-·30a*d·-·22b*d·-·30c*d·-·36d·,·a*b·-·41b*c·-
344 ······-----------------------------------------------------------------------344 ······-----------------------------------------------------------------------
345 ·········2······························2···2·············2··················345 ·········2······························2···2··············2················
346 ······11c··-·10a*d·+·21b*d·+·14c*d·+·19d·,·b··-·8b*c·+·24c··-·22a*d·-·36b*d·+346 ······11c··-·22a*d·-·40b*d·+·23c*d·+·42d·,·b··-·29b*c·+·24c··-·10a*d·-·36b*d
347 ······-----------------------------------------------------------------------347 ······-----------------------------------------------------------------------
348 ·················2···················2·····························2348 ···················2··················2······························2
349 ······19c*d·-·34d·,·a*c·-·29b*c·-·29c··-·29a*d·+·33b*d·+·19c*d·+·2d·)349 ······+·19c*d·+·23d·,·a*c·-·29b*c·-·8c··-·29a*d·+·33b*d·+·19c*d·+·16d·)
  
350 o14·:·Ideal·of·S</code></pre>350 o14·:·Ideal·of·S</code></pre>
351 </td>··········</tr>351 </td>··········</tr>
352 ··········<tr>352 ··········<tr>
353 <td>··············<pre><code·class="language-macaulay2">i15·:·F2·=·sub(F,·(vars·S)|pt2)353 <td>··············<pre><code·class="language-macaulay2">i15·:·F2·=·sub(F,·(vars·S)|pt2)
  
354 ··············2··············2······························2···············354 ··············2··············2·····························2···············
355 o15·=·ideal·(a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·,·a*b·+·22b*c·+355 o15·=·ideal·(a··-·17b*c·+·32c··-·17a*d·-·28b*d·-·7c*d·+·24d·,·a*b·-·41b*c·-
356 ······-----------------------------------------------------------------------356 ······-----------------------------------------------------------------------
357 ·········2······························2···2··············2················357 ········2······························2···2··············2··················
358 ······20c··-·27a*d·+·49b*d·-·11c*d·+·18d·,·b··+·19b*c·-·24c··-·38b*d·+·26c*d358 ······2c··-·21a*d·-·50b*d·+·13c*d·-·25d·,·b··+·34b*c·-·38c··-·24b*d·+·30c*d·-
359 ······-----------------------------------------------------------------------359 ······-----------------------------------------------------------------------
360 ··········2···················2······························2360 ·········2···················2······························2
361 ······+·3d·,·a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·)361 ······49d·,·a*c·-·16b*c·+·21c··-·47a*d·+·39b*d·+·19c*d·+·11d·)
  
362 o15·:·Ideal·of·S</code></pre>362 o15·:·Ideal·of·S</code></pre>
363 </td>··········</tr>363 </td>··········</tr>
364 ··········<tr>364 ··········<tr>
365 <td>··············<pre><code·class="language-macaulay2">i16·:·decompose·F1365 <td>··············<pre><code·class="language-macaulay2">i16·:·decompose·F1
  
366 ····································2·············2······················2366 ···································2··············2······················2
367 o16·=·{ideal·(a·-·29b·-·29c·-·14d,·b··-·8b*c·+·24c··+·33b*d·-·13c*d·-·39d·),367 o16·=·{ideal·(a·-·29b·-·8c·-·11d,·b··-·29b*c·+·24c··-·23b*d·+·40c*d·+·14d·),
368 ······-----------------------------------------------------------------------368 ······-----------------------------------------------------------------------
369 ······ideal·(c·-·29d,·b·+·22d,·a·+·4d)}369 ······ideal·(c·-·29d,·b·-·35d,·a·+·43d)}
  
370 o16·:·List</code></pre>370 o16·:·List</code></pre>
371 </td>··········</tr>371 </td>··········</tr>
372 ··········<tr>372 ··········<tr>
373 <td>··············<pre><code·class="language-macaulay2">i17·:·decompose·F2373 <td>··············<pre><code·class="language-macaulay2">i17·:·decompose·F2
  
 374 o17·=·{ideal·(b·-·6c·-·42d,·a·+·26c·-·5d),·ideal·(b·+·40c·+·18d,·a·-·46c·+
374 ·······························2······························2···2·········· 
375 o17·=·{ideal·(a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·,·b··+·19b*c·- 
376 ······----------------------------------------------------------------------- 
377 ·········2·····················2···················2························ 
378 ······24c··-·38b*d·+·26c*d·+·3d·,·a*b·+·22b*c·+·20c··-·27a*d·+·49b*d·-·11c*d 
379 ······-----------------------------------------------------------------------375 ······-----------------------------------------------------------------------
 376 ······19d)}
380 ···········2···2··············2······························2 
381 ······+·18d·,·a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·)} 
  
382 o17·:·List</code></pre>377 o17·:·List</code></pre>
383 </td>··········</tr>378 </td>··········</tr>
384 ········</table>379 ········</table>
385 ········<div>380 ········<div>
386 ··········<p>Note,·the·general·element·of·one·component·is·a·plane·conic·union·a·point.·(The·dimension·of·the·locus·of·all·such·is:·(choice·of·plane)·+·(choice·of·degree·2·in·plane)·+·choice·of·point.·This·is·3·+·5·+·3·=·11.</p>381 ··········<p>Note,·the·general·element·of·one·component·is·a·plane·conic·union·a·point.·(The·dimension·of·the·locus·of·all·such·is:·(choice·of·plane)·+·(choice·of·degree·2·in·plane)·+·choice·of·point.·This·is·3·+·5·+·3·=·11.</p>
387 ··········<p>The·other·component·consists·of·two·skew·lines.··This·has·dimension·(choice·of·line)·+·(choice·of·line).·This·is·4·+·4·=·8.··Also·notice·that·the·2·skew·lines·do·not·have·to·be·defined·over·the·base·field,·as·in·this·case.</p>382 ··········<p>The·other·component·consists·of·two·skew·lines.··This·has·dimension·(choice·of·line)·+·(choice·of·line).·This·is·4·+·4·=·8.··Also·notice·that·the·2·skew·lines·do·not·have·to·be·defined·over·the·base·field,·as·in·this·case.</p>
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Offset 424, 70 lines modifiedOffset 424, 65 lines modified
424 -----------------------------------------------------------+424 -----------------------------------------------------------+
425 This·tells·us·that·there·are·2·components·(at·least·over·the·given·field).425 This·tells·us·that·there·are·2·components·(at·least·over·the·given·field).
426 Their·dimensions·are·11,·8.426 Their·dimensions·are·11,·8.
427 We·can·find·random·points·on·each·component,·since·these·components·are427 We·can·find·random·points·on·each·component,·since·these·components·are
428 rational.428 rational.
429 i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0429 i12·:·pt1·=·randomPointOnRationalVariety·compsJ_0
  
430 o12·=·|·48·33·19·-49·36·14·-34·2·-27·-30·-11·21·19·19·42·-10·24·-36·-29·33·-8430 o12·=·|·-36·-30·42·21·-22·23·23·16·-7·-30·-11·-40·19·19·-41·-22·24·-36·-8·33
431 ······-----------------------------------------------------------------------431 ······-----------------------------------------------------------------------
432 ······-22·-29·-29·|432 ······-29·-10·-29·-29·|
  
433 ···············1·······24433 ···············1·······24
434 o12·:·Matrix·kk··<--·kk434 o12·:·Matrix·kk··<--·kk
435 i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1435 i13·:·pt2·=·randomPointOnRationalVariety·compsJ_1
  
436 o13·=·|·20·-50·18·-20·-48·-11·3·50·-19·38·20·49·26·39·22·-27·-24·-38·21·34·19436 o13·=·|·24·-7·-25·32·-28·13·-49·11·-17·-17·-2·-50·30·19·-41·-21·-38·-24·21·39
437 ······-----------------------------------------------------------------------437 ······-----------------------------------------------------------------------
438 ······0·-16·-47·|438 ······34·0·-16·-47·|
  
439 ···············1·······24439 ···············1·······24
440 o13·:·Matrix·kk··<--·kk440 o13·:·Matrix·kk··<--·kk
441 i14·:·F1·=·sub(F,·(vars·S)|pt1)441 i14·:·F1·=·sub(F,·(vars·S)|pt1)
  
442 ··············2··············2······························2442 ··············2·············2······························2
443 o14·=·ideal·(a··-·27b*c·-·49c··-·30a*d·+·36b*d·+·33c*d·+·48d·,·a*b·+·42b*c·-443 o14·=·ideal·(a··-·7b*c·+·21c··-·30a*d·-·22b*d·-·30c*d·-·36d·,·a*b·-·41b*c·-
444 ······-----------------------------------------------------------------------444 ······-----------------------------------------------------------------------
445 ·········2······························2···2·············2445 ·········2······························2···2··············2
446 ······11c··-·10a*d·+·21b*d·+·14c*d·+·19d·,·b··-·8b*c·+·24c··-·22a*d·-·36b*d·+446 ······11c··-·22a*d·-·40b*d·+·23c*d·+·42d·,·b··-·29b*c·+·24c··-·10a*d·-·36b*d
447 ······-----------------------------------------------------------------------447 ······-----------------------------------------------------------------------
448 ·················2···················2·····························2448 ···················2··················2······························2
449 ······19c*d·-·34d·,·a*c·-·29b*c·-·29c··-·29a*d·+·33b*d·+·19c*d·+·2d·)449 ······+·19c*d·+·23d·,·a*c·-·29b*c·-·8c··-·29a*d·+·33b*d·+·19c*d·+·16d·)
  
450 o14·:·Ideal·of·S450 o14·:·Ideal·of·S
451 i15·:·F2·=·sub(F,·(vars·S)|pt2)451 i15·:·F2·=·sub(F,·(vars·S)|pt2)
  
452 ··············2··············2······························2452 ··············2··············2·····························2
453 o15·=·ideal·(a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·,·a*b·+·22b*c·+453 o15·=·ideal·(a··-·17b*c·+·32c··-·17a*d·-·28b*d·-·7c*d·+·24d·,·a*b·-·41b*c·-
454 ······-----------------------------------------------------------------------454 ······-----------------------------------------------------------------------
455 ·········2······························2···2··············2455 ········2······························2···2··············2
456 ······20c··-·27a*d·+·49b*d·-·11c*d·+·18d·,·b··+·19b*c·-·24c··-·38b*d·+·26c*d456 ······2c··-·21a*d·-·50b*d·+·13c*d·-·25d·,·b··+·34b*c·-·38c··-·24b*d·+·30c*d·-
457 ······-----------------------------------------------------------------------457 ······-----------------------------------------------------------------------
458 ··········2···················2······························2458 ·········2···················2······························2
459 ······+·3d·,·a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·)459 ······49d·,·a*c·-·16b*c·+·21c··-·47a*d·+·39b*d·+·19c*d·+·11d·)
  
460 o15·:·Ideal·of·S460 o15·:·Ideal·of·S
461 i16·:·decompose·F1461 i16·:·decompose·F1
  
462 ····································2·············2······················2462 ···································2··············2······················2
463 o16·=·{ideal·(a·-·29b·-·29c·-·14d,·b··-·8b*c·+·24c··+·33b*d·-·13c*d·-·39d·),463 o16·=·{ideal·(a·-·29b·-·8c·-·11d,·b··-·29b*c·+·24c··-·23b*d·+·40c*d·+·14d·),
464 ······-----------------------------------------------------------------------464 ······-----------------------------------------------------------------------
465 ······ideal·(c·-·29d,·b·+·22d,·a·+·4d)}465 ······ideal·(c·-·29d,·b·-·35d,·a·+·43d)}
  
466 o16·:·List466 o16·:·List
467 i17·:·decompose·F2467 i17·:·decompose·F2
  
 468 o17·=·{ideal·(b·-·6c·-·42d,·a·+·26c·-·5d),·ideal·(b·+·40c·+·18d,·a·-·46c·+
468 ·······························2······························2···2 
469 o17·=·{ideal·(a*c·-·16b*c·+·21c··-·47a*d·+·34b*d·+·39c*d·+·50d·,·b··+·19b*c·- 
470 ······----------------------------------------------------------------------- 
471 ·········2·····················2···················2 
472 ······24c··-·38b*d·+·26c*d·+·3d·,·a*b·+·22b*c·+·20c··-·27a*d·+·49b*d·-·11c*d 
473 ······-----------------------------------------------------------------------469 ······-----------------------------------------------------------------------
 470 ······19d)}
474 ···········2···2··············2······························2 
475 ······+·18d·,·a··-·19b*c·-·20c··+·38a*d·-·48b*d·-·50c*d·+·20d·)} 
  
476 o17·:·List471 o17·:·List
477 Note,·the·general·element·of·one·component·is·a·plane·conic·union·a·point.·(The472 Note,·the·general·element·of·one·component·is·a·plane·conic·union·a·point.·(The
478 dimension·of·the·locus·of·all·such·is:·(choice·of·plane)·+·(choice·of·degree·2473 dimension·of·the·locus·of·all·such·is:·(choice·of·plane)·+·(choice·of·degree·2
479 in·plane)·+·choice·of·point.·This·is·3·+·5·+·3·=·11.474 in·plane)·+·choice·of·point.·This·is·3·+·5·+·3·=·11.
480 The·other·component·consists·of·two·skew·lines.·This·has·dimension·(choice·of475 The·other·component·consists·of·two·skew·lines.·This·has·dimension·(choice·of
481 line)·+·(choice·of·line).·This·is·4·+·4·=·8.·Also·notice·that·the·2·skew·lines476 line)·+·(choice·of·line).·This·is·4·+·4·=·8.·Also·notice·that·the·2·skew·lines
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./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=167 #:len=16
8 c2V0V2Fsa1RyYWNlKFpaKQ==8 c2V0V2Fsa1RyYWNlKFpaKQ==
9 #:len=2519 #:len=251
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo
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./usr/share/doc/Macaulay2/GroebnerWalk/example-output/___Groebner__Walk.out
    
Offset 11, 21 lines modifiedOffset 11, 21 lines modified
11 i3·:·R2·=·QQ[x,y,z,u,v,·MonomialOrder=>Weights=>{0,0,0,1,1}];11 i3·:·R2·=·QQ[x,y,z,u,v,·MonomialOrder=>Weights=>{0,0,0,1,1}];
  
12 i4·:·I2·=·sub(I1,·R2);12 i4·:·I2·=·sub(I1,·R2);
  
13 o4·:·Ideal·of·R213 o4·:·Ideal·of·R2
  
14 i5·:·elapsedTime·gb·I214 i5·:·elapsedTime·gb·I2
15 ·--·2.86182s·elapsed15 ·--·2.67378s·elapsed
  
16 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]16 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]
  
17 o5·:·GroebnerBasis17 o5·:·GroebnerBasis
  
18 i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)18 i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)
19 ·--·1.90874s·elapsed19 ·--·3.30948s·elapsed
  
20 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]20 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]
  
21 o6·:·GroebnerBasis21 o6·:·GroebnerBasis
  
22 i7·:·22 i7·:·
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./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html
    
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i4·:·I2·=·sub(I1,·R2);77 <td>··············<pre><code·class="language-macaulay2">i4·:·I2·=·sub(I1,·R2);
  
78 o4·:·Ideal·of·R2</code></pre>78 o4·:·Ideal·of·R2</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·gb·I281 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·gb·I2
82 ·--·2.86182s·elapsed82 ·--·2.67378s·elapsed
  
83 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]83 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]
  
84 o5·:·GroebnerBasis</code></pre>84 o5·:·GroebnerBasis</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ········</table>86 ········</table>
87 ········<div>87 ········<div>
88 ··········<p>but·it·is·faster·to·compute·directly·in·the·first·order·and·then·use·the·Groebner·walk.</p>88 ··········<p>but·it·is·faster·to·compute·directly·in·the·first·order·and·then·use·the·Groebner·walk.</p>
89 ········</div>89 ········</div>
90 ········<table·class="examples">90 ········<table·class="examples">
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)92 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)
93 ·--·1.90874s·elapsed93 ·--·3.30948s·elapsed
  
94 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]94 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]
  
95 o6·:·GroebnerBasis</code></pre>95 o6·:·GroebnerBasis</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ········</table>97 ········</table>
98 ······</div>98 ······</div>
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Offset 38, 23 lines modifiedOffset 38, 23 lines modified
38 using·a·different·weight·vector·and·then·graded·reverse·lexicographic·we·could38 using·a·different·weight·vector·and·then·graded·reverse·lexicographic·we·could
39 substitute·and·compute·directly,39 substitute·and·compute·directly,
40 i3·:·R2·=·QQ[x,y,z,u,v,·MonomialOrder=>Weights=>{0,0,0,1,1}];40 i3·:·R2·=·QQ[x,y,z,u,v,·MonomialOrder=>Weights=>{0,0,0,1,1}];
41 i4·:·I2·=·sub(I1,·R2);41 i4·:·I2·=·sub(I1,·R2);
  
42 o4·:·Ideal·of·R242 o4·:·Ideal·of·R2
43 i5·:·elapsedTime·gb·I243 i5·:·elapsedTime·gb·I2
44 ·--·2.86182s·elapsed44 ·--·2.67378s·elapsed
  
45 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]45 o5·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·16]
  
46 o5·:·GroebnerBasis46 o5·:·GroebnerBasis
47 but·it·is·faster·to·compute·directly·in·the·first·order·and·then·use·the47 but·it·is·faster·to·compute·directly·in·the·first·order·and·then·use·the
48 Groebner·walk.48 Groebner·walk.
49 i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)49 i6·:·elapsedTime·groebnerWalk(gb·I1,·R2)
50 ·--·1.90874s·elapsed50 ·--·3.30948s·elapsed
  
51 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]51 o6·=·GroebnerBasis[status:·done;·S-pairs·encountered·up·to·degree·0]
  
52 o6·:·GroebnerBasis52 o6·:·GroebnerBasis
53 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*53 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
54 The·target·ring·must·be·the·same·ring·as·the·ring·of·the·starting·ideal,·except54 The·target·ring·must·be·the·same·ring·as·the·ring·of·the·starting·ideal,·except
55 with·different·monomial·order.·The·ring·must·be·a·polynomial·ring·over·a·field.55 with·different·monomial·order.·The·ring·must·be·a·polynomial·ring·over·a·field.
⊟
734 B
./usr/share/doc/Macaulay2/Hadamard/dump/rawdocumentation.dump
    
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3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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./usr/share/doc/Macaulay2/HigherCIOperators/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj
11 b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu11 b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu
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729 B
./usr/share/doc/Macaulay2/HighestWeights/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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11 ZiB0aGUgZ3JvdXAgYWN0aW5nIG9uIGEgcmluZyIsICJsaW5lbnVtIiA9PiA4MywgU2VlQWxzbyA911 ZiB0aGUgZ3JvdXAgYWN0aW5nIG9uIGEgcmluZyIsICJsaW5lbnVtIiA9PiA4MywgU2VlQWxzbyA9
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721 B
./usr/share/doc/Macaulay2/HodgeIntegrals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=57 #:len=5
8 a2FwcGE=8 a2FwcGE=
9 #:len=13969 #:len=1396
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWlsbGVyLU1vcml0YS1NdW1mb3JkIGNs10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWlsbGVyLU1vcml0YS1NdW1mb3JkIGNs
11 YXNzZXMiLCAibGluZW51bSIgPT4gNzM4LCBJbnB1dHMgPT4ge1NQQU57VFR7ImEifSwiLCAiLFNQ11 YXNzZXMiLCAibGluZW51bSIgPT4gNzM4LCBJbnB1dHMgPT4ge1NQQU57VFR7ImEifSwiLCAiLFNQ
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738 B
./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=147 #:len=14
8 ZXVsZXJPcGVyYXRvcnM=8 ZXVsZXJPcGVyYXRvcnM=
9 #:len=19599 #:len=1959
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu
11 dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l11 dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l
⊟
925 B
./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out
    
Offset 42, 19 lines modifiedOffset 42, 19 lines modified
42 i5·:·w·=·{9,1,99999,·9999999,·3,·999}42 i5·:·w·=·{9,1,99999,·9999999,·3,·999}
  
43 o5·=·{9,·1,·99999,·9999999,·3,·999}43 o5·=·{9,·1,·99999,·9999999,·3,·999}
  
44 o5·:·List44 o5·:·List
  
45 i6·:·netList·cssLeadTerm(Hbeta,·w)45 i6·:·netList·cssLeadTerm(Hbeta,·w)
 46 ·--·.000003256s·elapsed
 47 ·--·.000003126s·elapsed
46 ·--·.000003094s·elapsed48 ·--·.000004609s·elapsed
47 ·--·.000003024s·elapsed49 ·--·.000003026s·elapsed
48 ·--·.000002514s·elapsed50 ·--·.000004409s·elapsed
49 ·--·.000003034s·elapsed 
50 ·--·.000002964s·elapsed 
51 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.51 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
52 Converting·to·Naive·algorithm.52 Converting·to·Naive·algorithm.
  
53 ·····+----------------------------------------------------+53 ·····+----------------------------------------------------+
54 ·····|···1·5···5·5········································|54 ·····|···1·5···5·5········································|
55 ·····|·-·-·-·-·-·-········································|55 ·····|·-·-·-·-·-·-········································|
56 ·····|···2·2···2·2········································|56 ·····|···2·2···2·2········································|
⊟
1.48 KB
./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
    
Offset 3, 15 lines modifiedOffset 3, 15 lines modified
3 i1·:·R·=·QQ[t_1..t_5];3 i1·:·R·=·QQ[t_1..t_5];
  
4 i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,·t_2*t_4);4 i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,·t_2*t_4);
  
5 o2·:·Ideal·of·R5 o2·:·Ideal·of·R
  
6 i3·:·solveFrobeniusIdeal·I6 i3·:·solveFrobeniusIdeal·I
7 ·--·.000003245s·elapsed7 ·--·.000005581s·elapsed
8 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.8 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
9 Converting·to·Naive·algorithm.9 Converting·to·Naive·algorithm.
  
10 ·············································································10 ·············································································
11 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,11 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
12 ················0········1········2·······3········0·······1·······2·······4·12 ················0········1········2·······3········0·······1·······2·······4·
13 ·····------------------------------------------------------------------------13 ·····------------------------------------------------------------------------
Offset 24, 15 lines modifiedOffset 24, 15 lines modified
24 ·······2····4····0···4····4····1···2····4····2···4····4····3·······424 ·······2····4····0···4····4····1···2····4····2···4····4····3·······4
  
25 o3·:·List25 o3·:·List
  
26 i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);26 i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);
  
27 i5·:·solveFrobeniusIdeal(I,·W)27 i5·:·solveFrobeniusIdeal(I,·W)
28 ·--·.000003665s·elapsed28 ·--·.000004309s·elapsed
29 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.29 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
30 Converting·to·Naive·algorithm.30 Converting·to·Naive·algorithm.
  
31 ·············································································31 ·············································································
32 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,32 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
33 ················0········1········2·······3········0·······1·······2·······4·33 ················0········1········2·······3········0·······1·······2·······4·
34 ·····------------------------------------------------------------------------34 ·····------------------------------------------------------------------------
⊟
1.7 KB
./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
    
Offset 122, 19 lines modifiedOffset 122, 19 lines modified
  
122 o5·=·{9,·1,·99999,·9999999,·3,·999}122 o5·=·{9,·1,·99999,·9999999,·3,·999}
  
123 o5·:·List</code></pre>123 o5·:·List</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i6·:·netList·cssLeadTerm(Hbeta,·w)126 <td>··············<pre><code·class="language-macaulay2">i6·:·netList·cssLeadTerm(Hbeta,·w)
 127 ·--·.000003256s·elapsed
 128 ·--·.000003126s·elapsed
127 ·--·.000003094s·elapsed129 ·--·.000004609s·elapsed
128 ·--·.000003024s·elapsed130 ·--·.000003026s·elapsed
129 ·--·.000002514s·elapsed131 ·--·.000004409s·elapsed
130 ·--·.000003034s·elapsed 
131 ·--·.000002964s·elapsed 
132 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.132 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
133 Converting·to·Naive·algorithm.133 Converting·to·Naive·algorithm.
  
134 ·····+----------------------------------------------------+134 ·····+----------------------------------------------------+
135 ·····|···1·5···5·5········································|135 ·····|···1·5···5·5········································|
136 ·····|·-·-·-·-·-·-········································|136 ·····|·-·-·-·-·-·-········································|
137 ·····|···2·2···2·2········································|137 ·····|···2·2···2·2········································|
⊟
764 B
html2text {}
    
Offset 55, 19 lines modifiedOffset 55, 19 lines modified
55 ··················1···6···1···655 ··················1···6···1···6
56 i5·:·w·=·{9,1,99999,·9999999,·3,·999}56 i5·:·w·=·{9,1,99999,·9999999,·3,·999}
  
57 o5·=·{9,·1,·99999,·9999999,·3,·999}57 o5·=·{9,·1,·99999,·9999999,·3,·999}
  
58 o5·:·List58 o5·:·List
59 i6·:·netList·cssLeadTerm(Hbeta,·w)59 i6·:·netList·cssLeadTerm(Hbeta,·w)
 60 ·--·.000003256s·elapsed
 61 ·--·.000003126s·elapsed
60 ·--·.000003094s·elapsed62 ·--·.000004609s·elapsed
61 ·--·.000003024s·elapsed63 ·--·.000003026s·elapsed
62 ·--·.000002514s·elapsed64 ·--·.000004409s·elapsed
63 ·--·.000003034s·elapsed 
64 ·--·.000002964s·elapsed 
65 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or65 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or
66 inhomogeneous·ideal.66 inhomogeneous·ideal.
67 Converting·to·Naive·algorithm.67 Converting·to·Naive·algorithm.
  
68 ·····+----------------------------------------------------+68 ·····+----------------------------------------------------+
69 ·····|···1·5···5·5········································|69 ·····|···1·5···5·5········································|
70 ·····|·-·-·-·-·-·-········································|70 ·····|·-·-·-·-·-·-········································|
⊟
2.95 KB
./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
    
Offset 79, 15 lines modifiedOffset 79, 15 lines modified
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,·t_2*t_4);80 <td>··············<pre><code·class="language-macaulay2">i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,·t_2*t_4);
  
81 o2·:·Ideal·of·R</code></pre>81 o2·:·Ideal·of·R</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i3·:·solveFrobeniusIdeal·I84 <td>··············<pre><code·class="language-macaulay2">i3·:·solveFrobeniusIdeal·I
85 ·--·.000003245s·elapsed85 ·--·.000005581s·elapsed
86 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.86 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
87 Converting·to·Naive·algorithm.87 Converting·to·Naive·algorithm.
  
88 ·············································································88 ·············································································
89 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,89 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
90 ················0········1········2·······3········0·······1·······2·······4·90 ················0········1········2·······3········0·······1·······2·······4·
91 ·····------------------------------------------------------------------------91 ·····------------------------------------------------------------------------
Offset 104, 15 lines modifiedOffset 104, 15 lines modified
104 ········</table>104 ········</table>
105 ········<table·class="examples">105 ········<table·class="examples">
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i5·:·solveFrobeniusIdeal(I,·W)110 <td>··············<pre><code·class="language-macaulay2">i5·:·solveFrobeniusIdeal(I,·W)
111 ·--·.000003665s·elapsed111 ·--·.000004309s·elapsed
112 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.112 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or·inhomogeneous·ideal.
113 Converting·to·Naive·algorithm.113 Converting·to·Naive·algorithm.
  
114 ·············································································114 ·············································································
115 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,115 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
116 ················0········1········2·······3········0·······1·······2·······4·116 ················0········1········2·······3········0·······1·······2·······4·
117 ·····------------------------------------------------------------------------117 ·····------------------------------------------------------------------------
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html2text {}
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 Here·is·[_\x8S_\x8S_\x8T,·Example·2.3.16]:17 Here·is·[_\x8S_\x8S_\x8T,·Example·2.3.16]:
18 i1·:·R·=·QQ[t_1..t_5];18 i1·:·R·=·QQ[t_1..t_5];
19 i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,19 i2·:·I·=·ideal(t_1+t_2+t_3+t_4+t_5,·t_1+t_2-t_4,·t_2+t_3-t_4,·t_1*t_3,
20 t_2*t_4);20 t_2*t_4);
  
21 o2·:·Ideal·of·R21 o2·:·Ideal·of·R
22 i3·:·solveFrobeniusIdeal·I22 i3·:·solveFrobeniusIdeal·I
23 ·--·.000003245s·elapsed23 ·--·.000005581s·elapsed
24 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or24 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or
25 inhomogeneous·ideal.25 inhomogeneous·ideal.
26 Converting·to·Naive·algorithm.26 Converting·to·Naive·algorithm.
  
  
27 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,27 o3·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
28 ················0········1········2·······3········0·······1·······2·······428 ················0········1········2·······3········0·······1·······2·······4
Offset 37, 15 lines modifiedOffset 37, 15 lines modified
37 ·······1·············1·············1·············3·················237 ·······1·············1·············1·············3·················2
38 ·····-·-logX·logX··-·-logX·logX··-·-logX·logX··-·-logX·logX··+·logX·}38 ·····-·-logX·logX··-·-logX·logX··-·-logX·logX··-·-logX·logX··+·logX·}
39 ·······2····4····0···4····4····1···2····4····2···4····4····3·······439 ·······2····4····0···4····4····1···2····4····2···4····4····3·······4
  
40 o3·:·List40 o3·:·List
41 i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);41 i4·:·W·=·makeWeylAlgebra(QQ[x_1..x_5]);
42 i5·:·solveFrobeniusIdeal(I,·W)42 i5·:·solveFrobeniusIdeal(I,·W)
43 ·--·.000003665s·elapsed43 ·--·.000004309s·elapsed
44 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or44 Warning:··F4·Algorithm·not·available·over·current·coefficient·ring·or
45 inhomogeneous·ideal.45 inhomogeneous·ideal.
46 Converting·to·Naive·algorithm.46 Converting·to·Naive·algorithm.
  
  
47 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,47 o5·=·{1,·-·2logX··+·3logX··-·2logX··+·logX·,·-·logX··+·logX··-·logX··+·logX·,
48 ················0········1········2·······3········0·······1·······2·······448 ················0········1········2·······3········0·······1·······2·······4
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./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
7 #:len=217 #:len=21
8 YWxsZ2VucyhER0FsZ2VicmEsWlop8 YWxsZ2VucyhER0FsZ2VicmEsWlop
9 #:len=2699 #:len=269
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbGxnZW5zLERHQWxnZWJyYSxaWiksImFsbGdlbnMo11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbGxnZW5zLERHQWxnZWJyYSxaWiksImFsbGdlbnMo
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./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
7 #:len=297 #:len=29
8 cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=8 cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=
9 #:len=3439 #:len=343
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3Nywgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3Nywgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks
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760 B
./usr/share/doc/Macaulay2/IntegralClosure/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=317 #:len=31
8 aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==8 aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==
9 #:len=11209 #:len=1120
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs
11 b3N1cmUiLCAibGluZW51bSIgPT4gMTQyOCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT11 b3N1cmUiLCAibGluZW51bSIgPT4gMTQyOCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
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./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.out
    
Offset 16, 15 lines modifiedOffset 16, 15 lines modified
16 i3·:·R·=·S/f16 i3·:·R·=·S/f
  
17 o3·=·R17 o3·=·R
  
18 o3·:·QuotientRing18 o3·:·QuotientRing
  
19 i4·:·time·R'·=·integralClosure·R19 i4·:·time·R'·=·integralClosure·R
20 ·--·used·0.517294s·(cpu);·0.326626s·(thread);·0s·(gc)20 ·--·used·0.700828s·(cpu);·0.390718s·(thread);·0s·(gc)
  
21 o4·=·R'21 o4·=·R'
  
22 o4·:·QuotientRing22 o4·:·QuotientRing
  
23 i5·:·netList·(ideal·R')_*23 i5·:·netList·(ideal·R')_*
  
Offset 83, 15 lines modifiedOffset 83, 15 lines modified
83 i9·:·R·=·S/f83 i9·:·R·=·S/f
  
84 o9·=·R84 o9·=·R
  
85 o9·:·QuotientRing85 o9·:·QuotientRing
  
86 i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)86 i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
87 ·--·used·0.596554s·(cpu);·0.367717s·(thread);·0s·(gc)87 ·--·used·0.669042s·(cpu);·0.365586s·(thread);·0s·(gc)
  
88 o10·=·R'88 o10·=·R'
  
89 o10·:·QuotientRing89 o10·:·QuotientRing
  
90 i11·:·netList·(ideal·R')_*90 i11·:·netList·(ideal·R')_*
  
Offset 150, 15 lines modifiedOffset 150, 15 lines modified
150 i15·:·R·=·S/f150 i15·:·R·=·S/f
  
151 o15·=·R151 o15·=·R
  
152 o15·:·QuotientRing152 o15·:·QuotientRing
  
153 i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)153 i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)
154 ·--·used·0.595118s·(cpu);·0.372909s·(thread);·0s·(gc)154 ·--·used·0.686389s·(cpu);·0.364268s·(thread);·0s·(gc)
  
155 o16·=·R'155 o16·=·R'
  
156 o16·:·QuotientRing156 o16·:·QuotientRing
  
157 i17·:·netList·(ideal·R')_*157 i17·:·netList·(ideal·R')_*
  
Offset 208, 15 lines modifiedOffset 208, 15 lines modified
208 i20·:·R·=·S/f208 i20·:·R·=·S/f
  
209 o20·=·R209 o20·=·R
  
210 o20·:·QuotientRing210 o20·:·QuotientRing
  
211 i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)211 i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)
212 ·--·used·0.607605s·(cpu);·0.372017s·(thread);·0s·(gc)212 ·--·used·0.832651s·(cpu);·0.422697s·(thread);·0s·(gc)
  
213 o21·=·R'213 o21·=·R'
  
214 o21·:·QuotientRing214 o21·:·QuotientRing
  
215 i22·:·netList·(ideal·R')_*215 i22·:·netList·(ideal·R')_*
  
Offset 266, 15 lines modifiedOffset 266, 15 lines modified
266 i25·:·R·=·S/f266 i25·:·R·=·S/f
  
267 o25·=·R267 o25·=·R
  
268 o25·:·QuotientRing268 o25·:·QuotientRing
  
269 i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)269 i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)
270 ·--·used·1.12494s·(cpu);·0.665136s·(thread);·0s·(gc)270 ·--·used·1.53817s·(cpu);·0.731348s·(thread);·0s·(gc)
  
271 o26·=·R'271 o26·=·R'
  
272 o26·:·QuotientRing272 o26·:·QuotientRing
  
273 i27·:·netList·(ideal·R')_*273 i27·:·netList·(ideal·R')_*
  
Offset 324, 15 lines modifiedOffset 324, 15 lines modified
324 i30·:·R·=·S/f324 i30·:·R·=·S/f
  
325 o30·=·R325 o30·=·R
  
326 o30·:·QuotientRing326 o30·:·QuotientRing
  
327 i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)327 i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)
328 ·--·used·0.678664s·(cpu);·0.396606s·(thread);·0s·(gc)328 ·--·used·0.893359s·(cpu);·0.440371s·(thread);·0s·(gc)
  
329 o31·=·R'329 o31·=·R'
  
330 o31·:·QuotientRing330 o31·:·QuotientRing
  
331 i32·:·netList·(ideal·R')_*331 i32·:·netList·(ideal·R')_*
  
Offset 382, 15 lines modifiedOffset 382, 15 lines modified
382 i35·:·R·=·S/f382 i35·:·R·=·S/f
  
383 o35·=·R383 o35·=·R
  
384 o35·:·QuotientRing384 o35·:·QuotientRing
  
385 i36·:·time·R'·=·integralClosure·R385 i36·:·time·R'·=·integralClosure·R
386 ·--·used·0.0981306s·(cpu);·0.0512022s·(thread);·0s·(gc)386 ·--·used·0.144615s·(cpu);·0.0580953s·(thread);·0s·(gc)
  
387 o36·=·R'387 o36·=·R'
  
388 o36·:·QuotientRing388 o36·:·QuotientRing
  
389 i37·:·netList·(ideal·R')_*389 i37·:·netList·(ideal·R')_*
  
Offset 432, 15 lines modifiedOffset 432, 15 lines modified
432 i40·:·R·=·S/I432 i40·:·R·=·S/I
  
433 o40·=·R433 o40·=·R
  
434 o40·:·QuotientRing434 o40·:·QuotientRing
  
435 i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)435 i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
436 ·--·used·0.100201s·(cpu);·0.0565278s·(thread);·0s·(gc)436 ·--·used·0.113547s·(cpu);·0.0617245s·(thread);·0s·(gc)
  
437 o41·=·R'437 o41·=·R'
  
438 o41·:·QuotientRing438 o41·:·QuotientRing
  
439 i42·:·icFractions·R439 i42·:·icFractions·R
  
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./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.out
    
Offset 1, 50 lines modifiedOffset 1, 50 lines modified
1 --·-*-·M2-comint·-*-·hash:·131779540694346152731 --·-*-·M2-comint·-*-·hash:·13177954069434615273
  
2 i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);2 i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
  
3 i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)3 i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)
4 ·[jacobian·time·0·sec·#minors·3]4 ·[jacobian·time·.000766717·sec·#minors·3]
5 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·25 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·2
  
6 ·[step·0:·6 ·[step·0:·
7 ······radical·(use·minprimes)·.00304692·seconds7 ······radical·(use·minprimes)·0·seconds
8 ······idlizer1:··.00643697·seconds8 ······idlizer1:··.00680135·seconds
9 ······idlizer2:··.00793008·seconds9 ······idlizer2:··.0110873·seconds
10 ······minpres:···.0079959·seconds10 ······minpres:···.00796023·seconds
11 ··time·.0855872·sec··#fractions·4]11 ··time·.129816·sec··#fractions·4]
12 ·[step·1:·12 ·[step·1:·
13 ······radical·(use·minprimes)·.0039771·seconds13 ······radical·(use·minprimes)·0·seconds
 14 ······idlizer1:··.0119441·seconds
14 ······idlizer1:··.00800509·seconds15 ······idlizer2:··.064009·seconds
15 ······idlizer2:··.0618165·seconds 
16 ······minpres:···.012009·seconds16 ······minpres:···.00798687·seconds
17 ··time·.0880962·sec··#fractions·4]17 ··time·.0961367·sec··#fractions·4]
18 ·[step·2:·18 ·[step·2:·
19 ······radical·(use·minprimes)·.00300098·seconds19 ······radical·(use·minprimes)·.00392575·seconds
20 ······idlizer1:··.0120029·seconds20 ······idlizer1:··.0104333·seconds
21 ······idlizer2:··.00895389·seconds21 ······idlizer2:··.011965·seconds
22 ······minpres:···.00834706·seconds22 ······minpres:···.0120078·seconds
23 ··time·.0906009·sec··#fractions·5]23 ··time·.094474·sec··#fractions·5]
24 ·[step·3:·24 ·[step·3:·
25 ······radical·(use·minprimes)·0·seconds25 ······radical·(use·minprimes)·.00399905·seconds
26 ······idlizer1:··.00920056·seconds26 ······idlizer1:··.0119994·seconds
27 ······idlizer2:··.0119983·seconds27 ······idlizer2:··.0119997·seconds
28 ······minpres:···.0668751·seconds28 ······minpres:···.0947087·seconds
29 ··time·.0988007·sec··#fractions·5]29 ··time·.130949·sec··#fractions·5]
30 ·[step·4:·30 ·[step·4:·
31 ······radical·(use·minprimes)·.00395202·seconds31 ······radical·(use·minprimes)·.000826229·seconds
32 ······idlizer1:··.00800066·seconds32 ······idlizer1:··.00673216·seconds
33 ······idlizer2:··.0119991·seconds33 ······idlizer2:··.0155193·seconds
34 ······minpres:···.008001·seconds34 ······minpres:···.0119941·seconds
35 ··time·.0399368·sec··#fractions·5]35 ··time·.0492054·sec··#fractions·5]
36 ·[step·5:·36 ·[step·5:·
37 ······radical·(use·minprimes)·.00400227·seconds37 ······radical·(use·minprimes)·0·seconds
38 ······idlizer1:···--·used·0.478393s·(cpu);·0.308821s·(thread);·0s·(gc)38 ······idlizer1:···--·used·0.577836s·(cpu);·0.301618s·(thread);·0s·(gc)
39 .00798507·seconds39 .00391349·seconds
40 ··time·.0713716·sec··#fractions·5]40 ··time·.0703759·sec··#fractions·5]
  
41 o2·=·R'41 o2·=·R'
  
42 o2·:·QuotientRing42 o2·:·QuotientRing
  
43 i3·:·trim·ideal·R'43 i3·:·trim·ideal·R'
  
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./usr/share/doc/Macaulay2/IntegralClosure/example-output/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.out
    
Offset 13, 26 lines modifiedOffset 13, 26 lines modified
  
13 ················2······2····2········2···2·2·····213 ················2······2····2········2···2·2·····2
14 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)14 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)
  
15 o3·:·Ideal·of·S15 o3·:·Ideal·of·S
  
16 i4·:·time·integralClosure·J16 i4·:·time·integralClosure·J
17 ·--·used·6.09789s·(cpu);·1.29964s·(thread);·0s·(gc)17 ·--·used·5.39518s·(cpu);·1.16459s·(thread);·0s·(gc)
  
18 ·············2·2··············2·2················2··········2···2·····18 ·············2·2··············2·2················2··········2···2·····
19 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-19 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ···········2···3···············2·2·····2···521 ···········2···3···············2·2·····2···5
22 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)22 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
23 o4·:·Ideal·of·S23 o4·:·Ideal·of·S
  
24 i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})24 i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})
25 ·--·used·9.42242s·(cpu);·1.59349s·(thread);·0s·(gc)25 ·--·used·7.48339s·(cpu);·1.23046s·(thread);·0s·(gc)
  
26 ·············2·2··············2·2················2··········2···2·····26 ·············2·2··············2·2················2··········2···2·····
27 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-27 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
28 ·····------------------------------------------------------------------------28 ·····------------------------------------------------------------------------
29 ···········2···3···············2·2·····2···529 ···········2···3···············2·2·····2···5
30 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)30 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
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Offset 90, 15 lines modifiedOffset 90, 15 lines modified
  
90 o3·=·R90 o3·=·R
  
91 o3·:·QuotientRing</code></pre>91 o3·:·QuotientRing</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·time·R'·=·integralClosure·R94 <td>··············<pre><code·class="language-macaulay2">i4·:·time·R'·=·integralClosure·R
95 ·--·used·0.517294s·(cpu);·0.326626s·(thread);·0s·(gc)95 ·--·used·0.700828s·(cpu);·0.390718s·(thread);·0s·(gc)
  
96 o4·=·R'96 o4·=·R'
  
97 o4·:·QuotientRing</code></pre>97 o4·:·QuotientRing</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i5·:·netList·(ideal·R')_*100 <td>··············<pre><code·class="language-macaulay2">i5·:·netList·(ideal·R')_*
Offset 165, 15 lines modifiedOffset 165, 15 lines modified
  
165 o9·=·R165 o9·=·R
  
166 o9·:·QuotientRing</code></pre>166 o9·:·QuotientRing</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)169 <td>··············<pre><code·class="language-macaulay2">i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
170 ·--·used·0.596554s·(cpu);·0.367717s·(thread);·0s·(gc)170 ·--·used·0.669042s·(cpu);·0.365586s·(thread);·0s·(gc)
  
171 o10·=·R'171 o10·=·R'
  
172 o10·:·QuotientRing</code></pre>172 o10·:·QuotientRing</code></pre>
173 </td>··········</tr>173 </td>··········</tr>
174 ··········<tr>174 ··········<tr>
175 <td>··············<pre><code·class="language-macaulay2">i11·:·netList·(ideal·R')_*175 <td>··············<pre><code·class="language-macaulay2">i11·:·netList·(ideal·R')_*
Offset 240, 15 lines modifiedOffset 240, 15 lines modified
  
240 o15·=·R240 o15·=·R
  
241 o15·:·QuotientRing</code></pre>241 o15·:·QuotientRing</code></pre>
242 </td>··········</tr>242 </td>··········</tr>
243 ··········<tr>243 ··········<tr>
244 <td>··············<pre><code·class="language-macaulay2">i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)244 <td>··············<pre><code·class="language-macaulay2">i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)
245 ·--·used·0.595118s·(cpu);·0.372909s·(thread);·0s·(gc)245 ·--·used·0.686389s·(cpu);·0.364268s·(thread);·0s·(gc)
  
246 o16·=·R'246 o16·=·R'
  
247 o16·:·QuotientRing</code></pre>247 o16·:·QuotientRing</code></pre>
248 </td>··········</tr>248 </td>··········</tr>
249 ··········<tr>249 ··········<tr>
250 <td>··············<pre><code·class="language-macaulay2">i17·:·netList·(ideal·R')_*250 <td>··············<pre><code·class="language-macaulay2">i17·:·netList·(ideal·R')_*
Offset 305, 15 lines modifiedOffset 305, 15 lines modified
  
305 o20·=·R305 o20·=·R
  
306 o20·:·QuotientRing</code></pre>306 o20·:·QuotientRing</code></pre>
307 </td>··········</tr>307 </td>··········</tr>
308 ··········<tr>308 ··········<tr>
309 <td>··············<pre><code·class="language-macaulay2">i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)309 <td>··············<pre><code·class="language-macaulay2">i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)
310 ·--·used·0.607605s·(cpu);·0.372017s·(thread);·0s·(gc)310 ·--·used·0.832651s·(cpu);·0.422697s·(thread);·0s·(gc)
  
311 o21·=·R'311 o21·=·R'
  
312 o21·:·QuotientRing</code></pre>312 o21·:·QuotientRing</code></pre>
313 </td>··········</tr>313 </td>··········</tr>
314 ··········<tr>314 ··········<tr>
315 <td>··············<pre><code·class="language-macaulay2">i22·:·netList·(ideal·R')_*315 <td>··············<pre><code·class="language-macaulay2">i22·:·netList·(ideal·R')_*
Offset 370, 15 lines modifiedOffset 370, 15 lines modified
  
370 o25·=·R370 o25·=·R
  
371 o25·:·QuotientRing</code></pre>371 o25·:·QuotientRing</code></pre>
372 </td>··········</tr>372 </td>··········</tr>
373 ··········<tr>373 ··········<tr>
374 <td>··············<pre><code·class="language-macaulay2">i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)374 <td>··············<pre><code·class="language-macaulay2">i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)
375 ·--·used·1.12494s·(cpu);·0.665136s·(thread);·0s·(gc)375 ·--·used·1.53817s·(cpu);·0.731348s·(thread);·0s·(gc)
  
376 o26·=·R'376 o26·=·R'
  
377 o26·:·QuotientRing</code></pre>377 o26·:·QuotientRing</code></pre>
378 </td>··········</tr>378 </td>··········</tr>
379 ··········<tr>379 ··········<tr>
380 <td>··············<pre><code·class="language-macaulay2">i27·:·netList·(ideal·R')_*380 <td>··············<pre><code·class="language-macaulay2">i27·:·netList·(ideal·R')_*
Offset 435, 15 lines modifiedOffset 435, 15 lines modified
  
435 o30·=·R435 o30·=·R
  
436 o30·:·QuotientRing</code></pre>436 o30·:·QuotientRing</code></pre>
437 </td>··········</tr>437 </td>··········</tr>
438 ··········<tr>438 ··········<tr>
439 <td>··············<pre><code·class="language-macaulay2">i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)439 <td>··············<pre><code·class="language-macaulay2">i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)
440 ·--·used·0.678664s·(cpu);·0.396606s·(thread);·0s·(gc)440 ·--·used·0.893359s·(cpu);·0.440371s·(thread);·0s·(gc)
  
441 o31·=·R'441 o31·=·R'
  
442 o31·:·QuotientRing</code></pre>442 o31·:·QuotientRing</code></pre>
443 </td>··········</tr>443 </td>··········</tr>
444 ··········<tr>444 ··········<tr>
445 <td>··············<pre><code·class="language-macaulay2">i32·:·netList·(ideal·R')_*445 <td>··············<pre><code·class="language-macaulay2">i32·:·netList·(ideal·R')_*
Offset 500, 15 lines modifiedOffset 500, 15 lines modified
  
500 o35·=·R500 o35·=·R
  
501 o35·:·QuotientRing</code></pre>501 o35·:·QuotientRing</code></pre>
502 </td>··········</tr>502 </td>··········</tr>
503 ··········<tr>503 ··········<tr>
504 <td>··············<pre><code·class="language-macaulay2">i36·:·time·R'·=·integralClosure·R504 <td>··············<pre><code·class="language-macaulay2">i36·:·time·R'·=·integralClosure·R
505 ·--·used·0.0981306s·(cpu);·0.0512022s·(thread);·0s·(gc)505 ·--·used·0.144615s·(cpu);·0.0580953s·(thread);·0s·(gc)
  
506 o36·=·R'506 o36·=·R'
  
507 o36·:·QuotientRing</code></pre>507 o36·:·QuotientRing</code></pre>
508 </td>··········</tr>508 </td>··········</tr>
509 ··········<tr>509 ··········<tr>
510 <td>··············<pre><code·class="language-macaulay2">i37·:·netList·(ideal·R')_*510 <td>··············<pre><code·class="language-macaulay2">i37·:·netList·(ideal·R')_*
Offset 560, 15 lines modifiedOffset 560, 15 lines modified
  
560 o40·=·R560 o40·=·R
  
561 o40·:·QuotientRing</code></pre>561 o40·:·QuotientRing</code></pre>
562 </td>··········</tr>562 </td>··········</tr>
563 ··········<tr>563 ··········<tr>
564 <td>··············<pre><code·class="language-macaulay2">i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)564 <td>··············<pre><code·class="language-macaulay2">i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
565 ·--·used·0.100201s·(cpu);·0.0565278s·(thread);·0s·(gc)565 ·--·used·0.113547s·(cpu);·0.0617245s·(thread);·0s·(gc)
  
566 o41·=·R'566 o41·=·R'
  
567 o41·:·QuotientRing</code></pre>567 o41·:·QuotientRing</code></pre>
568 </td>··········</tr>568 </td>··········</tr>
569 ··········<tr>569 ··········<tr>
570 <td>··············<pre><code·class="language-macaulay2">i42·:·icFractions·R570 <td>··············<pre><code·class="language-macaulay2">i42·:·icFractions·R
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Offset 49, 15 lines modifiedOffset 49, 15 lines modified
49 o2·:·Ideal·of·S49 o2·:·Ideal·of·S
50 i3·:·R·=·S/f50 i3·:·R·=·S/f
  
51 o3·=·R51 o3·=·R
  
52 o3·:·QuotientRing52 o3·:·QuotientRing
53 i4·:·time·R'·=·integralClosure·R53 i4·:·time·R'·=·integralClosure·R
54 ·--·used·0.517294s·(cpu);·0.326626s·(thread);·0s·(gc)54 ·--·used·0.700828s·(cpu);·0.390718s·(thread);·0s·(gc)
  
55 o4·=·R'55 o4·=·R'
  
56 o4·:·QuotientRing56 o4·:·QuotientRing
57 i5·:·netList·(ideal·R')_*57 i5·:·netList·(ideal·R')_*
  
58 ·····+------------------------------------------------------------------------+58 ·····+------------------------------------------------------------------------+
Offset 110, 15 lines modifiedOffset 110, 15 lines modified
110 o8·:·Ideal·of·S110 o8·:·Ideal·of·S
111 i9·:·R·=·S/f111 i9·:·R·=·S/f
  
112 o9·=·R112 o9·=·R
  
113 o9·:·QuotientRing113 o9·:·QuotientRing
114 i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)114 i10·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
115 ·--·used·0.596554s·(cpu);·0.367717s·(thread);·0s·(gc)115 ·--·used·0.669042s·(cpu);·0.365586s·(thread);·0s·(gc)
  
116 o10·=·R'116 o10·=·R'
  
117 o10·:·QuotientRing117 o10·:·QuotientRing
118 i11·:·netList·(ideal·R')_*118 i11·:·netList·(ideal·R')_*
  
119 ······+------------------------------------------------------------------------119 ······+------------------------------------------------------------------------
Offset 200, 15 lines modifiedOffset 200, 15 lines modified
200 o14·:·Ideal·of·S200 o14·:·Ideal·of·S
201 i15·:·R·=·S/f201 i15·:·R·=·S/f
  
202 o15·=·R202 o15·=·R
  
203 o15·:·QuotientRing203 o15·:·QuotientRing
204 i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)204 i16·:·time·R'·=·integralClosure(R,·Strategy·=>·AllCodimensions)
205 ·--·used·0.595118s·(cpu);·0.372909s·(thread);·0s·(gc)205 ·--·used·0.686389s·(cpu);·0.364268s·(thread);·0s·(gc)
  
206 o16·=·R'206 o16·=·R'
  
207 o16·:·QuotientRing207 o16·:·QuotientRing
208 i17·:·netList·(ideal·R')_*208 i17·:·netList·(ideal·R')_*
  
209 ······+------------------------------------------------------------------------209 ······+------------------------------------------------------------------------
Offset 282, 15 lines modifiedOffset 282, 15 lines modified
282 o19·:·Ideal·of·S282 o19·:·Ideal·of·S
283 i20·:·R·=·S/f283 i20·:·R·=·S/f
  
284 o20·=·R284 o20·=·R
  
285 o20·:·QuotientRing285 o20·:·QuotientRing
286 i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)286 i21·:·time·R'·=·integralClosure(R,·Strategy·=>·SimplifyFractions)
287 ·--·used·0.607605s·(cpu);·0.372017s·(thread);·0s·(gc)287 ·--·used·0.832651s·(cpu);·0.422697s·(thread);·0s·(gc)
  
288 o21·=·R'288 o21·=·R'
  
289 o21·:·QuotientRing289 o21·:·QuotientRing
290 i22·:·netList·(ideal·R')_*290 i22·:·netList·(ideal·R')_*
  
291 ······+------------------------------------------------------------------------291 ······+------------------------------------------------------------------------
Offset 364, 15 lines modifiedOffset 364, 15 lines modified
364 o24·:·Ideal·of·S364 o24·:·Ideal·of·S
365 i25·:·R·=·S/f365 i25·:·R·=·S/f
  
366 o25·=·R366 o25·=·R
  
367 o25·:·QuotientRing367 o25·:·QuotientRing
368 i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)368 i26·:·time·R'·=·integralClosure·(R,·Strategy·=>·RadicalCodim1)
369 ·--·used·1.12494s·(cpu);·0.665136s·(thread);·0s·(gc)369 ·--·used·1.53817s·(cpu);·0.731348s·(thread);·0s·(gc)
  
370 o26·=·R'370 o26·=·R'
  
371 o26·:·QuotientRing371 o26·:·QuotientRing
372 i27·:·netList·(ideal·R')_*372 i27·:·netList·(ideal·R')_*
  
373 ······+------------------------------------------------------------------------373 ······+------------------------------------------------------------------------
Offset 446, 15 lines modifiedOffset 446, 15 lines modified
446 o29·:·Ideal·of·S446 o29·:·Ideal·of·S
447 i30·:·R·=·S/f447 i30·:·R·=·S/f
  
448 o30·=·R448 o30·=·R
  
449 o30·:·QuotientRing449 o30·:·QuotientRing
450 i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)450 i31·:·time·R'·=·integralClosure·(R,·Strategy·=>·Vasconcelos)
451 ·--·used·0.678664s·(cpu);·0.396606s·(thread);·0s·(gc)451 ·--·used·0.893359s·(cpu);·0.440371s·(thread);·0s·(gc)
  
452 o31·=·R'452 o31·=·R'
  
453 o31·:·QuotientRing453 o31·:·QuotientRing
454 i32·:·netList·(ideal·R')_*454 i32·:·netList·(ideal·R')_*
  
455 ······+------------------------------------------------------------------------455 ······+------------------------------------------------------------------------
Offset 528, 15 lines modifiedOffset 528, 15 lines modified
528 o34·:·Ideal·of·S528 o34·:·Ideal·of·S
529 i35·:·R·=·S/f529 i35·:·R·=·S/f
  
530 o35·=·R530 o35·=·R
  
531 o35·:·QuotientRing531 o35·:·QuotientRing
532 i36·:·time·R'·=·integralClosure·R532 i36·:·time·R'·=·integralClosure·R
533 ·--·used·0.0981306s·(cpu);·0.0512022s·(thread);·0s·(gc)533 ·--·used·0.144615s·(cpu);·0.0580953s·(thread);·0s·(gc)
  
534 o36·=·R'534 o36·=·R'
  
535 o36·:·QuotientRing535 o36·:·QuotientRing
536 i37·:·netList·(ideal·R')_*536 i37·:·netList·(ideal·R')_*
  
537 ······+-----------+537 ······+-----------+
Offset 574, 15 lines modifiedOffset 574, 15 lines modified
574 o39·:·Ideal·of·S574 o39·:·Ideal·of·S
575 i40·:·R·=·S/I575 i40·:·R·=·S/I
  
576 o40·=·R576 o40·=·R
  
577 o40·:·QuotientRing577 o40·:·QuotientRing
578 i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)578 i41·:·time·R'·=·integralClosure(R,·Strategy·=>·Radical)
579 ·--·used·0.100201s·(cpu);·0.0565278s·(thread);·0s·(gc)579 ·--·used·0.113547s·(cpu);·0.0617245s·(thread);·0s·(gc)
  
580 o41·=·R'580 o41·=·R'
  
581 o41·:·QuotientRing581 o41·:·QuotientRing
582 i42·:·icFractions·R582 i42·:·icFractions·R
  
583 ········2583 ········2
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66 ········</div>66 ········</div>
67 ········<table·class="examples">67 ········<table·class="examples">
68 ··········<tr>68 ··········<tr>
69 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);</code></pre>69 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)72 <td>··············<pre><code·class="language-macaulay2">i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)
73 ·[jacobian·time·0·sec·#minors·3]73 ·[jacobian·time·.000766717·sec·#minors·3]
74 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·274 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·2
  
75 ·[step·0:·75 ·[step·0:·
76 ······radical·(use·minprimes)·.00304692·seconds76 ······radical·(use·minprimes)·0·seconds
77 ······idlizer1:··.00643697·seconds77 ······idlizer1:··.00680135·seconds
78 ······idlizer2:··.00793008·seconds78 ······idlizer2:··.0110873·seconds
79 ······minpres:···.0079959·seconds79 ······minpres:···.00796023·seconds
80 ··time·.0855872·sec··#fractions·4]80 ··time·.129816·sec··#fractions·4]
81 ·[step·1:·81 ·[step·1:·
82 ······radical·(use·minprimes)·.0039771·seconds82 ······radical·(use·minprimes)·0·seconds
 83 ······idlizer1:··.0119441·seconds
83 ······idlizer1:··.00800509·seconds84 ······idlizer2:··.064009·seconds
84 ······idlizer2:··.0618165·seconds 
85 ······minpres:···.012009·seconds85 ······minpres:···.00798687·seconds
86 ··time·.0880962·sec··#fractions·4]86 ··time·.0961367·sec··#fractions·4]
87 ·[step·2:·87 ·[step·2:·
88 ······radical·(use·minprimes)·.00300098·seconds88 ······radical·(use·minprimes)·.00392575·seconds
89 ······idlizer1:··.0120029·seconds89 ······idlizer1:··.0104333·seconds
90 ······idlizer2:··.00895389·seconds90 ······idlizer2:··.011965·seconds
91 ······minpres:···.00834706·seconds91 ······minpres:···.0120078·seconds
92 ··time·.0906009·sec··#fractions·5]92 ··time·.094474·sec··#fractions·5]
93 ·[step·3:·93 ·[step·3:·
94 ······radical·(use·minprimes)·0·seconds94 ······radical·(use·minprimes)·.00399905·seconds
95 ······idlizer1:··.00920056·seconds95 ······idlizer1:··.0119994·seconds
96 ······idlizer2:··.0119983·seconds96 ······idlizer2:··.0119997·seconds
97 ······minpres:···.0668751·seconds97 ······minpres:···.0947087·seconds
98 ··time·.0988007·sec··#fractions·5]98 ··time·.130949·sec··#fractions·5]
99 ·[step·4:·99 ·[step·4:·
100 ······radical·(use·minprimes)·.00395202·seconds100 ······radical·(use·minprimes)·.000826229·seconds
101 ······idlizer1:··.00800066·seconds101 ······idlizer1:··.00673216·seconds
102 ······idlizer2:··.0119991·seconds102 ······idlizer2:··.0155193·seconds
103 ······minpres:···.008001·seconds103 ······minpres:···.0119941·seconds
104 ··time·.0399368·sec··#fractions·5]104 ··time·.0492054·sec··#fractions·5]
105 ·[step·5:·105 ·[step·5:·
106 ······radical·(use·minprimes)·.00400227·seconds106 ······radical·(use·minprimes)·0·seconds
107 ······idlizer1:···--·used·0.478393s·(cpu);·0.308821s·(thread);·0s·(gc)107 ······idlizer1:···--·used·0.577836s·(cpu);·0.301618s·(thread);·0s·(gc)
108 .00798507·seconds108 .00391349·seconds
109 ··time·.0713716·sec··#fractions·5]109 ··time·.0703759·sec··#fractions·5]
  
110 o2·=·R'110 o2·=·R'
  
111 o2·:·QuotientRing</code></pre>111 o2·:·QuotientRing</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i3·:·trim·ideal·R'114 <td>··············<pre><code·class="language-macaulay2">i3·:·trim·ideal·R'
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Offset 13, 52 lines modifiedOffset 13, 52 lines modified
13 ············displayed.·A·value·of·0·means:·keep·quiet.13 ············displayed.·A·value·of·0·means:·keep·quiet.
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 When·the·computation·takes·a·considerable·time,·this·function·can·be·used·to15 When·the·computation·takes·a·considerable·time,·this·function·can·be·used·to
16 decide·if·it·will·ever·finish,·or·to·get·a·feel·for·what·is·happening·during16 decide·if·it·will·ever·finish,·or·to·get·a·feel·for·what·is·happening·during
17 the·computation.17 the·computation.
18 i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);18 i1·:·R·=·QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
19 i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)19 i2·:·time·R'·=·integralClosure(R,·Verbosity·=>·2)
20 ·[jacobian·time·0·sec·#minors·3]20 ·[jacobian·time·.000766717·sec·#minors·3]
21 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·221 integral·closure·nvars·3·numgens·1·is·S2·codim·1·codimJ·2
  
22 ·[step·0:22 ·[step·0:
23 ······radical·(use·minprimes)·.00304692·seconds23 ······radical·(use·minprimes)·0·seconds
24 ······idlizer1:··.00643697·seconds24 ······idlizer1:··.00680135·seconds
25 ······idlizer2:··.00793008·seconds25 ······idlizer2:··.0110873·seconds
26 ······minpres:···.0079959·seconds26 ······minpres:···.00796023·seconds
27 ··time·.0855872·sec··#fractions·4]27 ··time·.129816·sec··#fractions·4]
28 ·[step·1:28 ·[step·1:
29 ······radical·(use·minprimes)·.0039771·seconds29 ······radical·(use·minprimes)·0·seconds
 30 ······idlizer1:··.0119441·seconds
30 ······idlizer1:··.00800509·seconds31 ······idlizer2:··.064009·seconds
31 ······idlizer2:··.0618165·seconds 
32 ······minpres:···.012009·seconds32 ······minpres:···.00798687·seconds
33 ··time·.0880962·sec··#fractions·4]33 ··time·.0961367·sec··#fractions·4]
34 ·[step·2:34 ·[step·2:
35 ······radical·(use·minprimes)·.00300098·seconds35 ······radical·(use·minprimes)·.00392575·seconds
36 ······idlizer1:··.0120029·seconds36 ······idlizer1:··.0104333·seconds
37 ······idlizer2:··.00895389·seconds37 ······idlizer2:··.011965·seconds
38 ······minpres:···.00834706·seconds38 ······minpres:···.0120078·seconds
39 ··time·.0906009·sec··#fractions·5]39 ··time·.094474·sec··#fractions·5]
40 ·[step·3:40 ·[step·3:
41 ······radical·(use·minprimes)·0·seconds41 ······radical·(use·minprimes)·.00399905·seconds
42 ······idlizer1:··.00920056·seconds42 ······idlizer1:··.0119994·seconds
43 ······idlizer2:··.0119983·seconds43 ······idlizer2:··.0119997·seconds
44 ······minpres:···.0668751·seconds44 ······minpres:···.0947087·seconds
45 ··time·.0988007·sec··#fractions·5]45 ··time·.130949·sec··#fractions·5]
46 ·[step·4:46 ·[step·4:
47 ······radical·(use·minprimes)·.00395202·seconds47 ······radical·(use·minprimes)·.000826229·seconds
48 ······idlizer1:··.00800066·seconds48 ······idlizer1:··.00673216·seconds
49 ······idlizer2:··.0119991·seconds49 ······idlizer2:··.0155193·seconds
50 ······minpres:···.008001·seconds50 ······minpres:···.0119941·seconds
51 ··time·.0399368·sec··#fractions·5]51 ··time·.0492054·sec··#fractions·5]
52 ·[step·5:52 ·[step·5:
53 ······radical·(use·minprimes)·.00400227·seconds53 ······radical·(use·minprimes)·0·seconds
54 ······idlizer1:···--·used·0.478393s·(cpu);·0.308821s·(thread);·0s·(gc)54 ······idlizer1:···--·used·0.577836s·(cpu);·0.301618s·(thread);·0s·(gc)
55 .00798507·seconds55 .00391349·seconds
56 ··time·.0713716·sec··#fractions·5]56 ··time·.0703759·sec··#fractions·5]
  
57 o2·=·R'57 o2·=·R'
  
58 o2·:·QuotientRing58 o2·:·QuotientRing
59 i3·:·trim·ideal·R'59 i3·:·trim·ideal·R'
  
60 ·····················3···2·····················2·2····4···········460 ·····················3···2·····················2·2····4···········4
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./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html
    
Offset 111, 27 lines modifiedOffset 111, 27 lines modified
111 ················2······2····2········2···2·2·····2111 ················2······2····2········2···2·2·····2
112 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)112 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)
  
113 o3·:·Ideal·of·S</code></pre>113 o3·:·Ideal·of·S</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i4·:·time·integralClosure·J116 <td>··············<pre><code·class="language-macaulay2">i4·:·time·integralClosure·J
117 ·--·used·6.09789s·(cpu);·1.29964s·(thread);·0s·(gc)117 ·--·used·5.39518s·(cpu);·1.16459s·(thread);·0s·(gc)
  
118 ·············2·2··············2·2················2··········2···2·····118 ·············2·2··············2·2················2··········2···2·····
119 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-119 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
120 ·····------------------------------------------------------------------------120 ·····------------------------------------------------------------------------
121 ···········2···3···············2·2·····2···5121 ···········2···3···············2·2·····2···5
122 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)122 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
123 o4·:·Ideal·of·S</code></pre>123 o4·:·Ideal·of·S</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})126 <td>··············<pre><code·class="language-macaulay2">i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})
127 ·--·used·9.42242s·(cpu);·1.59349s·(thread);·0s·(gc)127 ·--·used·7.48339s·(cpu);·1.23046s·(thread);·0s·(gc)
  
128 ·············2·2··············2·2················2··········2···2·····128 ·············2·2··············2·2················2··········2···2·····
129 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-129 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
130 ·····------------------------------------------------------------------------130 ·····------------------------------------------------------------------------
131 ···········2···3···············2·2·····2···5131 ···········2···3···············2·2·····2···5
132 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)132 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
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Offset 47, 25 lines modifiedOffset 47, 25 lines modified
47 i3·:·J·=·ideal·jacobian·ideal·F47 i3·:·J·=·ideal·jacobian·ideal·F
  
48 ················2······2····2········2···2·2·····248 ················2······2····2········2···2·2·····2
49 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)49 o3·=·ideal·(2a*b·c·+·3a·,·2a·b*c·+·3b·,·a·b··+·3c·)
  
50 o3·:·Ideal·of·S50 o3·:·Ideal·of·S
51 i4·:·time·integralClosure·J51 i4·:·time·integralClosure·J
52 ·--·used·6.09789s·(cpu);·1.29964s·(thread);·0s·(gc)52 ·--·used·5.39518s·(cpu);·1.16459s·(thread);·0s·(gc)
  
53 ·············2·2··············2·2················2··········2···253 ·············2·2··············2·2················2··········2···2
54 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-54 o4·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ···········2···3···············2·2·····2···556 ···········2···3···············2·2·····2···5
57 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)57 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
58 o4·:·Ideal·of·S58 o4·:·Ideal·of·S
59 i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})59 i5·:·time·integralClosure(J,·Strategy=>{RadicalCodim1})
60 ·--·used·9.42242s·(cpu);·1.59349s·(thread);·0s·(gc)60 ·--·used·7.48339s·(cpu);·1.23046s·(thread);·0s·(gc)
  
61 ·············2·2··············2·2················2··········2···261 ·············2·2··············2·2················2··········2···2
62 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-62 o5·=·ideal·(b·c··-·16000a*c,·a·c··-·16000b*c,·a*b·c·-·16000a·,·a·b*c·-
63 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
64 ···········2···3···············2·2·····2···564 ···········2···3···············2·2·····2···5
65 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)65 ·····16000b·,·a·c·-·16000a*b,·a·b··+·3c·,·a·b·+·15997a*c)
  
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=137 #:len=13
8 c2NocmVpZXJHcmFwaA==8 c2NocmVpZXJHcmFwaA==
9 #:len=17129 #:len=1712
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p
11 dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYxLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi11 dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYxLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi
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./usr/share/doc/Macaulay2/InvariantRing/example-output/_equivariant__Hilbert.out
    
Offset 25, 15 lines modifiedOffset 25, 15 lines modified
25 o3·:·DiagonalAction25 o3·:·DiagonalAction
  
26 i4·:·T.cache.?equivariantHilbert26 i4·:·T.cache.?equivariantHilbert
  
27 o4·=·false27 o4·=·false
  
28 i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)28 i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)
29 ·--·.00203254s·elapsed29 ·--·.00352027s·elapsed
  
30 ··················-1····-1·······2·2··············-2····-1·-1····-2··2··30 ··················-1····-1·······2·2··············-2····-1·-1····-2··2··
31 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+31 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+
32 ···········0·1····1·····0········0·1····0····1····1·····0··1·····0······32 ···········0·1····1·····0········0·1····0····1····1·····0··1·····0······
33 ·····------------------------------------------------------------------------33 ·····------------------------------------------------------------------------
34 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··334 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··3
35 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T·35 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T·
Offset 51, 10 lines modifiedOffset 51, 10 lines modified
51 ·········0···151 ·········0···1
  
52 i6·:·T.cache.?equivariantHilbert52 i6·:·T.cache.?equivariantHilbert
  
53 o6·=·true53 o6·=·true
  
54 i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);54 i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);
55 ·--·.000367185s·elapsed55 ·--·.000677821s·elapsed
  
56 i8·:·56 i8·:·
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./usr/share/doc/Macaulay2/InvariantRing/example-output/_hsop_spalgorithms.out
    
Offset 23, 23 lines modifiedOffset 23, 23 lines modified
23 o3·=·QQ[x..z]·<-·{|·0·-1·0··|,·|·0·-1·0·|}23 o3·=·QQ[x..z]·<-·{|·0·-1·0··|,·|·0·-1·0·|}
24 ··················|·1·0··0··|··|·1·0··0·|24 ··················|·1·0··0··|··|·1·0··0·|
25 ··················|·0·0··-1·|··|·0·0··1·|25 ··················|·0·0··-1·|··|·0·0··1·|
  
26 o3·:·FiniteGroupAction26 o3·:·FiniteGroupAction
  
27 i4·:·time·P1=primaryInvariants·C4xC227 i4·:·time·P1=primaryInvariants·C4xC2
28 ·--·used·0.519864s·(cpu);·0.397396s·(thread);·0s·(gc)28 ·--·used·0.614351s·(cpu);·0.448703s·(thread);·0s·(gc)
  
29 ·······2···2····2···2·229 ·······2···2····2···2·2
30 o4·=·{z·,·x··+·y·,·x·y·}30 o4·=·{z·,·x··+·y·,·x·y·}
  
31 o4·:·List31 o4·:·List
  
32 i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)32 i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)
33 ·--·used·0.441691s·(cpu);·0.288825s·(thread);·0s·(gc)33 ·--·used·0.630299s·(cpu);·0.419936s·(thread);·0s·(gc)
  
34 ···········8·········7··········6·2·········5·3·········4·4·········3·5··34 ···········8·········7··········6·2·········5·3·········4·4·········3·5··
35 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+35 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+
36 ·····------------------------------------------------------------------------36 ·····------------------------------------------------------------------------
37 ···········2·6···········7········8·········6·2·········5···2·········4·2·2··37 ···········2·6···········7········8·········6·2·········5···2·········4·2·2··
38 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-38 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
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81 ·····------------------------------------------------------------------------81 ·····------------------------------------------------------------------------
82 ···············2·6·············2·6············882 ···············2·6·············2·6············8
83 ·····348625296x·z··-·348625296y·z··+·43046721z·}83 ·····348625296x·z··-·348625296y·z··+·43046721z·}
  
84 o5·:·List84 o5·:·List
  
85 i6·:·time·secondaryInvariants(P1,C4xC2)85 i6·:·time·secondaryInvariants(P1,C4xC2)
86 ·--·used·0.069519s·(cpu);·0.0434899s·(thread);·0s·(gc)86 ·--·used·0.0850753s·(cpu);·0.04805s·(thread);·0s·(gc)
  
87 ··········3·······387 ··········3·······3
88 o6·=·{1,·x·y·-·x*y·}88 o6·=·{1,·x·y·-·x*y·}
  
89 o6·:·List89 o6·:·List
  
90 i7·:·time·secondaryInvariants(P2,C4xC2)90 i7·:·time·secondaryInvariants(P2,C4xC2)
91 ·--·used·1.29892s·(cpu);·1.01851s·(thread);·0s·(gc)91 ·--·used·1.65608s·(cpu);·1.18215s·(thread);·0s·(gc)
  
92 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4··92 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4··
93 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+93 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+
94 ·····------------------------------------------------------------------------94 ·····------------------------------------------------------------------------
95 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···695 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···6
96 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x·96 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x·
97 ·····------------------------------------------------------------------------97 ·····------------------------------------------------------------------------
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22 ·····|·0·1·-1·1··|22 ·····|·0·1·-1·1··|
23 ·····|·1·0·-1·-1·|23 ·····|·1·0·-1·-1·|
  
24 o3·:·DiagonalAction24 o3·:·DiagonalAction
  
25 i4·:·S·=·invariantRing·T25 i4·:·S·=·invariantRing·T
  
 26 o4·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5·
 27 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 28 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3·
 29 ·····------------------------------------------------------------------------
26 o4·=·············230 ··············2
27 ·····QQ[x·x·x·,·x·x·x·]31 ·····x·x·x·,·x·x·x·]
28 ·········1·2·3···1·3·432 ······1·2·3···1·3·4
  
29 o4·:·RingOfInvariants33 o4·:·RingOfInvariants
  
30 i5·:·S·=·R^T34 i5·:·S·=·R^T
  
 35 o5·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5·
 36 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 37 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3·
 38 ·····------------------------------------------------------------------------
31 o5·=·············239 ··············2
32 ·····QQ[x·x·x·,·x·x·x·]40 ·····x·x·x·,·x·x·x·]
33 ·········1·2·3···1·3·441 ······1·2·3···1·3·4
  
34 o5·:·RingOfInvariants42 o5·:·RingOfInvariants
  
35 i6·:·43 i6·:·
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14 ···········|·1·0·0·0·|··|·1·0·0·0·|14 ···········|·1·0·0·0·|··|·1·0·0·0·|
15 ···········|·0·0·1·0·|··|·0·1·0·0·|15 ···········|·0·0·1·0·|··|·0·1·0·0·|
16 ···········|·0·0·0·1·|··|·0·0·1·0·|16 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
17 o3·:·FiniteGroupAction17 o3·:·FiniteGroupAction
  
18 i4·:·elapsedTime·invariants·S418 i4·:·elapsedTime·invariants·S4
19 ·--·.757738s·elapsed19 ·--·.639789s·elapsed
  
20 ··························2····2····2····2···3····3····3····3···4····4····4··20 ··························2····2····2····2···3····3····3····3···4····4····4··
21 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+21 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
22 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··22 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··
23 ·····------------------------------------------------------------------------23 ·····------------------------------------------------------------------------
24 ······424 ······4
25 ·····x·}25 ·····x·}
26 ······426 ······4
  
27 o4·:·List27 o4·:·List
  
28 i5·:·elapsedTime·invariants(S4,DegreeBound=>4)28 i5·:·elapsedTime·invariants(S4,DegreeBound=>4)
29 ·--·.632792s·elapsed29 ·--·.399958s·elapsed
  
30 Warning:·stopping·condition·not·met!30 Warning:·stopping·condition·not·met!
31 Output·may·not·generate·the·entire·ring·of·invariants.31 Output·may·not·generate·the·entire·ring·of·invariants.
32 Increase·value·of·DegreeBound.32 Increase·value·of·DegreeBound.
  
  
33 ··························2····2····2····2···3····3····3····3···4····4····4··33 ··························2····2····2····2···3····3····3····3···4····4····4··
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14 ···········|·1·0·0·0·|··|·1·0·0·0·|14 ···········|·1·0·0·0·|··|·1·0·0·0·|
15 ···········|·0·0·1·0·|··|·0·1·0·0·|15 ···········|·0·0·1·0·|··|·0·1·0·0·|
16 ···········|·0·0·0·1·|··|·0·0·1·0·|16 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
17 o3·:·FiniteGroupAction17 o3·:·FiniteGroupAction
  
18 i4·:·elapsedTime·invariants·S418 i4·:·elapsedTime·invariants·S4
19 ·--·.551284s·elapsed19 ·--·.997107s·elapsed
  
20 ··························2····2····2····2···3····3····3····3···4····4····4··20 ··························2····2····2····2···3····3····3····3···4····4····4··
21 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+21 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
22 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··22 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··
23 ·····------------------------------------------------------------------------23 ·····------------------------------------------------------------------------
24 ······424 ······4
25 ·····x·}25 ·····x·}
26 ······426 ······4
  
27 o4·:·List27 o4·:·List
  
28 i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)28 i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)
29 ·--·.084845s·elapsed29 ·--·.0674603s·elapsed
  
30 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+30 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+
31 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·3··31 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·3··
32 ·····------------------------------------------------------------------------32 ·····------------------------------------------------------------------------
33 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}33 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}
34 ······1·2·4····1·3·4····2·3·4···1·2·3·434 ······1·2·4····1·3·4····2·3·4···1·2·3·4
  
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80 ············680 ············6
81 ·····104976z·}81 ·····104976z·}
  
82 o4·:·List82 o4·:·List
  
83 i5·:·primaryInvariants(S3)83 i5·:·primaryInvariants(S3)
  
84 ···································3····3····3 
85 o5·=·{x·+·y·+·z,·x*y·+·x*z·+·y*z,·x··+·y··+·z·}84 ··················2····2····2···2·······2····2·····2·······2······2
 85 o5·=·{x·+·y·+·z,·x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·}
  
86 o5·:·List86 o5·:·List
  
87 i6·:·K=GF(101)87 i6·:·K=GF(101)
  
88 o6·=·K88 o6·=·K
  
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Offset 16, 13 lines modifiedOffset 16, 13 lines modified
16 ··················|·0·0·1·|··|·1·0·0·|16 ··················|·0·0·1·|··|·1·0·0·|
17 ··················|·1·0·0·|··|·0·0·1·|17 ··················|·1·0·0·|··|·0·0·1·|
  
18 o3·:·FiniteGroupAction18 o3·:·FiniteGroupAction
  
19 i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})19 i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})
  
20 ·······3····3····3···2·······2····2·····2·······2······2···4····4····420 ·······2·······2····2·····2·······2······2··········4····4····4
21 o4·=·{x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x··+·y··+·z·}21 o4·=·{x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x*y*z,·x··+·y··+·z·}
  
22 o4·:·List22 o4·:·List
  
23 i5·:·23 i5·:·
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i4·:·T.cache.?equivariantHilbert77 <td>··············<pre><code·class="language-macaulay2">i4·:·T.cache.?equivariantHilbert
  
78 o4·=·false</code></pre>78 o4·=·false</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)81 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)
82 ·--·.00203254s·elapsed82 ·--·.00352027s·elapsed
  
83 ··················-1····-1·······2·2··············-2····-1·-1····-2··2··83 ··················-1····-1·······2·2··············-2····-1·-1····-2··2··
84 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+84 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+
85 ···········0·1····1·····0········0·1····0····1····1·····0··1·····0······85 ···········0·1····1·····0········0·1····0····1····1·····0··1·····0······
86 ·····------------------------------------------------------------------------86 ·····------------------------------------------------------------------------
87 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··387 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··3
88 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T·88 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T·
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104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i6·:·T.cache.?equivariantHilbert105 <td>··············<pre><code·class="language-macaulay2">i6·:·T.cache.?equivariantHilbert
  
106 o6·=·true</code></pre>106 o6·=·true</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);109 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);
110 ·--·.000367185s·elapsed</code></pre>110 ·--·.000677821s·elapsed</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ········</table>112 ········</table>
113 ······</div>113 ······</div>
114 ······<div·class="waystouse">114 ······<div·class="waystouse">
115 ········<h2>For·the·programmer</h2>115 ········<h2>For·the·programmer</h2>
116 ········<p>The·object·<a·title="stores·equivariant·Hilbert·series·expansions"·href="_equivariant__Hilbert.html">equivariantHilbert</a>·is·<span>a·<a·title="the·class·of·all·symbols"·href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p>116 ········<p>The·object·<a·title="stores·equivariant·Hilbert·series·expansions"·href="_equivariant__Hilbert.html">equivariantHilbert</a>·is·<span>a·<a·title="the·class·of·all·symbols"·href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p>
117 ······</div>117 ······</div>
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30 ·····|·0··-1·1·|30 ·····|·0··-1·1·|
  
31 o3·:·DiagonalAction31 o3·:·DiagonalAction
32 i4·:·T.cache.?equivariantHilbert32 i4·:·T.cache.?equivariantHilbert
  
33 o4·=·false33 o4·=·false
34 i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)34 i5·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5)
35 ·--·.00203254s·elapsed35 ·--·.00352027s·elapsed
  
36 ··················-1····-1·······2·2··············-2····-1·-1····-2··236 ··················-1····-1·······2·2··············-2····-1·-1····-2··2
37 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+37 o5·=·1·+·(z·z··+·z···+·z··)T·+·(z·z··+·z··+·z··+·z···+·z··z···+·z··)T··+
38 ···········0·1····1·····0········0·1····0····1····1·····0··1·····038 ···········0·1····1·····0········0·1····0····1····1·····0··1·····0
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··340 ·······3·3····2········2······-1········-3····-1······-1·-2····-2·-1····-3··3
41 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T41 ·····(z·z··+·z·z··+·z·z··+·z·z···+·1·+·z···+·z··z··+·z··z···+·z··z···+·z··)T
Offset 54, 10 lines modifiedOffset 54, 10 lines modified
  
54 o5·:·ZZ[z·..z·][T]54 o5·:·ZZ[z·..z·][T]
55 ·········0···155 ·········0···1
56 i6·:·T.cache.?equivariantHilbert56 i6·:·T.cache.?equivariantHilbert
  
57 o6·=·true57 o6·=·true
58 i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);58 i7·:·elapsedTime·equivariantHilbertSeries(T,·Order·=>·5);
59 ·--·.000367185s·elapsed59 ·--·.000677821s·elapsed
60 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*60 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
61 The·object·_\x8e_\x8q_\x8u_\x8i_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8H_\x8i_\x8l_\x8b_\x8e_\x8r_\x8t·is·a·_\x8s_\x8y_\x8m_\x8b_\x8o_\x8l.61 The·object·_\x8e_\x8q_\x8u_\x8i_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8H_\x8i_\x8l_\x8b_\x8e_\x8r_\x8t·is·a·_\x8s_\x8y_\x8m_\x8b_\x8o_\x8l.
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78 o3·:·FiniteGroupAction</code></pre>78 o3·:·FiniteGroupAction</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ········</table>80 ········</table>
81 ········<p>The·two·algorithms·used·in·<a·title="computes·a·list·of·primary·invariants·for·the·invariant·ring·of·a·finite·group"·href="_primary__Invariants.html">primaryInvariants</a>·are·timed.·One·sees·that·the·Dade·algorithm·is·faster,·however·the·primary·invariants·output·are·all·of·degree·8·and·have·ugly·coefficients.</p>81 ········<p>The·two·algorithms·used·in·<a·title="computes·a·list·of·primary·invariants·for·the·invariant·ring·of·a·finite·group"·href="_primary__Invariants.html">primaryInvariants</a>·are·timed.·One·sees·that·the·Dade·algorithm·is·faster,·however·the·primary·invariants·output·are·all·of·degree·8·and·have·ugly·coefficients.</p>
82 ········<table·class="examples">82 ········<table·class="examples">
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·time·P1=primaryInvariants·C4xC284 <td>··············<pre><code·class="language-macaulay2">i4·:·time·P1=primaryInvariants·C4xC2
85 ·--·used·0.519864s·(cpu);·0.397396s·(thread);·0s·(gc)85 ·--·used·0.614351s·(cpu);·0.448703s·(thread);·0s·(gc)
  
86 ·······2···2····2···2·286 ·······2···2····2···2·2
87 o4·=·{z·,·x··+·y·,·x·y·}87 o4·=·{z·,·x··+·y·,·x·y·}
  
88 o4·:·List</code></pre>88 o4·:·List</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)
92 ·--·used·0.441691s·(cpu);·0.288825s·(thread);·0s·(gc)92 ·--·used·0.630299s·(cpu);·0.419936s·(thread);·0s·(gc)
  
93 ···········8·········7··········6·2·········5·3·········4·4·········3·5··93 ···········8·········7··········6·2·········5·3·········4·4·········3·5··
94 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+94 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+
95 ·····------------------------------------------------------------------------95 ·····------------------------------------------------------------------------
96 ···········2·6···········7········8·········6·2·········5···2·········4·2·2··96 ···········2·6···········7········8·········6·2·········5···2·········4·2·2··
97 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-97 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-
98 ·····------------------------------------------------------------------------98 ·····------------------------------------------------------------------------
Offset 141, 24 lines modifiedOffset 141, 24 lines modified
141 o5·:·List</code></pre>141 o5·:·List</code></pre>
142 </td>··········</tr>142 </td>··········</tr>
143 ········</table>143 ········</table>
144 ········<p>The·extra·work·done·by·the·default·algorithm·to·ensure·an·optimal·hsop·is·rewarded·by·needing·to·calculate·a·smaller·collection·of·corresponding·secondary·invariants.··In·fact,·it·has·proved·quicker·overall·to·calculate·the·invariant·ring·based·on·the·optimal·algorithm·rather·than·the·Dade·algorithm.</p>144 ········<p>The·extra·work·done·by·the·default·algorithm·to·ensure·an·optimal·hsop·is·rewarded·by·needing·to·calculate·a·smaller·collection·of·corresponding·secondary·invariants.··In·fact,·it·has·proved·quicker·overall·to·calculate·the·invariant·ring·based·on·the·optimal·algorithm·rather·than·the·Dade·algorithm.</p>
145 ········<table·class="examples">145 ········<table·class="examples">
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i6·:·time·secondaryInvariants(P1,C4xC2)147 <td>··············<pre><code·class="language-macaulay2">i6·:·time·secondaryInvariants(P1,C4xC2)
148 ·--·used·0.069519s·(cpu);·0.0434899s·(thread);·0s·(gc)148 ·--·used·0.0850753s·(cpu);·0.04805s·(thread);·0s·(gc)
  
149 ··········3·······3149 ··········3·······3
150 o6·=·{1,·x·y·-·x*y·}150 o6·=·{1,·x·y·-·x*y·}
  
151 o6·:·List</code></pre>151 o6·:·List</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i7·:·time·secondaryInvariants(P2,C4xC2)154 <td>··············<pre><code·class="language-macaulay2">i7·:·time·secondaryInvariants(P2,C4xC2)
155 ·--·used·1.29892s·(cpu);·1.01851s·(thread);·0s·(gc)155 ·--·used·1.65608s·(cpu);·1.18215s·(thread);·0s·(gc)
  
156 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4··156 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4··
157 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+157 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+
158 ·····------------------------------------------------------------------------158 ·····------------------------------------------------------------------------
159 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···6159 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···6
160 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x·160 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x·
161 ·····------------------------------------------------------------------------161 ·····------------------------------------------------------------------------
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69 ··················|·0·0··-1·|··|·0·0··1·|69 ··················|·0·0··-1·|··|·0·0··1·|
  
70 o3·:·FiniteGroupAction70 o3·:·FiniteGroupAction
71 The·two·algorithms·used·in·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8I_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s·are·timed.·One·sees·that·the·Dade71 The·two·algorithms·used·in·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8I_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s·are·timed.·One·sees·that·the·Dade
72 algorithm·is·faster,·however·the·primary·invariants·output·are·all·of·degree·872 algorithm·is·faster,·however·the·primary·invariants·output·are·all·of·degree·8
73 and·have·ugly·coefficients.73 and·have·ugly·coefficients.
74 i4·:·time·P1=primaryInvariants·C4xC274 i4·:·time·P1=primaryInvariants·C4xC2
75 ·--·used·0.519864s·(cpu);·0.397396s·(thread);·0s·(gc)75 ·--·used·0.614351s·(cpu);·0.448703s·(thread);·0s·(gc)
  
76 ·······2···2····2···2·276 ·······2···2····2···2·2
77 o4·=·{z·,·x··+·y·,·x·y·}77 o4·=·{z·,·x··+·y·,·x·y·}
  
78 o4·:·List78 o4·:·List
79 i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)79 i5·:·time·P2=primaryInvariants(C4xC2,Dade=>true)
80 ·--·used·0.441691s·(cpu);·0.288825s·(thread);·0s·(gc)80 ·--·used·0.630299s·(cpu);·0.419936s·(thread);·0s·(gc)
  
81 ···········8·········7··········6·2·········5·3·········4·4·········3·581 ···········8·········7··········6·2·········5·3·········4·4·········3·5
82 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+82 o5·=·{4096x··+·24576x·y·+·38912x·y··-·18432x·y··-·65280x·y··+·18432x·y··+
83 ·····------------------------------------------------------------------------83 ·····------------------------------------------------------------------------
84 ···········2·6···········7········8·········6·2·········5···2·········4·2·284 ···········2·6···········7········8·········6·2·········5···2·········4·2·2
85 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-85 ·····38912x·y··-·24576x*y··+·4096y··-·23040x·z··-·69120x·y*z··-·28800x·y·z··-
86 ·····------------------------------------------------------------------------86 ·····------------------------------------------------------------------------
Offset 129, 22 lines modifiedOffset 129, 22 lines modified
  
129 o5·:·List129 o5·:·List
130 The·extra·work·done·by·the·default·algorithm·to·ensure·an·optimal·hsop·is130 The·extra·work·done·by·the·default·algorithm·to·ensure·an·optimal·hsop·is
131 rewarded·by·needing·to·calculate·a·smaller·collection·of·corresponding131 rewarded·by·needing·to·calculate·a·smaller·collection·of·corresponding
132 secondary·invariants.·In·fact,·it·has·proved·quicker·overall·to·calculate·the132 secondary·invariants.·In·fact,·it·has·proved·quicker·overall·to·calculate·the
133 invariant·ring·based·on·the·optimal·algorithm·rather·than·the·Dade·algorithm.133 invariant·ring·based·on·the·optimal·algorithm·rather·than·the·Dade·algorithm.
134 i6·:·time·secondaryInvariants(P1,C4xC2)134 i6·:·time·secondaryInvariants(P1,C4xC2)
135 ·--·used·0.069519s·(cpu);·0.0434899s·(thread);·0s·(gc)135 ·--·used·0.0850753s·(cpu);·0.04805s·(thread);·0s·(gc)
  
136 ··········3·······3136 ··········3·······3
137 o6·=·{1,·x·y·-·x*y·}137 o6·=·{1,·x·y·-·x*y·}
  
138 o6·:·List138 o6·:·List
139 i7·:·time·secondaryInvariants(P2,C4xC2)139 i7·:·time·secondaryInvariants(P2,C4xC2)
140 ·--·used·1.29892s·(cpu);·1.01851s·(thread);·0s·(gc)140 ·--·used·1.65608s·(cpu);·1.18215s·(thread);·0s·(gc)
  
141 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4141 ··········2···2····2···4···2·2····2·2···2·2···3·······3···4····4···6···2·4
142 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+142 o7·=·{1,·z·,·x··+·y·,·z·,·x·z··+·y·z·,·x·y·,·x·y·-·x*y·,·x··+·y·,·z·,·x·z··+
143 ·····------------------------------------------------------------------------143 ·····------------------------------------------------------------------------
144 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···6144 ······2·4···2·2·2···3···2······3·2···4·2····4·2···4·2····2·4···5·······5···6
145 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x145 ·····y·z·,·x·y·z·,·x·y*z··-·x*y·z·,·x·z··+·y·z·,·x·y··+·x·y·,·x·y·-·x*y·,·x
146 ·····------------------------------------------------------------------------146 ·····------------------------------------------------------------------------
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110 ·····|·1·0·-1·-1·|110 ·····|·1·0·-1·-1·|
  
111 o3·:·DiagonalAction</code></pre>111 o3·:·DiagonalAction</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i4·:·S·=·invariantRing·T114 <td>··············<pre><code·class="language-macaulay2">i4·:·S·=·invariantRing·T
  
 115 o4·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5·
 116 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 117 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3·
 118 ·····------------------------------------------------------------------------
115 o4·=·············2119 ··············2
116 ·····QQ[x·x·x·,·x·x·x·]120 ·····x·x·x·,·x·x·x·]
117 ·········1·2·3···1·3·4121 ······1·2·3···1·3·4
  
118 o4·:·RingOfInvariants</code></pre>122 o4·:·RingOfInvariants</code></pre>
119 </td>··········</tr>123 </td>··········</tr>
120 ········</table>124 ········</table>
121 ········<p>The·algebra·generators·for·the·ring·of·invariants·are·computed·upon·initialization·by·the·method·<a·title="computes·the·generating·invariants·of·a·group·action"·href="_invariants.html">invariants</a>.</p>125 ········<p>The·algebra·generators·for·the·ring·of·invariants·are·computed·upon·initialization·by·the·method·<a·title="computes·the·generating·invariants·of·a·group·action"·href="_invariants.html">invariants</a>.</p>
122 ········<p>Alternatively,·we·can·use·the·following·shortcut·to·construct·a·ring·of·invariants.</p>126 ········<p>Alternatively,·we·can·use·the·following·shortcut·to·construct·a·ring·of·invariants.</p>
123 ········<table·class="examples">127 ········<table·class="examples">
124 ··········<tr>128 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i5·:·S·=·R^T129 <td>··············<pre><code·class="language-macaulay2">i5·:·S·=·R^T
  
 130 o5·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5·
 131 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 132 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3·
 133 ·····------------------------------------------------------------------------
126 o5·=·············2134 ··············2
127 ·····QQ[x·x·x·,·x·x·x·]135 ·····x·x·x·,·x·x·x·]
128 ·········1·2·3···1·3·4136 ······1·2·3···1·3·4
  
129 o5·:·RingOfInvariants</code></pre>137 o5·:·RingOfInvariants</code></pre>
130 </td>··········</tr>138 </td>··········</tr>
131 ········</table>139 ········</table>
132 ······</div>140 ······</div>
133 ······<div>141 ······<div>
134 ········<h2>Caveat</h2>142 ········<h2>Caveat</h2>
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46 ·····|·0·1·-1·1··|46 ·····|·0·1·-1·1··|
47 ·····|·1·0·-1·-1·|47 ·····|·1·0·-1·-1·|
  
48 o3·:·DiagonalAction48 o3·:·DiagonalAction
49 i4·:·S·=·invariantRing·T49 i4·:·S·=·invariantRing·T
  
 50 o4·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5
 51 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 52 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3
 53 ·····------------------------------------------------------------------------
50 o4·=·············254 ··············2
51 ·····QQ[x·x·x·,·x·x·x·]55 ·····x·x·x·,·x·x·x·]
52 ·········1·2·3···1·3·456 ······1·2·3···1·3·4
  
53 o4·:·RingOfInvariants57 o4·:·RingOfInvariants
54 The·algebra·generators·for·the·ring·of·invariants·are·computed·upon58 The·algebra·generators·for·the·ring·of·invariants·are·computed·upon
55 initialization·by·the·method·_\x8i_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s.59 initialization·by·the·method·_\x8i_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s.
56 Alternatively,·we·can·use·the·following·shortcut·to·construct·a·ring·of60 Alternatively,·we·can·use·the·following·shortcut·to·construct·a·ring·of
57 invariants.61 invariants.
58 i5·:·S·=·R^T62 i5·:·S·=·R^T
  
 63 o5·=·····5···3·2···5·3·4·····4·2·3·····6·3·3···3···2·····4·2·2···5·5·5
 64 ·····QQ[x·x·x·x·,·x·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·x·,·x·x·x·,·x·x·x·,
 65 ·········1·2·3·4···1·2·3·4···1·2·3·4···1·3·4···1·2·3·4···1·3·4···1·2·3
 66 ·····------------------------------------------------------------------------
59 o5·=·············267 ··············2
60 ·····QQ[x·x·x·,·x·x·x·]68 ·····x·x·x·,·x·x·x·]
61 ·········1·2·3···1·3·469 ······1·2·3···1·3·4
  
62 o5·:·RingOfInvariants70 o5·:·RingOfInvariants
63 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*71 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
64 By·default,·the·invariants·of·a·diagonal·group·action·are·computed·over·an72 By·default,·the·invariants·of·a·diagonal·group·action·are·computed·over·an
65 infinite·extension·of·the·ground·field·specified·by·the·user·over·which·the73 infinite·extension·of·the·ground·field·specified·by·the·user·over·which·the
66 action·is·defined.·Provided·the·action·is·defined,·it·is·possible·to·compute74 action·is·defined.·Provided·the·action·is·defined,·it·is·possible·to·compute
67 invariants·literally·over·the·specified·ground·field·in·prime·characteristic75 invariants·literally·over·the·specified·ground·field·in·prime·characteristic
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89 ···········|·0·0·1·0·|··|·0·1·0·0·|89 ···········|·0·0·1·0·|··|·0·1·0·0·|
90 ···········|·0·0·0·1·|··|·0·0·1·0·|90 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
91 o3·:·FiniteGroupAction</code></pre>91 o3·:·FiniteGroupAction</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·invariants·S494 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·invariants·S4
95 ·--·.757738s·elapsed95 ·--·.639789s·elapsed
  
96 ··························2····2····2····2···3····3····3····3···4····4····4··96 ··························2····2····2····2···3····3····3····3···4····4····4··
97 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+97 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
98 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··98 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ······4100 ······4
101 ·····x·}101 ·····x·}
102 ······4102 ······4
  
103 o4·:·List</code></pre>103 o4·:·List</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·invariants(S4,DegreeBound=>4)106 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·invariants(S4,DegreeBound=>4)
107 ·--·.632792s·elapsed107 ·--·.399958s·elapsed
  
108 Warning:·stopping·condition·not·met!108 Warning:·stopping·condition·not·met!
109 Output·may·not·generate·the·entire·ring·of·invariants.109 Output·may·not·generate·the·entire·ring·of·invariants.
110 Increase·value·of·DegreeBound.110 Increase·value·of·DegreeBound.
  
  
111 ··························2····2····2····2···3····3····3····3···4····4····4··111 ··························2····2····2····2···3····3····3····3···4····4····4··
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34 o3·=·R·<-·{|·0·1·0·0·|,·|·0·0·0·1·|}34 o3·=·R·<-·{|·0·1·0·0·|,·|·0·0·0·1·|}
35 ···········|·1·0·0·0·|··|·1·0·0·0·|35 ···········|·1·0·0·0·|··|·1·0·0·0·|
36 ···········|·0·0·1·0·|··|·0·1·0·0·|36 ···········|·0·0·1·0·|··|·0·1·0·0·|
37 ···········|·0·0·0·1·|··|·0·0·1·0·|37 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
38 o3·:·FiniteGroupAction38 o3·:·FiniteGroupAction
39 i4·:·elapsedTime·invariants·S439 i4·:·elapsedTime·invariants·S4
40 ·--·.757738s·elapsed40 ·--·.639789s·elapsed
  
41 ··························2····2····2····2···3····3····3····3···4····4····441 ··························2····2····2····2···3····3····3····3···4····4····4
42 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+42 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
43 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····343 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3
44 ·····------------------------------------------------------------------------44 ·····------------------------------------------------------------------------
45 ······445 ······4
46 ·····x·}46 ·····x·}
47 ······447 ······4
  
48 o4·:·List48 o4·:·List
49 i5·:·elapsedTime·invariants(S4,DegreeBound=>4)49 i5·:·elapsedTime·invariants(S4,DegreeBound=>4)
50 ·--·.632792s·elapsed50 ·--·.399958s·elapsed
  
51 Warning:·stopping·condition·not·met!51 Warning:·stopping·condition·not·met!
52 Output·may·not·generate·the·entire·ring·of·invariants.52 Output·may·not·generate·the·entire·ring·of·invariants.
53 Increase·value·of·DegreeBound.53 Increase·value·of·DegreeBound.
  
  
54 ··························2····2····2····2···3····3····3····3···4····4····454 ··························2····2····2····2···3····3····3····3···4····4····4
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89 ···········|·0·0·1·0·|··|·0·1·0·0·|89 ···········|·0·0·1·0·|··|·0·1·0·0·|
90 ···········|·0·0·0·1·|··|·0·0·1·0·|90 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
91 o3·:·FiniteGroupAction</code></pre>91 o3·:·FiniteGroupAction</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·invariants·S494 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·invariants·S4
95 ·--·.551284s·elapsed95 ·--·.997107s·elapsed
  
96 ··························2····2····2····2···3····3····3····3···4····4····4··96 ··························2····2····2····2···3····3····3····3···4····4····4··
97 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+97 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
98 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··98 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3··
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ······4100 ······4
101 ·····x·}101 ·····x·}
102 ······4102 ······4
  
103 o4·:·List</code></pre>103 o4·:·List</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)106 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)
107 ·--·.084845s·elapsed107 ·--·.0674603s·elapsed
  
108 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+108 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+
109 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·3··109 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·3··
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
111 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}111 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}
112 ······1·2·4····1·3·4····2·3·4···1·2·3·4112 ······1·2·4····1·3·4····2·3·4···1·2·3·4
  
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36 o3·=·R·<-·{|·0·1·0·0·|,·|·0·0·0·1·|}36 o3·=·R·<-·{|·0·1·0·0·|,·|·0·0·0·1·|}
37 ···········|·1·0·0·0·|··|·1·0·0·0·|37 ···········|·1·0·0·0·|··|·1·0·0·0·|
38 ···········|·0·0·1·0·|··|·0·1·0·0·|38 ···········|·0·0·1·0·|··|·0·1·0·0·|
39 ···········|·0·0·0·1·|··|·0·0·1·0·|39 ···········|·0·0·0·1·|··|·0·0·1·0·|
  
40 o3·:·FiniteGroupAction40 o3·:·FiniteGroupAction
41 i4·:·elapsedTime·invariants·S441 i4·:·elapsedTime·invariants·S4
42 ·--·.551284s·elapsed42 ·--·.997107s·elapsed
  
43 ··························2····2····2····2···3····3····3····3···4····4····443 ··························2····2····2····2···3····3····3····3···4····4····4
44 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+44 o4·=·{x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+·x·,·x··+·x··+·x··+
45 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····345 ·······1····2····3····4···1····2····3····4···1····2····3····4···1····2····3
46 ·····------------------------------------------------------------------------46 ·····------------------------------------------------------------------------
47 ······447 ······4
48 ·····x·}48 ·····x·}
49 ······449 ······4
  
50 o4·:·List50 o4·:·List
51 i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)51 i5·:·elapsedTime·invariants(S4,UseLinearAlgebra=>true)
52 ·--·.084845s·elapsed52 ·--·.0674603s·elapsed
  
53 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+53 o5·=·{x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x··+·x·x··+·x·x··+·x·x·,·x·x·x··+
54 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·354 ·······1····2····3····4···1·2····1·3····2·3····1·4····2·4····3·4···1·2·3
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}56 ·····x·x·x··+·x·x·x··+·x·x·x·,·x·x·x·x·}
57 ······1·2·4····1·3·4····2·3·4···1·2·3·457 ······1·2·4····1·3·4····2·3·4···1·2·3·4
  
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158 </td>··········</tr>158 </td>··········</tr>
159 ········</table>159 ········</table>
160 ········<p>Compare·this·result·to·the·hsop·output·when·Dade·is·left·to·its·default·value·<a·href="../../Macaulay2Doc/html/_false.html">false</a>.</p>160 ········<p>Compare·this·result·to·the·hsop·output·when·Dade·is·left·to·its·default·value·<a·href="../../Macaulay2Doc/html/_false.html">false</a>.</p>
161 ········<table·class="examples">161 ········<table·class="examples">
162 ··········<tr>162 ··········<tr>
163 <td>··············<pre><code·class="language-macaulay2">i5·:·primaryInvariants(S3)163 <td>··············<pre><code·class="language-macaulay2">i5·:·primaryInvariants(S3)
  
164 ···································3····3····3 
165 o5·=·{x·+·y·+·z,·x*y·+·x*z·+·y*z,·x··+·y··+·z·}164 ··················2····2····2···2·······2····2·····2·······2······2
 165 o5·=·{x·+·y·+·z,·x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·}
  
166 o5·:·List</code></pre>166 o5·:·List</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ········</table>168 ········</table>
169 ········<p>Below,·the·invariant·ring·<span·class="tt">QQ[x,y,z]</span><sup><span·class="tt">S3</span></sup>·is·calculated·with·<span·class="tt">K</span>·being·the·field·with·101·elements.</p>169 ········<p>Below,·the·invariant·ring·<span·class="tt">QQ[x,y,z]</span><sup><span·class="tt">S3</span></sup>·is·calculated·with·<span·class="tt">K</span>·being·the·field·with·101·elements.</p>
170 ········<table·class="examples">170 ········<table·class="examples">
171 ··········<tr>171 ··········<tr>
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98 ·····104976z·}98 ·····104976z·}
  
99 o4·:·List99 o4·:·List
100 Compare·this·result·to·the·hsop·output·when·Dade·is·left·to·its·default·value100 Compare·this·result·to·the·hsop·output·when·Dade·is·left·to·its·default·value
101 _\x8f_\x8a_\x8l_\x8s_\x8e.101 _\x8f_\x8a_\x8l_\x8s_\x8e.
102 i5·:·primaryInvariants(S3)102 i5·:·primaryInvariants(S3)
  
103 ···································3····3····3 
104 o5·=·{x·+·y·+·z,·x*y·+·x*z·+·y*z,·x··+·y··+·z·}103 ··················2····2····2···2·······2····2·····2·······2······2
 104 o5·=·{x·+·y·+·z,·x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·}
  
105 o5·:·List105 o5·:·List
106 Below,·the·invariant·ring·QQ[x,y,z]S3·is·calculated·with·K·being·the·field·with106 Below,·the·invariant·ring·QQ[x,y,z]S3·is·calculated·with·K·being·the·field·with
107 101·elements.107 101·elements.
108 i6·:·K=GF(101)108 i6·:·K=GF(101)
  
109 o6·=·K109 o6·=·K
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90 ··················|·1·0·0·|··|·0·0·1·|90 ··················|·1·0·0·|··|·0·0·1·|
  
91 o3·:·FiniteGroupAction</code></pre>91 o3·:·FiniteGroupAction</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})94 <td>··············<pre><code·class="language-macaulay2">i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})
  
95 ·······3····3····3···2·······2····2·····2·······2······2···4····4····495 ·······2·······2····2·····2·······2······2··········4····4····4
96 o4·=·{x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x··+·y··+·z·}96 o4·=·{x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x*y*z,·x··+·y··+·z·}
  
97 o4·:·List</code></pre>97 o4·:·List</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ········</table>99 ········</table>
100 ······</div>100 ······</div>
101 ······<div>101 ······<div>
102 ········<h2>Further·information</h2>102 ········<h2>Further·information</h2>
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35 o3·=·QQ[x..z]·<-·{|·0·1·0·|,·|·0·1·0·|}35 o3·=·QQ[x..z]·<-·{|·0·1·0·|,·|·0·1·0·|}
36 ··················|·0·0·1·|··|·1·0·0·|36 ··················|·0·0·1·|··|·1·0·0·|
37 ··················|·1·0·0·|··|·0·0·1·|37 ··················|·1·0·0·|··|·0·0·1·|
  
38 o3·:·FiniteGroupAction38 o3·:·FiniteGroupAction
39 i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})39 i4·:·primaryInvariants(S3,DegreeVector=>{3,3,4})
  
40 ·······3····3····3···2·······2····2·····2·······2······2···4····4····440 ·······2·······2····2·····2·······2······2··········4····4····4
41 o4·=·{x··+·y··+·z·,·x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x··+·y··+·z·}41 o4·=·{x·y·+·x*y··+·x·z·+·y·z·+·x*z··+·y*z·,·x*y*z,·x··+·y··+·z·}
  
42 o4·:·List42 o4·:·List
43 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*43 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
44 ····*·Default·value:·044 ····*·Default·value:·0
45 ····*·Function:·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8I_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s·--·computes·a·list·of·primary·invariants·for45 ····*·Function:·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8I_\x8n_\x8v_\x8a_\x8r_\x8i_\x8a_\x8n_\x8t_\x8s·--·computes·a·list·of·primary·invariants·for
46 ······the·invariant·ring·of·a·finite·group46 ······the·invariant·ring·of·a·finite·group
47 ····*·Option·key:·_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8V_\x8e_\x8c_\x8t_\x8o_\x8r·--·an·optional·argument·for·primaryInvariants47 ····*·Option·key:·_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8V_\x8e_\x8c_\x8t_\x8o_\x8r·--·an·optional·argument·for·primaryInvariants
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=277 #:len=27
8 aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp8 aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp
9 #:len=2709 #:len=270
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDA2LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDA2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz
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713 B
./usr/share/doc/Macaulay2/JSON/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=137 #:len=13
8 TmFtZVNlcGFyYXRvcg==8 TmFtZVNlcGFyYXRvcg==
9 #:len=2109 #:len=210
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc5LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs
⊟
483 B
./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
    
Offset 39, 19 lines modifiedOffset 39, 19 lines modified
  
39 o8·=·{1,·2,·3}39 o8·=·{1,·2,·3}
  
40 o8·:·List40 o8·:·List
  
41 i9·:·jsonFile·=·temporaryFileName()·|·".json"41 i9·:·jsonFile·=·temporaryFileName()·|·".json"
  
42 o9·=·/tmp/M2-3971259-0/0.json42 o9·=·/tmp/M2-2989647-0/0.json
  
43 i10·:·jsonFile·<<·"[1,·2,·3]"·<<·endl·<<·close43 i10·:·jsonFile·<<·"[1,·2,·3]"·<<·endl·<<·close
  
44 o10·=·/tmp/M2-3971259-0/0.json44 o10·=·/tmp/M2-2989647-0/0.json
  
45 o10·:·File45 o10·:·File
  
46 i11·:·fromJSON·openIn·jsonFile46 i11·:·fromJSON·openIn·jsonFile
  
47 o11·=·{1,·2,·3}47 o11·=·{1,·2,·3}
  
⊟
1.3 KB
./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
    
Offset 150, 20 lines modifiedOffset 150, 20 lines modified
150 ········<div>150 ········<div>
151 ··········<p>The·input·may·also·be·a·file·containing·JSON·data.</p>151 ··········<p>The·input·may·also·be·a·file·containing·JSON·data.</p>
152 ········</div>152 ········</div>
153 ········<table·class="examples">153 ········<table·class="examples">
154 ··········<tr>154 ··········<tr>
155 <td>··············<pre><code·class="language-macaulay2">i9·:·jsonFile·=·temporaryFileName()·|·&quot;.json&quot;155 <td>··············<pre><code·class="language-macaulay2">i9·:·jsonFile·=·temporaryFileName()·|·&quot;.json&quot;
  
156 o9·=·/tmp/M2-3971259-0/0.json</code></pre>156 o9·=·/tmp/M2-2989647-0/0.json</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i10·:·jsonFile·&lt;&lt;·&quot;[1,·2,·3]&quot;·&lt;&lt;·endl·&lt;&lt;·close159 <td>··············<pre><code·class="language-macaulay2">i10·:·jsonFile·&lt;&lt;·&quot;[1,·2,·3]&quot;·&lt;&lt;·endl·&lt;&lt;·close
  
160 o10·=·/tmp/M2-3971259-0/0.json160 o10·=·/tmp/M2-2989647-0/0.json
  
161 o10·:·File</code></pre>161 o10·:·File</code></pre>
162 </td>··········</tr>162 </td>··········</tr>
163 ··········<tr>163 ··········<tr>
164 <td>··············<pre><code·class="language-macaulay2">i11·:·fromJSON·openIn·jsonFile164 <td>··············<pre><code·class="language-macaulay2">i11·:·fromJSON·openIn·jsonFile
  
165 o11·=·{1,·2,·3}165 o11·=·{1,·2,·3}
⊟
431 B
html2text {}
    
Offset 54, 18 lines modifiedOffset 54, 18 lines modified
  
54 o8·=·{1,·2,·3}54 o8·=·{1,·2,·3}
  
55 o8·:·List55 o8·:·List
56 The·input·may·also·be·a·file·containing·JSON·data.56 The·input·may·also·be·a·file·containing·JSON·data.
57 i9·:·jsonFile·=·temporaryFileName()·|·".json"57 i9·:·jsonFile·=·temporaryFileName()·|·".json"
  
58 o9·=·/tmp/M2-3971259-0/0.json58 o9·=·/tmp/M2-2989647-0/0.json
59 i10·:·jsonFile·<<·"[1,·2,·3]"·<<·endl·<<·close59 i10·:·jsonFile·<<·"[1,·2,·3]"·<<·endl·<<·close
  
60 o10·=·/tmp/M2-3971259-0/0.json60 o10·=·/tmp/M2-2989647-0/0.json
  
61 o10·:·File61 o10·:·File
62 i11·:·fromJSON·openIn·jsonFile62 i11·:·fromJSON·openIn·jsonFile
  
63 o11·=·{1,·2,·3}63 o11·=·{1,·2,·3}
  
64 o11·:·List64 o11·:·List
⊟
726 B
./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=227 #:len=22
8 amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==8 amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==
9 #:len=24699 #:len=2469
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh
11 cmlldHkiLCAibGluZW51bSIgPT4gMTY2NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT11 cmlldHkiLCAibGluZW51bSIgPT4gMTY2NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
⊟
1.06 KB
./usr/share/doc/Macaulay2/Jets/example-output/___Example_sp1.out
    
Offset 17, 24 lines modifiedOffset 17, 24 lines modified
17 o3·=·ideal·(y0*z0*x2·+·x0*z0*y2·+·x0*y0*z2·+·z0*x1*y1·+·y0*x1*z1·+·x0*y1*z1,17 o3·=·ideal·(y0*z0*x2·+·x0*z0*y2·+·x0*y0*z2·+·z0*x1*y1·+·y0*x1*z1·+·x0*y1*z1,
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ·····y0*z0*x1·+·x0*z0*y1·+·x0*y0*z1,·x0*y0*z0)19 ·····y0*z0*x1·+·x0*z0*y1·+·x0*y0*z1,·x0*y0*z0)
  
20 o3·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]20 o3·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]
  
21 i4·:·elapsedTime·jetsRadical(2,I)21 i4·:·elapsedTime·jetsRadical(2,I)
22 ·--·.00166588s·elapsed22 ·--·.00247732s·elapsed
  
23 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,23 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,
24 ·····------------------------------------------------------------------------24 ·····------------------------------------------------------------------------
25 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)25 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)
  
26 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]26 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]
  
27 i5·:·elapsedTime·radical·J2I27 i5·:·elapsedTime·radical·J2I
28 ·--·.311432s·elapsed28 ·--·.299999s·elapsed
  
29 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,29 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,
30 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
31 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)31 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)
  
32 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]32 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]
  
⊟
2.78 KB
./usr/share/doc/Macaulay2/Jets/example-output/___Storing_sp__Computations.out
    
Offset 33, 15 lines modifiedOffset 33, 15 lines modified
33 o6·:·Ideal·of·R33 o6·:·Ideal·of·R
  
34 i7·:·I.cache.?jet34 i7·:·I.cache.?jet
  
35 o7·=·false35 o7·=·false
  
36 i8·:·elapsedTime·jets(3,I)36 i8·:·elapsedTime·jets(3,I)
37 ·--·.00723962s·elapsed37 ·--·.00812489s·elapsed
  
38 ··················································2·················238 ··················································2·················2
39 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)39 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
40 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]40 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
  
41 i9·:·I.cache.?jet41 i9·:·I.cache.?jet
Offset 53, 23 lines modifiedOffset 53, 23 lines modified
53 o10·=·CacheTable{jetsMatrix·=>·|·2x0x3-y3+2x1x2·|}53 o10·=·CacheTable{jetsMatrix·=>·|·2x0x3-y3+2x1x2·|}
54 ·······························|·2x0x2-y2+x1^2··|54 ·······························|·2x0x2-y2+x1^2··|
55 ·······························|·2x0x1-y1·······|55 ·······························|·2x0x1-y1·······|
56 ·······························|·x0^2-y0········|56 ·······························|·x0^2-y0········|
57 ·················jetsMaxOrder·=>·357 ·················jetsMaxOrder·=>·3
  
58 i11·:·elapsedTime·jets(3,I)58 i11·:·elapsedTime·jets(3,I)
59 ·--·.00319752s·elapsed59 ·--·.00214649s·elapsed
  
60 ···················································2·················260 ···················································2·················2
61 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)61 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
62 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]62 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
  
63 i12·:·elapsedTime·jets(2,I)63 i12·:·elapsedTime·jets(2,I)
64 ·--·.00205611s·elapsed64 ·--·.00185383s·elapsed
  
65 ·····························2·················265 ·····························2·················2
66 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)66 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
67 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]67 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]
  
68 i13·:·Q·=·R/I68 i13·:·Q·=·R/I
Offset 148, 15 lines modifiedOffset 148, 15 lines modified
148 o22·=·true148 o22·=·true
  
149 i23·:·f.cache.?jet149 i23·:·f.cache.?jet
  
150 o23·=·false150 o23·=·false
  
151 i24·:·elapsedTime·jets(3,f)151 i24·:·elapsedTime·jets(3,f)
152 ·--·.00962748s·elapsed152 ·--·.0103725s·elapsed
  
153 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]······················································2····················2153 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]······················································2····················2
154 o24·=·map·(QQ[t0][t1][t2][t3],·----------------------------------------------------------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})154 o24·=·map·(QQ[t0][t1][t2][t3],·----------------------------------------------------------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
155 ······································································2·················2155 ······································································2·················2
156 ·······························(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)156 ·······························(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
157 ····················································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]157 ····················································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
Offset 173, 15 lines modifiedOffset 173, 15 lines modified
173 o26·=·CacheTable{jetsMatrix·=>·|·t3·2t0t3+2t1t2·|}173 o26·=·CacheTable{jetsMatrix·=>·|·t3·2t0t3+2t1t2·|}
174 ·······························|·t2·2t0t2+t1^2··|174 ·······························|·t2·2t0t2+t1^2··|
175 ·······························|·t1·2t0t1·······|175 ·······························|·t1·2t0t1·······|
176 ·······························|·t0·t0^2········|176 ·······························|·t0·t0^2········|
177 ·················jetsMaxOrder·=>·3177 ·················jetsMaxOrder·=>·3
  
178 i27·:·elapsedTime·jets(2,f)178 i27·:·elapsedTime·jets(2,f)
179 ·--·.000626447s·elapsed179 ·--·.000594244s·elapsed
  
180 ···································QQ[x0,·y0][x1,·y1][x2,·y2]··························2····················2180 ···································QQ[x0,·y0][x1,·y1][x2,·y2]··························2····················2
181 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})181 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
182 ············································2·················2182 ············································2·················2
183 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)183 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
184 ·········································QQ[x0,·y0][x1,·y1][x2,·y2]184 ·········································QQ[x0,·y0][x1,·y1][x2,·y2]
⊟
2.65 KB
./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
    
Offset 73, 25 lines modifiedOffset 73, 25 lines modified
73 ········</table>73 ········</table>
74 ········<div>74 ········<div>
75 ··········<p>However,·by·[GS06,·Theorem·3.1],·the·radical·is·always·a·(squarefree)·monomial·ideal.·In·fact,·the·proof·of·[GS06,·Theorem·3.2]·shows·that·the·radical·is·generated·by·the·individual·terms·in·the·generators·of·the·ideal·of·jets.·This·observation·provides·an·alternative·algorithm·for·computing·radicals·of·jets·of·monomial·ideals,·which·can·be·faster·than·the·default·radical·computation·in·Macaulay2.</p>75 ··········<p>However,·by·[GS06,·Theorem·3.1],·the·radical·is·always·a·(squarefree)·monomial·ideal.·In·fact,·the·proof·of·[GS06,·Theorem·3.2]·shows·that·the·radical·is·generated·by·the·individual·terms·in·the·generators·of·the·ideal·of·jets.·This·observation·provides·an·alternative·algorithm·for·computing·radicals·of·jets·of·monomial·ideals,·which·can·be·faster·than·the·default·radical·computation·in·Macaulay2.</p>
76 ········</div>76 ········</div>
77 ········<table·class="examples">77 ········<table·class="examples">
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·jetsRadical(2,I)79 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·jetsRadical(2,I)
80 ·--·.00166588s·elapsed80 ·--·.00247732s·elapsed
  
81 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,81 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,
82 ·····------------------------------------------------------------------------82 ·····------------------------------------------------------------------------
83 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)83 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)
  
84 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]</code></pre>84 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·radical·J2I87 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·radical·J2I
88 ·--·.311432s·elapsed88 ·--·.299999s·elapsed
  
89 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,89 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,
90 ·····------------------------------------------------------------------------90 ·····------------------------------------------------------------------------
91 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)91 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)
  
92 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]</code></pre>92 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
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Offset 27, 23 lines modifiedOffset 27, 23 lines modified
27 However,·by·[GS06,·Theorem·3.1],·the·radical·is·always·a·(squarefree)·monomial27 However,·by·[GS06,·Theorem·3.1],·the·radical·is·always·a·(squarefree)·monomial
28 ideal.·In·fact,·the·proof·of·[GS06,·Theorem·3.2]·shows·that·the·radical·is28 ideal.·In·fact,·the·proof·of·[GS06,·Theorem·3.2]·shows·that·the·radical·is
29 generated·by·the·individual·terms·in·the·generators·of·the·ideal·of·jets.·This29 generated·by·the·individual·terms·in·the·generators·of·the·ideal·of·jets.·This
30 observation·provides·an·alternative·algorithm·for·computing·radicals·of·jets·of30 observation·provides·an·alternative·algorithm·for·computing·radicals·of·jets·of
31 monomial·ideals,·which·can·be·faster·than·the·default·radical·computation·in31 monomial·ideals,·which·can·be·faster·than·the·default·radical·computation·in
32 Macaulay2.32 Macaulay2.
33 i4·:·elapsedTime·jetsRadical(2,I)33 i4·:·elapsedTime·jetsRadical(2,I)
34 ·--·.00166588s·elapsed34 ·--·.00247732s·elapsed
  
35 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,35 o4·=·ideal·(y0*z0*x2,·x0*z0*y2,·x0*y0*z2,·z0*x1*y1,·y0*x1*z1,·x0*y1*z1,
36 ·····------------------------------------------------------------------------36 ·····------------------------------------------------------------------------
37 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)37 ·····y0*z0*x1,·x0*z0*y1,·x0*y0*z1,·x0*y0*z0)
  
38 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]38 o4·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]
39 i5·:·elapsedTime·radical·J2I39 i5·:·elapsedTime·radical·J2I
40 ·--·.311432s·elapsed40 ·--·.299999s·elapsed
  
41 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,41 o5·=·ideal·(x0*y0*z0,·x0*y0*z1,·x0*z0*y1,·y0*z0*x1,·x0*y1*z1,·y0*x1*z1,
42 ·····------------------------------------------------------------------------42 ·····------------------------------------------------------------------------
43 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)43 ·····z0*x1*y1,·x0*y0*z2,·x0*z0*y2,·y0*z0*x2)
  
44 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]44 o5·:·Ideal·of·QQ[x0,·y0,·z0][x1,·y1,·z1][x2,·y2,·z2]
45 For·a·monomial·hypersurface,·[GS06,·Theorem·3.2]·describes·the·minimal·primes45 For·a·monomial·hypersurface,·[GS06,·Theorem·3.2]·describes·the·minimal·primes
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Offset 95, 15 lines modifiedOffset 95, 15 lines modified
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i7·:·I.cache.?jet96 <td>··············<pre><code·class="language-macaulay2">i7·:·I.cache.?jet
  
97 o7·=·false</code></pre>97 o7·=·false</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·jets(3,I)100 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·jets(3,I)
101 ·--·.00723962s·elapsed101 ·--·.00812489s·elapsed
  
102 ··················································2·················2102 ··················································2·················2
103 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)103 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
104 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]</code></pre>104 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
Offset 118, 24 lines modifiedOffset 118, 24 lines modified
118 ·······························|·2x0x2-y2+x1^2··|118 ·······························|·2x0x2-y2+x1^2··|
119 ·······························|·2x0x1-y1·······|119 ·······························|·2x0x1-y1·······|
120 ·······························|·x0^2-y0········|120 ·······························|·x0^2-y0········|
121 ·················jetsMaxOrder·=>·3</code></pre>121 ·················jetsMaxOrder·=>·3</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·jets(3,I)124 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·jets(3,I)
125 ·--·.00319752s·elapsed125 ·--·.00214649s·elapsed
  
126 ···················································2·················2126 ···················································2·················2
127 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)127 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
128 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]</code></pre>128 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·jets(2,I)131 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·jets(2,I)
132 ·--·.00205611s·elapsed132 ·--·.00185383s·elapsed
  
133 ·····························2·················2133 ·····························2·················2
134 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)134 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
135 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]</code></pre>135 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]</code></pre>
136 </td>··········</tr>136 </td>··········</tr>
137 ········</table>137 ········</table>
Offset 236, 15 lines modifiedOffset 236, 15 lines modified
236 ··········<tr>236 ··········<tr>
237 <td>··············<pre><code·class="language-macaulay2">i23·:·f.cache.?jet237 <td>··············<pre><code·class="language-macaulay2">i23·:·f.cache.?jet
  
238 o23·=·false</code></pre>238 o23·=·false</code></pre>
239 </td>··········</tr>239 </td>··········</tr>
240 ··········<tr>240 ··········<tr>
241 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·jets(3,f)241 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·jets(3,f)
242 ·--·.00962748s·elapsed242 ·--·.0103725s·elapsed
  
243 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]······················································2····················2243 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]······················································2····················2
244 o24·=·map·(QQ[t0][t1][t2][t3],·----------------------------------------------------------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})244 o24·=·map·(QQ[t0][t1][t2][t3],·----------------------------------------------------------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
245 ······································································2·················2245 ······································································2·················2
246 ·······························(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)246 ·······························(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
247 ····················································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]247 ····················································QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
Offset 264, 15 lines modifiedOffset 264, 15 lines modified
264 ·······························|·t2·2t0t2+t1^2··|264 ·······························|·t2·2t0t2+t1^2··|
265 ·······························|·t1·2t0t1·······|265 ·······························|·t1·2t0t1·······|
266 ·······························|·t0·t0^2········|266 ·······························|·t0·t0^2········|
267 ·················jetsMaxOrder·=>·3</code></pre>267 ·················jetsMaxOrder·=>·3</code></pre>
268 </td>··········</tr>268 </td>··········</tr>
269 ··········<tr>269 ··········<tr>
270 <td>··············<pre><code·class="language-macaulay2">i27·:·elapsedTime·jets(2,f)270 <td>··············<pre><code·class="language-macaulay2">i27·:·elapsedTime·jets(2,f)
271 ·--·.000626447s·elapsed271 ·--·.000594244s·elapsed
  
272 ···································QQ[x0,·y0][x1,·y1][x2,·y2]··························2····················2272 ···································QQ[x0,·y0][x1,·y1][x2,·y2]··························2····················2
273 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})273 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
274 ············································2·················2274 ············································2·················2
275 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)275 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
276 ·········································QQ[x0,·y0][x1,·y1][x2,·y2]276 ·········································QQ[x0,·y0][x1,·y1][x2,·y2]
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Offset 41, 15 lines modifiedOffset 41, 15 lines modified
41 o6·=·ideal(x··-·y)41 o6·=·ideal(x··-·y)
  
42 o6·:·Ideal·of·R42 o6·:·Ideal·of·R
43 i7·:·I.cache.?jet43 i7·:·I.cache.?jet
  
44 o7·=·false44 o7·=·false
45 i8·:·elapsedTime·jets(3,I)45 i8·:·elapsedTime·jets(3,I)
46 ·--·.00723962s·elapsed46 ·--·.00812489s·elapsed
  
47 ··················································2·················247 ··················································2·················2
48 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)48 o8·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
49 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]49 o8·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
50 i9·:·I.cache.?jet50 i9·:·I.cache.?jet
  
Offset 58, 22 lines modifiedOffset 58, 22 lines modified
  
58 o10·=·CacheTable{jetsMatrix·=>·|·2x0x3-y3+2x1x2·|}58 o10·=·CacheTable{jetsMatrix·=>·|·2x0x3-y3+2x1x2·|}
59 ·······························|·2x0x2-y2+x1^2··|59 ·······························|·2x0x2-y2+x1^2··|
60 ·······························|·2x0x1-y1·······|60 ·······························|·2x0x1-y1·······|
61 ·······························|·x0^2-y0········|61 ·······························|·x0^2-y0········|
62 ·················jetsMaxOrder·=>·362 ·················jetsMaxOrder·=>·3
63 i11·:·elapsedTime·jets(3,I)63 i11·:·elapsedTime·jets(3,I)
64 ·--·.00319752s·elapsed64 ·--·.00214649s·elapsed
  
65 ···················································2·················265 ···················································2·················2
66 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)66 o11·=·ideal·(2x0*x3·-·y3·+·2x1*x2,·2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
67 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]67 o11·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2][x3,·y3]
68 i12·:·elapsedTime·jets(2,I)68 i12·:·elapsedTime·jets(2,I)
69 ·--·.00205611s·elapsed69 ·--·.00185383s·elapsed
  
70 ·····························2·················270 ·····························2·················2
71 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)71 o12·=·ideal·(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
  
72 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]72 o12·:·Ideal·of·QQ[x0,·y0][x1,·y1][x2,·y2]
73 For·quotient·rings,·data·is·stored·under·*.jet.·Each·jets·order·gives·rise·to·a73 For·quotient·rings,·data·is·stored·under·*.jet.·Each·jets·order·gives·rise·to·a
74 different·quotient·that·is·stored·separately·under·*.jet.jetsRing·(order·zero74 different·quotient·that·is·stored·separately·under·*.jet.jetsRing·(order·zero
Offset 153, 15 lines modifiedOffset 153, 15 lines modified
153 i22·:·isWellDefined·f153 i22·:·isWellDefined·f
  
154 o22·=·true154 o22·=·true
155 i23·:·f.cache.?jet155 i23·:·f.cache.?jet
  
156 o23·=·false156 o23·=·false
157 i24·:·elapsedTime·jets(3,f)157 i24·:·elapsedTime·jets(3,f)
158 ·--·.00962748s·elapsed158 ·--·.0103725s·elapsed
  
159 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,159 ··············································QQ[x0,·y0][x1,·y1][x2,·y2][x3,
160 y3]······················································2····················2160 y3]······················································2····················2
161 o24·=·map·(QQ[t0][t1][t2][t3],·------------------------------------------------161 o24·=·map·(QQ[t0][t1][t2][t3],·------------------------------------------------
162 ----------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})162 ----------------,·{t3,·2t0*t3·+·2t1*t2,·t2,·2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
163 ······································································2163 ······································································2
164 2164 2
Offset 183, 15 lines modifiedOffset 183, 15 lines modified
  
183 o26·=·CacheTable{jetsMatrix·=>·|·t3·2t0t3+2t1t2·|}183 o26·=·CacheTable{jetsMatrix·=>·|·t3·2t0t3+2t1t2·|}
184 ·······························|·t2·2t0t2+t1^2··|184 ·······························|·t2·2t0t2+t1^2··|
185 ·······························|·t1·2t0t1·······|185 ·······························|·t1·2t0t1·······|
186 ·······························|·t0·t0^2········|186 ·······························|·t0·t0^2········|
187 ·················jetsMaxOrder·=>·3187 ·················jetsMaxOrder·=>·3
188 i27·:·elapsedTime·jets(2,f)188 i27·:·elapsedTime·jets(2,f)
189 ·--·.000626447s·elapsed189 ·--·.000594244s·elapsed
  
190 ···································QQ[x0,·y0][x1,·y1][x2,·y2]190 ···································QQ[x0,·y0][x1,·y1][x2,·y2]
191 2····················2191 2····················2
192 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,192 o27·=·map·(QQ[t0][t1][t2],·------------------------------------------,·{t2,
193 2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})193 2t0*t2·+·t1·,·t1,·2t0*t1,·t0,·t0·})
194 ············································2·················2194 ············································2·················2
195 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)195 ···························(2x0*x2·-·y2·+·x1·,·2x0*x1·-·y1,·x0··-·y0)
⊟
759 B
./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=417 #:len=41
8 Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=8 Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=
9 #:len=2989 #:len=298
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU4NSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU4NSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu
⊟
1.95 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out
    
Offset 19, 15 lines modifiedOffset 19, 15 lines modified
19 ······32003··0···5···0···5·········32003··0···5···0···5··········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5··········32003··0···5···0···5·········19 ······32003··0···5···0···5·········32003··0···5···0···5··········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5···········32003··0···5···0···5··········32003··0···5···0···5·········
20 ·····················································································································································································································································································································1020 ·····················································································································································································································································································································10
21 ·····0····························1·····························2······························3······························4······························5······························6······························7······························8·····························921 ·····0····························1·····························2······························3······························4······························5······························6······························7······························8·····························9
  
22 o3·:·ChainComplex22 o3·:·ChainComplex
  
23 i4·:·L·=·analyzeStrand(F,a);·#L23 i4·:·L·=·analyzeStrand(F,a);·#L
24 ·--·.0287247s·elapsed24 ·--·.0225582s·elapsed
  
25 o5·=·35025 o5·=·350
  
26 i6·:·betti·F_a,·betti·F26 i6·:·betti·F_a,·betti·F
  
27 ···············0·········0··1···2···3···4···5···6···7··8·927 ···············0·········0··1···2···3···4···5···6···7··8·9
28 o6·=·(total:·833,·total:·1·36·187·491·793·833·573·250·63·7)28 o6·=·(total:·833,·total:·1·36·187·491·793·833·573·250·63·7)
Offset 46, 19 lines modifiedOffset 46, 19 lines modified
46 o7·:·Expression·of·class·Product46 o7·:·Expression·of·class·Product
  
47 i8·:·L3·=·select(L,c->c%3==0);·#L347 i8·:·L3·=·select(L,c->c%3==0);·#L3
  
48 o9·=·1448 o9·=·14
  
49 i10·:·carpetBettiTable(a,b,3)49 i10·:·carpetBettiTable(a,b,3)
50 ·--·.00289955s·elapsed50 ·--·.0018994s·elapsed
 51 ·--·.00522643s·elapsed
 52 ·--·.0256953s·elapsed
 53 ·--·.0121517s·elapsed
51 ·--·.00460872s·elapsed54 ·--·.00408654s·elapsed
52 ·--·.0196895s·elapsed 
53 ·--·.0087345s·elapsed 
54 ·--·.0031281s·elapsed 
  
55 ·············0··1···2···3···4···5···6···7··8·955 ·············0··1···2···3···4···5···6···7··8·9
56 o10·=·total:·1·36·160·315·302·302·315·160·36·156 o10·=·total:·1·36·160·315·302·302·315·160·36·1
57 ··········0:·1··.···.···.···.···.···.···.··.·.57 ··········0:·1··.···.···.···.···.···.···.··.·.
58 ··········1:·.·36·160·315·288··14···.···.··.·.58 ··········1:·.·36·160·315·288··14···.···.··.·.
59 ··········2:·.··.···.···.··14·288·315·160·36·.59 ··········2:·.··.···.···.··14·288·315·160·36·.
60 ··········3:·.··.···.···.···.···.···.···.··.·160 ··········3:·.··.···.···.···.···.···.···.··.·1
⊟
2.11 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out
    
Offset 3, 20 lines modifiedOffset 3, 20 lines modified
3 i1·:·a=5,b=53 i1·:·a=5,b=5
  
4 o1·=·(5,·5)4 o1·=·(5,·5)
  
5 o1·:·Sequence5 o1·:·Sequence
  
6 i2·:·elapsedTime·T=carpetBettiTable(a,b,3)6 i2·:·elapsedTime·T=carpetBettiTable(a,b,3)
7 ·--·.00204935s·elapsed 
8 ·--·.0050294s·elapsed 
9 ·--·.0319282s·elapsed 
10 ·--·.00807892s·elapsed 
11 ·--·.00291233s·elapsed7 ·--·.00296162s·elapsed
 8 ·--·.00701579s·elapsed
 9 ·--·.0254555s·elapsed
 10 ·--·.0108626s·elapsed
12 ·--·.309191s·elapsed11 ·--·.0039095s·elapsed
 12 ·--·.373471s·elapsed
  
13 ············0··1···2···3···4···5···6···7··8·913 ············0··1···2···3···4···5···6···7··8·9
14 o2·=·total:·1·36·160·315·302·302·315·160·36·114 o2·=·total:·1·36·160·315·302·302·315·160·36·1
15 ·········0:·1··.···.···.···.···.···.···.··.·.15 ·········0:·1··.···.···.···.···.···.···.··.·.
16 ·········1:·.·36·160·315·288··14···.···.··.·.16 ·········1:·.·36·160·315·288··14···.···.··.·.
17 ·········2:·.··.···.···.··14·288·315·160·36·.17 ·········2:·.··.···.···.··14·288·315·160·36·.
18 ·········3:·.··.···.···.···.···.···.···.··.·118 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 26, 15 lines modifiedOffset 26, 15 lines modified
26 i3·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);26 i3·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);
  
27 ··············ZZ27 ··············ZZ
28 o3·:·Ideal·of·--[x·..x·,·y·..y·]28 o3·:·Ideal·of·--[x·..x·,·y·..y·]
29 ···············3··0···5···0···529 ···············3··0···5···0···5
  
30 i4·:·elapsedTime·T'=minimalBetti·J30 i4·:·elapsedTime·T'=minimalBetti·J
31 ·--·.266565s·elapsed31 ·--·.244737s·elapsed
  
32 ············0··1···2···3···4···5···6···7··8·932 ············0··1···2···3···4···5···6···7··8·9
33 o4·=·total:·1·36·160·315·302·302·315·160·36·133 o4·=·total:·1·36·160·315·302·302·315·160·36·1
34 ·········0:·1··.···.···.···.···.···.···.··.·.34 ·········0:·1··.···.···.···.···.···.···.··.·.
35 ·········1:·.·36·160·315·288··14···.···.··.·.35 ·········1:·.·36·160·315·288··14···.···.··.·.
36 ·········2:·.··.···.···.··14·288·315·160·36·.36 ·········2:·.··.···.···.··14·288·315·160·36·.
37 ·········3:·.··.···.···.···.···.···.···.··.·137 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 48, 22 lines modifiedOffset 48, 22 lines modified
48 ·········1:·.·.·.·.·.·.·.·.·.·.48 ·········1:·.·.·.·.·.·.·.·.·.·.
49 ·········2:·.·.·.·.·.·.·.·.·.·.49 ·········2:·.·.·.·.·.·.·.·.·.·.
50 ·········3:·.·.·.·.·.·.·.·.·.·.50 ·········3:·.·.·.·.·.·.·.·.·.·.
  
51 o5·:·BettiTally51 o5·:·BettiTally
  
52 i6·:·elapsedTime·h=carpetBettiTables(6,6);52 i6·:·elapsedTime·h=carpetBettiTables(6,6);
 53 ·--·.00373134s·elapsed
53 ·--·.00834527s·elapsed54 ·--·.0150234s·elapsed
54 ·--·.0299358s·elapsed 
55 ·--·.102968s·elapsed55 ·--·.132568s·elapsed
56 ·--·.8111s·elapsed56 ·--·1.17797s·elapsed
 57 ·--·.338035s·elapsed
57 ·--·.251842s·elapsed58 ·--·.0335583s·elapsed
 59 ·--·.00565548s·elapsed
58 ·--·.0437321s·elapsed60 ·--·5.74672s·elapsed
59 ·--·.01127s·elapsed 
60 ·--·4.58719s·elapsed 
  
61 i7·:·carpetBettiTable(h,7)61 i7·:·carpetBettiTable(h,7)
  
62 ············0··1···2···3····4····5····6····7···8···9·10·1162 ············0··1···2···3····4····5····6····7···8···9·10·11
63 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··163 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··1
64 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.64 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.
65 ·········1:·.·55·320·891·1408·1155····.····.···.···.··.··.65 ·········1:·.·55·320·891·1408·1155····.····.···.···.··.··.
⊟
1.94 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out
    
Offset 3, 19 lines modifiedOffset 3, 19 lines modified
3 i1·:·a=5,b=53 i1·:·a=5,b=5
  
4 o1·=·(5,·5)4 o1·=·(5,·5)
  
5 o1·:·Sequence5 o1·:·Sequence
  
6 i2·:·h=carpetBettiTables(a,b)6 i2·:·h=carpetBettiTables(a,b)
7 ·--·.00197174s·elapsed7 ·--·.00371274s·elapsed
 8 ·--·.00990599s·elapsed
8 ·--·.00621081s·elapsed9 ·--·.0361867s·elapsed
9 ·--·.0194232s·elapsed10 ·--·.0152813s·elapsed
10 ·--·.0165629s·elapsed 
11 ·--·.00296745s·elapsed11 ·--·.00361937s·elapsed
  
12 ···························0··1···2···3···4···5···6···7··8·912 ···························0··1···2···3···4···5···6···7··8·9
13 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}13 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
14 ························0:·1··.···.···.···.···.···.···.··.·.14 ························0:·1··.···.···.···.···.···.···.··.·.
15 ························1:·.·36·160·315·288···.···.···.··.·.15 ························1:·.·36·160·315·288···.···.···.··.·.
16 ························2:·.··.···.···.···.·288·315·160·36·.16 ························2:·.··.···.···.···.·288·315·160·36·.
17 ························3:·.··.···.···.···.···.···.···.··.·117 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 48, 15 lines modifiedOffset 48, 15 lines modified
48 i4·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);48 i4·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);
  
49 ··············ZZ49 ··············ZZ
50 o4·:·Ideal·of·--[x·..x·,·y·..y·]50 o4·:·Ideal·of·--[x·..x·,·y·..y·]
51 ···············3··0···5···0···551 ···············3··0···5···0···5
  
52 i5·:·elapsedTime·T'=minimalBetti·J52 i5·:·elapsedTime·T'=minimalBetti·J
53 ·--·.506272s·elapsed53 ·--·.251137s·elapsed
  
54 ············0··1···2···3···4···5···6···7··8·954 ············0··1···2···3···4···5···6···7··8·9
55 o5·=·total:·1·36·160·315·302·302·315·160·36·155 o5·=·total:·1·36·160·315·302·302·315·160·36·1
56 ·········0:·1··.···.···.···.···.···.···.··.·.56 ·········0:·1··.···.···.···.···.···.···.··.·.
57 ·········1:·.·36·160·315·288··14···.···.··.·.57 ·········1:·.·36·160·315·288··14···.···.··.·.
58 ·········2:·.··.···.···.··14·288·315·160·36·.58 ·········2:·.··.···.···.··14·288·315·160·36·.
59 ·········3:·.··.···.···.···.···.···.···.··.·159 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 70, 22 lines modifiedOffset 70, 22 lines modified
70 ·········1:·.·.·.·.·.·.·.·.·.·.70 ·········1:·.·.·.·.·.·.·.·.·.·.
71 ·········2:·.·.·.·.·.·.·.·.·.·.71 ·········2:·.·.·.·.·.·.·.·.·.·.
72 ·········3:·.·.·.·.·.·.·.·.·.·.72 ·········3:·.·.·.·.·.·.·.·.·.·.
  
73 o6·:·BettiTally73 o6·:·BettiTally
  
74 i7·:·elapsedTime·h=carpetBettiTables(6,6);74 i7·:·elapsedTime·h=carpetBettiTables(6,6);
75 ·--·.00367266s·elapsed75 ·--·.0222809s·elapsed
76 ·--·.0144045s·elapsed76 ·--·.0154033s·elapsed
77 ·--·.099294s·elapsed 
78 ·--·2.22771s·elapsed 
79 ·--·1.13192s·elapsed77 ·--·.116889s·elapsed
80 ·--·.0818742s·elapsed78 ·--·1.68376s·elapsed
81 ·--·.00998064s·elapsed79 ·--·.503276s·elapsed
82 ·--·8.74567s·elapsed80 ·--·.0343167s·elapsed
 81 ·--·.00572571s·elapsed
 82 ·--·6.39374s·elapsed
  
83 i8·:·keys·h83 i8·:·keys·h
  
84 o8·=·{0,·2,·3,·5}84 o8·=·{0,·2,·3,·5}
  
85 o8·:·List85 o8·:·List
  
⊟
1.62 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out
    
Offset 3, 42 lines modifiedOffset 3, 42 lines modified
3 i1·:·a=4,b=43 i1·:·a=4,b=4
  
4 o1·=·(4,·4)4 o1·=·(4,·4)
  
5 o1·:·Sequence5 o1·:·Sequence
  
6 i2·:·d=carpetDet(a,b)6 i2·:·d=carpetDet(a,b)
7 ·--·.00509704s·elapsed 
8 ·--·.0283743s·elapsed7 ·--·.0213466s·elapsed
 8 ·--·.014109s·elapsed
 9 ·--·.000238217s·elapsed
9 ·--·.000197989s·elapsed10 ·--·.000159759s·elapsed
10 ·--·.000128293s·elapsed 
11 ·--·.000113561s·elapsed 
12 ·--·.000129816s·elapsed 
13 ·--·.000126621s·elapsed11 ·--·.000142628s·elapsed
 12 ·--·.000131887s·elapsed
 13 ·--·.000153778s·elapsed
 14 ·--·.000141074s·elapsed
 15 ·--·.000215554s·elapsed
 16 ·--·.000246212s·elapsed
 17 ·--·.000387176s·elapsed
 18 ·--·.000124644s·elapsed
 19 ·--·.000137688s·elapsed
 20 ·--·.000116799s·elapsed
14 ·--·.000152901s·elapsed21 ·--·.000112801s·elapsed
15 ·--·.000147313s·elapsed 
16 ·--·.000135714s·elapsed22 ·--·.000105277s·elapsed
 23 ·--·.000142197s·elapsed
 24 ·--·.000111389s·elapsed
 25 ·--·.000115326s·elapsed
 26 ·--·.000121107s·elapsed
 27 ·--·.000125665s·elapsed
 28 ·--·.000119213s·elapsed
 29 ·--·.00017078s·elapsed
 30 ·--·.000141155s·elapsed
 31 ·--·.000158727s·elapsed
 32 ·--·.00195276s·elapsed
 33 ·--·.000119484s·elapsed
17 ·--·.000247303s·elapsed34 ·--·.000247334s·elapsed
18 ·--·.000122275s·elapsed 
19 ·--·.000111357s·elapsed 
20 ·--·.00818544s·elapsed 
21 ·--·.000126722s·elapsed 
22 ·--·.000103167s·elapsed 
23 ·--·.000101773s·elapsed 
24 ·--·.000191048s·elapsed 
25 ·--·.000127052s·elapsed 
26 ·--·.000124678s·elapsed 
27 ·--·.000174403s·elapsed 
28 ·--·.000123465s·elapsed 
29 ·--·.000143016s·elapsed 
30 ·--·.000107172s·elapsed 
31 ·--·.00010521s·elapsed 
32 ·--·.000116836s·elapsed 
33 ·--·.000128103s·elapsed 
34 ·--·.000107351s·elapsed 
35 (number·Of·blocks,·26)35 (number·Of·blocks,·26)
36 136 1
37 137 1
38 138 1
39 139 1
40 240 2
41 ·241 ·2
⊟
528 B
./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out
    
Offset 3, 17 lines modifiedOffset 3, 17 lines modified
3 i1·:·(a,b)=computeBound(6,4,3)3 i1·:·(a,b)=computeBound(6,4,3)
  
4 o1·=·(9,·7)4 o1·=·(9,·7)
  
5 o1·:·Sequence5 o1·:·Sequence
  
6 i2·:·computeBound·36 i2·:·computeBound·3
7 ·--·1.34545s·elapsed 
8 ·--·1.07082s·elapsed 
9 ·--·.504563s·elapsed7 ·--·.760467s·elapsed
 8 ·--·.409083s·elapsed
10 ·--·1.22392s·elapsed9 ·--·.220772s·elapsed
11 ·--·.834693s·elapsed 
12 ·--·1.37163s·elapsed10 ·--·.767807s·elapsed
 11 ·--·.445704s·elapsed
 12 ·--·.672478s·elapsed
  
13 o2·=·613 o2·=·6
  
14 i3·:·14 i3·:·
⊟
2.03 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out
    
Offset 9, 19 lines modifiedOffset 9, 19 lines modified
9 i2·:·e=(-1,5)9 i2·:·e=(-1,5)
  
10 o2·=·(-1,·5)10 o2·=·(-1,·5)
  
11 o2·:·Sequence11 o2·:·Sequence
  
12 i3·:·h=degenerateK3BettiTables(a,b,e)12 i3·:·h=degenerateK3BettiTables(a,b,e)
13 ·--·.00189157s·elapsed 
14 ·--·.00468804s·elapsed 
15 ·--·.0195169s·elapsed13 ·--·.00237356s·elapsed
16 ·--·.00721757s·elapsed14 ·--·.0052173s·elapsed
 15 ·--·.0205611s·elapsed
17 ·--·.00284809s·elapsed16 ·--·.0084417s·elapsed
 17 ·--·.00336082s·elapsed
  
18 ···························0··1···2···3···4···5···6···7··8·918 ···························0··1···2···3···4···5···6···7··8·9
19 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}19 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
20 ························0:·1··.···.···.···.···.···.···.··.·.20 ························0:·1··.···.···.···.···.···.···.··.·.
21 ························1:·.·36·160·315·288···.···.···.··.·.21 ························1:·.·36·160·315·288···.···.···.··.·.
22 ························2:·.··.···.···.···.·288·315·160·36·.22 ························2:·.··.···.···.···.·288·315·160·36·.
23 ························3:·.··.···.···.···.···.···.···.··.·123 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 49, 15 lines modifiedOffset 49, 15 lines modified
49 i4·:·keys·h49 i4·:·keys·h
  
50 o4·=·{0,·2,·3,·5}50 o4·=·{0,·2,·3,·5}
  
51 o4·:·List51 o4·:·List
  
52 i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)52 i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)
53 ·--·.424093s·elapsed53 ·--·.366147s·elapsed
  
54 ············0··1···2···3···4···5···6···7··8·954 ············0··1···2···3···4···5···6···7··8·9
55 o5·=·total:·1·36·167·370·476·476·370·167·36·155 o5·=·total:·1·36·167·370·476·476·370·167·36·1
56 ·········0:·1··.···.···.···.···.···.···.··.·.56 ·········0:·1··.···.···.···.···.···.···.··.·.
57 ·········1:·.·36·160·322·336·140··48···7··.·.57 ·········1:·.·36·160·322·336·140··48···7··.·.
58 ·········2:·.··.···7··48·140·336·322·160·36·.58 ·········2:·.··.···7··48·140·336·322·160·36·.
59 ·········3:·.··.···.···.···.···.···.···.··.·159 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 77, 19 lines modifiedOffset 77, 19 lines modified
77 i7·:·e=(-1,5^2)77 i7·:·e=(-1,5^2)
  
78 o7·=·(-1,·25)78 o7·=·(-1,·25)
  
79 o7·:·Sequence79 o7·:·Sequence
  
80 i8·:·h=degenerateK3BettiTables(a,b,e)80 i8·:·h=degenerateK3BettiTables(a,b,e)
81 ·--·.00413622s·elapsed 
82 ·--·.00480755s·elapsed 
83 ·--·.0205215s·elapsed81 ·--·.00212253s·elapsed
 82 ·--·.00539577s·elapsed
 83 ·--·.032081s·elapsed
84 ·--·.012116s·elapsed84 ·--·.0155421s·elapsed
85 ·--·.00333302s·elapsed85 ·--·.00382758s·elapsed
  
86 ···························0··1···2···3···4···5···6···7··8·986 ···························0··1···2···3···4···5···6···7··8·9
87 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}87 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}
88 ························0:·1··.···.···.···.···.···.···.··.·.88 ························0:·1··.···.···.···.···.···.···.··.·.
89 ························1:·.·36·160·315·288···.···.···.··.·.89 ························1:·.·36·160·315·288···.···.···.··.·.
90 ························2:·.··.···.···.···.·288·315·160·36·.90 ························2:·.··.···.···.···.·288·315·160·36·.
91 ························3:·.··.···.···.···.···.···.···.··.·191 ························3:·.··.···.···.···.···.···.···.··.·1
⊟
1.37 KB
./usr/share/doc/Macaulay2/K3Carpets/example-output/_resonance__Det.out
    
Offset 1, 33 lines modifiedOffset 1, 33 lines modified
1 --·-*-·M2-comint·-*-·hash:·17291828916907047381 --·-*-·M2-comint·-*-·hash:·1729182891690704738
  
2 i1·:·a=42 i1·:·a=4
  
3 o1·=·43 o1·=·4
  
4 i2·:·(d1,d2)=resonanceDet(a)4 i2·:·(d1,d2)=resonanceDet(a)
5 ·--·.0474775s·elapsed5 ·--·.0141837s·elapsed
 6 ·--·.000035988s·elapsed
 7 ·--·.000093776s·elapsed
 8 ·--·.000089919s·elapsed
 9 ·--·.0000907s·elapsed
 10 ·--·.000113693s·elapsed
 11 ·--·.000100879s·elapsed
6 ·--·.000030627s·elapsed12 ·--·.000030677s·elapsed
7 ·--·.000069194s·elapsed13 ·--·.000092614s·elapsed
8 ·--·.000050365s·elapsed14 ·--·.000155772s·elapsed
9 ·--·.000058698s·elapsed 
10 ·--·.000063466s·elapsed 
11 ·--·.000199841s·elapsed15 ·--·.000128591s·elapsed
12 ·--·.000022634s·elapsed16 ·--·.000079249s·elapsed
13 ·--·.0526269s·elapsed 
14 ·--·.000108815s·elapsed 
15 ·--·.000124086s·elapsed 
16 ·--·.000068743s·elapsed 
17 ·--·.000058549s·elapsed 
18 ·--·.000059089s·elapsed17 ·--·.000097002s·elapsed
19 ·--·.000044588s·elapsed 
20 ·--·.00765186s·elapsed 
21 ·--·.000029114s·elapsed18 ·--·.000265427s·elapsed
22 ·--·.000063997s·elapsed19 ·--·.000067967s·elapsed
23 ·--·.000017897s·elapsed20 ·--·.000074299s·elapsed
 21 ·--·.000023474s·elapsed
 22 ·--·.000265718s·elapsed
 23 ·--·.000091552s·elapsed
24 (number·of·blocks=·,·18)24 (number·of·blocks=·,·18)
25 (size·of·the·matrices,·Tally{1·=>·4})25 (size·of·the·matrices,·Tally{1·=>·4})
26 ·····························2·=>·626 ·····························2·=>·6
27 ·····························3·=>·227 ·····························3·=>·2
28 ·····························4·=>·628 ·····························4·=>·6
29 ·······0·129 ·······0·1
30 total:·1·130 total:·1·1
⊟
3.12 KB
./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
    
Offset 96, 15 lines modifiedOffset 96, 15 lines modified
96 ·····················································································································································································································································································································1096 ·····················································································································································································································································································································10
97 ·····0····························1·····························2······························3······························4······························5······························6······························7······························8·····························997 ·····0····························1·····························2······························3······························4······························5······························6······························7······························8·····························9
  
98 o3·:·ChainComplex</code></pre>98 o3·:·ChainComplex</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i4·:·L·=·analyzeStrand(F,a);·#L101 <td>··············<pre><code·class="language-macaulay2">i4·:·L·=·analyzeStrand(F,a);·#L
102 ·--·.0287247s·elapsed102 ·--·.0225582s·elapsed
  
103 o5·=·350</code></pre>103 o5·=·350</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i6·:·betti·F_a,·betti·F106 <td>··············<pre><code·class="language-macaulay2">i6·:·betti·F_a,·betti·F
  
107 ···············0·········0··1···2···3···4···5···6···7··8·9107 ···············0·········0··1···2···3···4···5···6···7··8·9
Offset 127, 19 lines modifiedOffset 127, 19 lines modified
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i8·:·L3·=·select(L,c->c%3==0);·#L3128 <td>··············<pre><code·class="language-macaulay2">i8·:·L3·=·select(L,c->c%3==0);·#L3
  
129 o9·=·14</code></pre>129 o9·=·14</code></pre>
130 </td>··········</tr>130 </td>··········</tr>
131 ··········<tr>131 ··········<tr>
132 <td>··············<pre><code·class="language-macaulay2">i10·:·carpetBettiTable(a,b,3)132 <td>··············<pre><code·class="language-macaulay2">i10·:·carpetBettiTable(a,b,3)
133 ·--·.00289955s·elapsed133 ·--·.0018994s·elapsed
 134 ·--·.00522643s·elapsed
 135 ·--·.0256953s·elapsed
 136 ·--·.0121517s·elapsed
134 ·--·.00460872s·elapsed137 ·--·.00408654s·elapsed
135 ·--·.0196895s·elapsed 
136 ·--·.0087345s·elapsed 
137 ·--·.0031281s·elapsed 
  
138 ·············0··1···2···3···4···5···6···7··8·9138 ·············0··1···2···3···4···5···6···7··8·9
139 o10·=·total:·1·36·160·315·302·302·315·160·36·1139 o10·=·total:·1·36·160·315·302·302·315·160·36·1
140 ··········0:·1··.···.···.···.···.···.···.··.·.140 ··········0:·1··.···.···.···.···.···.···.··.·.
141 ··········1:·.·36·160·315·288··14···.···.··.·.141 ··········1:·.·36·160·315·288··14···.···.··.·.
142 ··········2:·.··.···.···.··14·288·315·160·36·.142 ··········2:·.··.···.···.··14·288·315·160·36·.
143 ··········3:·.··.···.···.···.···.···.···.··.·1143 ··········3:·.··.···.···.···.···.···.···.··.·1
⊟
1.21 KB
html2text {}
    
Offset 50, 15 lines modifiedOffset 50, 15 lines modified
50 ·····0····························1·····························250 ·····0····························1·····························2
51 3······························4······························551 3······························4······························5
52 6······························7······························852 6······························7······························8
53 953 9
  
54 o3·:·ChainComplex54 o3·:·ChainComplex
55 i4·:·L·=·analyzeStrand(F,a);·#L55 i4·:·L·=·analyzeStrand(F,a);·#L
56 ·--·.0287247s·elapsed56 ·--·.0225582s·elapsed
  
57 o5·=·35057 o5·=·350
58 i6·:·betti·F_a,·betti·F58 i6·:·betti·F_a,·betti·F
  
59 ···············0·········0··1···2···3···4···5···6···7··8·959 ···············0·········0··1···2···3···4···5···6···7··8·9
60 o6·=·(total:·833,·total:·1·36·187·491·793·833·573·250·63·7)60 o6·=·(total:·833,·total:·1·36·187·491·793·833·573·250·63·7)
61 ··········6:·350······0:·1··.···.···.···.···.···.···.··.·.61 ··········6:·350······0:·1··.···.···.···.···.···.···.··.·.
Offset 73, 19 lines modifiedOffset 73, 19 lines modified
73 o7·=·2···373 o7·=·2···3
  
74 o7·:·Expression·of·class·Product74 o7·:·Expression·of·class·Product
75 i8·:·L3·=·select(L,c->c%3==0);·#L375 i8·:·L3·=·select(L,c->c%3==0);·#L3
  
76 o9·=·1476 o9·=·14
77 i10·:·carpetBettiTable(a,b,3)77 i10·:·carpetBettiTable(a,b,3)
78 ·--·.00289955s·elapsed78 ·--·.0018994s·elapsed
 79 ·--·.00522643s·elapsed
 80 ·--·.0256953s·elapsed
 81 ·--·.0121517s·elapsed
79 ·--·.00460872s·elapsed82 ·--·.00408654s·elapsed
80 ·--·.0196895s·elapsed 
81 ·--·.0087345s·elapsed 
82 ·--·.0031281s·elapsed 
  
83 ·············0··1···2···3···4···5···6···7··8·983 ·············0··1···2···3···4···5···6···7··8·9
84 o10·=·total:·1·36·160·315·302·302·315·160·36·184 o10·=·total:·1·36·160·315·302·302·315·160·36·1
85 ··········0:·1··.···.···.···.···.···.···.··.·.85 ··········0:·1··.···.···.···.···.···.···.··.·.
86 ··········1:·.·36·160·315·288··14···.···.··.·.86 ··········1:·.·36·160·315·288··14···.···.··.·.
87 ··········2:·.··.···.···.··14·288·315·160·36·.87 ··········2:·.··.···.···.··14·288·315·160·36·.
88 ··········3:·.··.···.···.···.···.···.···.··.·188 ··········3:·.··.···.···.···.···.···.···.··.·1
⊟
4.53 KB
./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
    
Offset 83, 20 lines modifiedOffset 83, 20 lines modified
  
83 o1·=·(5,·5)83 o1·=·(5,·5)
  
84 o1·:·Sequence</code></pre>84 o1·:·Sequence</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i2·:·elapsedTime·T=carpetBettiTable(a,b,3)87 <td>··············<pre><code·class="language-macaulay2">i2·:·elapsedTime·T=carpetBettiTable(a,b,3)
88 ·--·.00204935s·elapsed 
89 ·--·.0050294s·elapsed 
90 ·--·.0319282s·elapsed 
91 ·--·.00807892s·elapsed 
92 ·--·.00291233s·elapsed88 ·--·.00296162s·elapsed
 89 ·--·.00701579s·elapsed
 90 ·--·.0254555s·elapsed
 91 ·--·.0108626s·elapsed
93 ·--·.309191s·elapsed92 ·--·.0039095s·elapsed
 93 ·--·.373471s·elapsed
  
94 ············0··1···2···3···4···5···6···7··8·994 ············0··1···2···3···4···5···6···7··8·9
95 o2·=·total:·1·36·160·315·302·302·315·160·36·195 o2·=·total:·1·36·160·315·302·302·315·160·36·1
96 ·········0:·1··.···.···.···.···.···.···.··.·.96 ·········0:·1··.···.···.···.···.···.···.··.·.
97 ·········1:·.·36·160·315·288··14···.···.··.·.97 ·········1:·.·36·160·315·288··14···.···.··.·.
98 ·········2:·.··.···.···.··14·288·315·160·36·.98 ·········2:·.··.···.···.··14·288·315·160·36·.
99 ·········3:·.··.···.···.···.···.···.···.··.·199 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 108, 15 lines modifiedOffset 108, 15 lines modified
  
108 ··············ZZ108 ··············ZZ
109 o3·:·Ideal·of·--[x·..x·,·y·..y·]109 o3·:·Ideal·of·--[x·..x·,·y·..y·]
110 ···············3··0···5···0···5</code></pre>110 ···············3··0···5···0···5</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·T'=minimalBetti·J113 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·T'=minimalBetti·J
114 ·--·.266565s·elapsed114 ·--·.244737s·elapsed
  
115 ············0··1···2···3···4···5···6···7··8·9115 ············0··1···2···3···4···5···6···7··8·9
116 o4·=·total:·1·36·160·315·302·302·315·160·36·1116 o4·=·total:·1·36·160·315·302·302·315·160·36·1
117 ·········0:·1··.···.···.···.···.···.···.··.·.117 ·········0:·1··.···.···.···.···.···.···.··.·.
118 ·········1:·.·36·160·315·288··14···.···.··.·.118 ·········1:·.·36·160·315·288··14···.···.··.·.
119 ·········2:·.··.···.···.··14·288·315·160·36·.119 ·········2:·.··.···.···.··14·288·315·160·36·.
120 ·········3:·.··.···.···.···.···.···.···.··.·1120 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 132, 22 lines modifiedOffset 132, 22 lines modified
132 ·········2:·.·.·.·.·.·.·.·.·.·.132 ·········2:·.·.·.·.·.·.·.·.·.·.
133 ·········3:·.·.·.·.·.·.·.·.·.·.133 ·········3:·.·.·.·.·.·.·.·.·.·.
  
134 o5·:·BettiTally</code></pre>134 o5·:·BettiTally</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·h=carpetBettiTables(6,6);137 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·h=carpetBettiTables(6,6);
 138 ·--·.00373134s·elapsed
138 ·--·.00834527s·elapsed139 ·--·.0150234s·elapsed
139 ·--·.0299358s·elapsed 
140 ·--·.102968s·elapsed140 ·--·.132568s·elapsed
141 ·--·.8111s·elapsed 
142 ·--·.251842s·elapsed 
143 ·--·.0437321s·elapsed 
144 ·--·.01127s·elapsed141 ·--·1.17797s·elapsed
 142 ·--·.338035s·elapsed
 143 ·--·.0335583s·elapsed
 144 ·--·.00565548s·elapsed
145 ·--·4.58719s·elapsed</code></pre>145 ·--·5.74672s·elapsed</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i7·:·carpetBettiTable(h,7)148 <td>··············<pre><code·class="language-macaulay2">i7·:·carpetBettiTable(h,7)
  
149 ············0··1···2···3····4····5····6····7···8···9·10·11149 ············0··1···2···3····4····5····6····7···8···9·10·11
150 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··1150 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··1
151 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.151 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.
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Offset 26, 20 lines modifiedOffset 26, 20 lines modified
26 resulting·data·allow·us·to·compute·the·Betti·tables·for·arbitrary·primes.26 resulting·data·allow·us·to·compute·the·Betti·tables·for·arbitrary·primes.
27 i1·:·a=5,b=527 i1·:·a=5,b=5
  
28 o1·=·(5,·5)28 o1·=·(5,·5)
  
29 o1·:·Sequence29 o1·:·Sequence
30 i2·:·elapsedTime·T=carpetBettiTable(a,b,3)30 i2·:·elapsedTime·T=carpetBettiTable(a,b,3)
31 ·--·.00204935s·elapsed 
32 ·--·.0050294s·elapsed 
33 ·--·.0319282s·elapsed 
34 ·--·.00807892s·elapsed 
35 ·--·.00291233s·elapsed31 ·--·.00296162s·elapsed
 32 ·--·.00701579s·elapsed
 33 ·--·.0254555s·elapsed
 34 ·--·.0108626s·elapsed
36 ·--·.309191s·elapsed35 ·--·.0039095s·elapsed
 36 ·--·.373471s·elapsed
  
37 ············0··1···2···3···4···5···6···7··8·937 ············0··1···2···3···4···5···6···7··8·9
38 o2·=·total:·1·36·160·315·302·302·315·160·36·138 o2·=·total:·1·36·160·315·302·302·315·160·36·1
39 ·········0:·1··.···.···.···.···.···.···.··.·.39 ·········0:·1··.···.···.···.···.···.···.··.·.
40 ·········1:·.·36·160·315·288··14···.···.··.·.40 ·········1:·.·36·160·315·288··14···.···.··.·.
41 ·········2:·.··.···.···.··14·288·315·160·36·.41 ·········2:·.··.···.···.··14·288·315·160·36·.
42 ·········3:·.··.···.···.···.···.···.···.··.·142 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 47, 15 lines modifiedOffset 47, 15 lines modified
47 o2·:·BettiTally47 o2·:·BettiTally
48 i3·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);48 i3·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);
  
49 ··············ZZ49 ··············ZZ
50 o3·:·Ideal·of·--[x·..x·,·y·..y·]50 o3·:·Ideal·of·--[x·..x·,·y·..y·]
51 ···············3··0···5···0···551 ···············3··0···5···0···5
52 i4·:·elapsedTime·T'=minimalBetti·J52 i4·:·elapsedTime·T'=minimalBetti·J
53 ·--·.266565s·elapsed53 ·--·.244737s·elapsed
  
54 ············0··1···2···3···4···5···6···7··8·954 ············0··1···2···3···4···5···6···7··8·9
55 o4·=·total:·1·36·160·315·302·302·315·160·36·155 o4·=·total:·1·36·160·315·302·302·315·160·36·1
56 ·········0:·1··.···.···.···.···.···.···.··.·.56 ·········0:·1··.···.···.···.···.···.···.··.·.
57 ·········1:·.·36·160·315·288··14···.···.··.·.57 ·········1:·.·36·160·315·288··14···.···.··.·.
58 ·········2:·.··.···.···.··14·288·315·160·36·.58 ·········2:·.··.···.···.··14·288·315·160·36·.
59 ·········3:·.··.···.···.···.···.···.···.··.·159 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 67, 22 lines modifiedOffset 67, 22 lines modified
67 o5·=·total:·.·.·.·.·.·.·.·.·.·.67 o5·=·total:·.·.·.·.·.·.·.·.·.·.
68 ·········1:·.·.·.·.·.·.·.·.·.·.68 ·········1:·.·.·.·.·.·.·.·.·.·.
69 ·········2:·.·.·.·.·.·.·.·.·.·.69 ·········2:·.·.·.·.·.·.·.·.·.·.
70 ·········3:·.·.·.·.·.·.·.·.·.·.70 ·········3:·.·.·.·.·.·.·.·.·.·.
  
71 o5·:·BettiTally71 o5·:·BettiTally
72 i6·:·elapsedTime·h=carpetBettiTables(6,6);72 i6·:·elapsedTime·h=carpetBettiTables(6,6);
 73 ·--·.00373134s·elapsed
73 ·--·.00834527s·elapsed74 ·--·.0150234s·elapsed
74 ·--·.0299358s·elapsed 
75 ·--·.102968s·elapsed75 ·--·.132568s·elapsed
76 ·--·.8111s·elapsed76 ·--·1.17797s·elapsed
 77 ·--·.338035s·elapsed
77 ·--·.251842s·elapsed78 ·--·.0335583s·elapsed
 79 ·--·.00565548s·elapsed
78 ·--·.0437321s·elapsed80 ·--·5.74672s·elapsed
79 ·--·.01127s·elapsed 
80 ·--·4.58719s·elapsed 
81 i7·:·carpetBettiTable(h,7)81 i7·:·carpetBettiTable(h,7)
  
82 ············0··1···2···3····4····5····6····7···8···9·10·1182 ············0··1···2···3····4····5····6····7···8···9·10·11
83 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··183 o7·=·total:·1·55·320·891·1408·1155·1155·1408·891·320·55··1
84 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.84 ·········0:·1··.···.···.····.····.····.····.···.···.··.··.
85 ·········1:·.·55·320·891·1408·1155····.····.···.···.··.··.85 ·········1:·.·55·320·891·1408·1155····.····.···.···.··.··.
86 ·········2:·.··.···.···.····.····.·1155·1408·891·320·55··.86 ·········2:·.··.···.···.····.····.·1155·1408·891·320·55··.
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Offset 78, 19 lines modifiedOffset 78, 19 lines modified
  
78 o1·=·(5,·5)78 o1·=·(5,·5)
  
79 o1·:·Sequence</code></pre>79 o1·:·Sequence</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i2·:·h=carpetBettiTables(a,b)82 <td>··············<pre><code·class="language-macaulay2">i2·:·h=carpetBettiTables(a,b)
83 ·--·.00197174s·elapsed83 ·--·.00371274s·elapsed
 84 ·--·.00990599s·elapsed
84 ·--·.00621081s·elapsed85 ·--·.0361867s·elapsed
85 ·--·.0194232s·elapsed86 ·--·.0152813s·elapsed
86 ·--·.0165629s·elapsed 
87 ·--·.00296745s·elapsed87 ·--·.00361937s·elapsed
  
88 ···························0··1···2···3···4···5···6···7··8·988 ···························0··1···2···3···4···5···6···7··8·9
89 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}89 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
90 ························0:·1··.···.···.···.···.···.···.··.·.90 ························0:·1··.···.···.···.···.···.···.··.·.
91 ························1:·.·36·160·315·288···.···.···.··.·.91 ························1:·.·36·160·315·288···.···.···.··.·.
92 ························2:·.··.···.···.···.·288·315·160·36·.92 ························2:·.··.···.···.···.·288·315·160·36·.
93 ························3:·.··.···.···.···.···.···.···.··.·193 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 126, 15 lines modifiedOffset 126, 15 lines modified
  
126 ··············ZZ126 ··············ZZ
127 o4·:·Ideal·of·--[x·..x·,·y·..y·]127 o4·:·Ideal·of·--[x·..x·,·y·..y·]
128 ···············3··0···5···0···5</code></pre>128 ···············3··0···5···0···5</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·T'=minimalBetti·J131 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·T'=minimalBetti·J
132 ·--·.506272s·elapsed132 ·--·.251137s·elapsed
  
133 ············0··1···2···3···4···5···6···7··8·9133 ············0··1···2···3···4···5···6···7··8·9
134 o5·=·total:·1·36·160·315·302·302·315·160·36·1134 o5·=·total:·1·36·160·315·302·302·315·160·36·1
135 ·········0:·1··.···.···.···.···.···.···.··.·.135 ·········0:·1··.···.···.···.···.···.···.··.·.
136 ·········1:·.·36·160·315·288··14···.···.··.·.136 ·········1:·.·36·160·315·288··14···.···.··.·.
137 ·········2:·.··.···.···.··14·288·315·160·36·.137 ·········2:·.··.···.···.··14·288·315·160·36·.
138 ·········3:·.··.···.···.···.···.···.···.··.·1138 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 150, 22 lines modifiedOffset 150, 22 lines modified
150 ·········2:·.·.·.·.·.·.·.·.·.·.150 ·········2:·.·.·.·.·.·.·.·.·.·.
151 ·········3:·.·.·.·.·.·.·.·.·.·.151 ·········3:·.·.·.·.·.·.·.·.·.·.
  
152 o6·:·BettiTally</code></pre>152 o6·:·BettiTally</code></pre>
153 </td>··········</tr>153 </td>··········</tr>
154 ··········<tr>154 ··········<tr>
155 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·h=carpetBettiTables(6,6);155 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·h=carpetBettiTables(6,6);
156 ·--·.00367266s·elapsed156 ·--·.0222809s·elapsed
157 ·--·.0144045s·elapsed157 ·--·.0154033s·elapsed
158 ·--·.099294s·elapsed 
159 ·--·2.22771s·elapsed 
160 ·--·1.13192s·elapsed158 ·--·.116889s·elapsed
 159 ·--·1.68376s·elapsed
 160 ·--·.503276s·elapsed
161 ·--·.0818742s·elapsed161 ·--·.0343167s·elapsed
162 ·--·.00998064s·elapsed162 ·--·.00572571s·elapsed
163 ·--·8.74567s·elapsed</code></pre>163 ·--·6.39374s·elapsed</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ··········<tr>165 ··········<tr>
166 <td>··············<pre><code·class="language-macaulay2">i8·:·keys·h166 <td>··············<pre><code·class="language-macaulay2">i8·:·keys·h
  
167 o8·=·{0,·2,·3,·5}167 o8·=·{0,·2,·3,·5}
  
168 o8·:·List</code></pre>168 o8·:·List</code></pre>
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22 resulting·data·allow·us·to·compute·the·Betti·tables·for·arbitrary·primes.22 resulting·data·allow·us·to·compute·the·Betti·tables·for·arbitrary·primes.
23 i1·:·a=5,b=523 i1·:·a=5,b=5
  
24 o1·=·(5,·5)24 o1·=·(5,·5)
  
25 o1·:·Sequence25 o1·:·Sequence
26 i2·:·h=carpetBettiTables(a,b)26 i2·:·h=carpetBettiTables(a,b)
27 ·--·.00197174s·elapsed27 ·--·.00371274s·elapsed
 28 ·--·.00990599s·elapsed
28 ·--·.00621081s·elapsed29 ·--·.0361867s·elapsed
29 ·--·.0194232s·elapsed30 ·--·.0152813s·elapsed
30 ·--·.0165629s·elapsed 
31 ·--·.00296745s·elapsed31 ·--·.00361937s·elapsed
  
32 ···························0··1···2···3···4···5···6···7··8·932 ···························0··1···2···3···4···5···6···7··8·9
33 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}33 o2·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
34 ························0:·1··.···.···.···.···.···.···.··.·.34 ························0:·1··.···.···.···.···.···.···.··.·.
35 ························1:·.·36·160·315·288···.···.···.··.·.35 ························1:·.·36·160·315·288···.···.···.··.·.
36 ························2:·.··.···.···.···.·288·315·160·36·.36 ························2:·.··.···.···.···.·288·315·160·36·.
37 ························3:·.··.···.···.···.···.···.···.··.·137 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 64, 15 lines modifiedOffset 64, 15 lines modified
64 o3·:·BettiTally64 o3·:·BettiTally
65 i4·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);65 i4·:·J=canonicalCarpet(a+b+1,b,Characteristic=>3);
  
66 ··············ZZ66 ··············ZZ
67 o4·:·Ideal·of·--[x·..x·,·y·..y·]67 o4·:·Ideal·of·--[x·..x·,·y·..y·]
68 ···············3··0···5···0···568 ···············3··0···5···0···5
69 i5·:·elapsedTime·T'=minimalBetti·J69 i5·:·elapsedTime·T'=minimalBetti·J
70 ·--·.506272s·elapsed70 ·--·.251137s·elapsed
  
71 ············0··1···2···3···4···5···6···7··8·971 ············0··1···2···3···4···5···6···7··8·9
72 o5·=·total:·1·36·160·315·302·302·315·160·36·172 o5·=·total:·1·36·160·315·302·302·315·160·36·1
73 ·········0:·1··.···.···.···.···.···.···.··.·.73 ·········0:·1··.···.···.···.···.···.···.··.·.
74 ·········1:·.·36·160·315·288··14···.···.··.·.74 ·········1:·.·36·160·315·288··14···.···.··.·.
75 ·········2:·.··.···.···.··14·288·315·160·36·.75 ·········2:·.··.···.···.··14·288·315·160·36·.
76 ·········3:·.··.···.···.···.···.···.···.··.·176 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 84, 22 lines modifiedOffset 84, 22 lines modified
84 o6·=·total:·.·.·.·.·.·.·.·.·.·.84 o6·=·total:·.·.·.·.·.·.·.·.·.·.
85 ·········1:·.·.·.·.·.·.·.·.·.·.85 ·········1:·.·.·.·.·.·.·.·.·.·.
86 ·········2:·.·.·.·.·.·.·.·.·.·.86 ·········2:·.·.·.·.·.·.·.·.·.·.
87 ·········3:·.·.·.·.·.·.·.·.·.·.87 ·········3:·.·.·.·.·.·.·.·.·.·.
  
88 o6·:·BettiTally88 o6·:·BettiTally
89 i7·:·elapsedTime·h=carpetBettiTables(6,6);89 i7·:·elapsedTime·h=carpetBettiTables(6,6);
90 ·--·.00367266s·elapsed90 ·--·.0222809s·elapsed
91 ·--·.0144045s·elapsed91 ·--·.0154033s·elapsed
92 ·--·.099294s·elapsed 
93 ·--·2.22771s·elapsed 
94 ·--·1.13192s·elapsed92 ·--·.116889s·elapsed
95 ·--·.0818742s·elapsed93 ·--·1.68376s·elapsed
96 ·--·.00998064s·elapsed94 ·--·.503276s·elapsed
97 ·--·8.74567s·elapsed95 ·--·.0343167s·elapsed
 96 ·--·.00572571s·elapsed
 97 ·--·6.39374s·elapsed
98 i8·:·keys·h98 i8·:·keys·h
  
99 o8·=·{0,·2,·3,·5}99 o8·=·{0,·2,·3,·5}
  
100 o8·:·List100 o8·:·List
101 i9·:·carpetBettiTable(h,7)101 i9·:·carpetBettiTable(h,7)
  
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Offset 78, 42 lines modifiedOffset 78, 42 lines modified
  
78 o1·=·(4,·4)78 o1·=·(4,·4)
  
79 o1·:·Sequence</code></pre>79 o1·:·Sequence</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i2·:·d=carpetDet(a,b)82 <td>··············<pre><code·class="language-macaulay2">i2·:·d=carpetDet(a,b)
83 ·--·.00509704s·elapsed 
84 ·--·.0283743s·elapsed83 ·--·.0213466s·elapsed
 84 ·--·.014109s·elapsed
 85 ·--·.000238217s·elapsed
85 ·--·.000197989s·elapsed86 ·--·.000159759s·elapsed
86 ·--·.000128293s·elapsed 
87 ·--·.000113561s·elapsed 
88 ·--·.000129816s·elapsed 
89 ·--·.000126621s·elapsed87 ·--·.000142628s·elapsed
 88 ·--·.000131887s·elapsed
 89 ·--·.000153778s·elapsed
 90 ·--·.000141074s·elapsed
 91 ·--·.000215554s·elapsed
 92 ·--·.000246212s·elapsed
 93 ·--·.000387176s·elapsed
 94 ·--·.000124644s·elapsed
 95 ·--·.000137688s·elapsed
 96 ·--·.000116799s·elapsed
90 ·--·.000152901s·elapsed97 ·--·.000112801s·elapsed
91 ·--·.000147313s·elapsed 
92 ·--·.000135714s·elapsed98 ·--·.000105277s·elapsed
 99 ·--·.000142197s·elapsed
 100 ·--·.000111389s·elapsed
 101 ·--·.000115326s·elapsed
 102 ·--·.000121107s·elapsed
 103 ·--·.000125665s·elapsed
 104 ·--·.000119213s·elapsed
 105 ·--·.00017078s·elapsed
 106 ·--·.000141155s·elapsed
 107 ·--·.000158727s·elapsed
 108 ·--·.00195276s·elapsed
 109 ·--·.000119484s·elapsed
93 ·--·.000247303s·elapsed110 ·--·.000247334s·elapsed
94 ·--·.000122275s·elapsed 
95 ·--·.000111357s·elapsed 
96 ·--·.00818544s·elapsed 
97 ·--·.000126722s·elapsed 
98 ·--·.000103167s·elapsed 
99 ·--·.000101773s·elapsed 
100 ·--·.000191048s·elapsed 
101 ·--·.000127052s·elapsed 
102 ·--·.000124678s·elapsed 
103 ·--·.000174403s·elapsed 
104 ·--·.000123465s·elapsed 
105 ·--·.000143016s·elapsed 
106 ·--·.000107172s·elapsed 
107 ·--·.00010521s·elapsed 
108 ·--·.000116836s·elapsed 
109 ·--·.000128103s·elapsed 
110 ·--·.000107351s·elapsed 
111 (number·Of·blocks,·26)111 (number·Of·blocks,·26)
112 1112 1
113 1113 1
114 1114 1
115 1115 1
116 2116 2
117 ·2117 ·2
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Offset 20, 42 lines modifiedOffset 20, 42 lines modified
20 determinants·and·return·their·product.20 determinants·and·return·their·product.
21 i1·:·a=4,b=421 i1·:·a=4,b=4
  
22 o1·=·(4,·4)22 o1·=·(4,·4)
  
23 o1·:·Sequence23 o1·:·Sequence
24 i2·:·d=carpetDet(a,b)24 i2·:·d=carpetDet(a,b)
25 ·--·.00509704s·elapsed 
26 ·--·.0283743s·elapsed25 ·--·.0213466s·elapsed
 26 ·--·.014109s·elapsed
 27 ·--·.000238217s·elapsed
27 ·--·.000197989s·elapsed28 ·--·.000159759s·elapsed
28 ·--·.000128293s·elapsed 
29 ·--·.000113561s·elapsed 
30 ·--·.000129816s·elapsed 
31 ·--·.000126621s·elapsed29 ·--·.000142628s·elapsed
 30 ·--·.000131887s·elapsed
 31 ·--·.000153778s·elapsed
 32 ·--·.000141074s·elapsed
 33 ·--·.000215554s·elapsed
 34 ·--·.000246212s·elapsed
 35 ·--·.000387176s·elapsed
 36 ·--·.000124644s·elapsed
 37 ·--·.000137688s·elapsed
 38 ·--·.000116799s·elapsed
32 ·--·.000152901s·elapsed39 ·--·.000112801s·elapsed
33 ·--·.000147313s·elapsed 
34 ·--·.000135714s·elapsed40 ·--·.000105277s·elapsed
 41 ·--·.000142197s·elapsed
 42 ·--·.000111389s·elapsed
 43 ·--·.000115326s·elapsed
 44 ·--·.000121107s·elapsed
 45 ·--·.000125665s·elapsed
 46 ·--·.000119213s·elapsed
 47 ·--·.00017078s·elapsed
 48 ·--·.000141155s·elapsed
 49 ·--·.000158727s·elapsed
 50 ·--·.00195276s·elapsed
 51 ·--·.000119484s·elapsed
35 ·--·.000247303s·elapsed52 ·--·.000247334s·elapsed
36 ·--·.000122275s·elapsed 
37 ·--·.000111357s·elapsed 
38 ·--·.00818544s·elapsed 
39 ·--·.000126722s·elapsed 
40 ·--·.000103167s·elapsed 
41 ·--·.000101773s·elapsed 
42 ·--·.000191048s·elapsed 
43 ·--·.000127052s·elapsed 
44 ·--·.000124678s·elapsed 
45 ·--·.000174403s·elapsed 
46 ·--·.000123465s·elapsed 
47 ·--·.000143016s·elapsed 
48 ·--·.000107172s·elapsed 
49 ·--·.00010521s·elapsed 
50 ·--·.000116836s·elapsed 
51 ·--·.000128103s·elapsed 
52 ·--·.000107351s·elapsed 
53 (number·Of·blocks,·26)53 (number·Of·blocks,·26)
54 154 1
55 155 1
56 156 1
57 157 1
58 258 2
59 ·259 ·2
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Offset 86, 20 lines modifiedOffset 86, 20 lines modified
  
86 o1·=·(9,·7)86 o1·=·(9,·7)
  
87 o1·:·Sequence</code></pre>87 o1·:·Sequence</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i2·:·computeBound·390 <td>··············<pre><code·class="language-macaulay2">i2·:·computeBound·3
91 ·--·1.34545s·elapsed 
92 ·--·1.07082s·elapsed 
93 ·--·.504563s·elapsed91 ·--·.760467s·elapsed
 92 ·--·.409083s·elapsed
94 ·--·1.22392s·elapsed93 ·--·.220772s·elapsed
95 ·--·.834693s·elapsed 
96 ·--·1.37163s·elapsed94 ·--·.767807s·elapsed
 95 ·--·.445704s·elapsed
 96 ·--·.672478s·elapsed
  
97 o2·=·6</code></pre>97 o2·=·6</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ········</table>99 ········</table>
100 ······</div>100 ······</div>
101 ······<div>101 ······<div>
102 ········<h2>See·also</h2>102 ········<h2>See·also</h2>
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Offset 26, 20 lines modifiedOffset 26, 20 lines modified
26 classes·mod·k.·We·conjecture·that·c=k^2-k.26 classes·mod·k.·We·conjecture·that·c=k^2-k.
27 i1·:·(a,b)=computeBound(6,4,3)27 i1·:·(a,b)=computeBound(6,4,3)
  
28 o1·=·(9,·7)28 o1·=·(9,·7)
  
29 o1·:·Sequence29 o1·:·Sequence
30 i2·:·computeBound·330 i2·:·computeBound·3
31 ·--·1.34545s·elapsed 
32 ·--·1.07082s·elapsed 
33 ·--·.504563s·elapsed31 ·--·.760467s·elapsed
 32 ·--·.409083s·elapsed
34 ·--·1.22392s·elapsed33 ·--·.220772s·elapsed
35 ·--·.834693s·elapsed 
36 ·--·1.37163s·elapsed34 ·--·.767807s·elapsed
 35 ·--·.445704s·elapsed
 36 ·--·.672478s·elapsed
  
37 o2·=·637 o2·=·6
38 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*38 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
39 ····*·_\x8r_\x8e_\x8l_\x8a_\x8t_\x8i_\x8v_\x8e_\x8E_\x8q_\x8u_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·compute·the·relative·quadrics39 ····*·_\x8r_\x8e_\x8l_\x8a_\x8t_\x8i_\x8v_\x8e_\x8E_\x8q_\x8u_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·compute·the·relative·quadrics
40 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8co\x8om\x8mp\x8pu\x8ut\x8te\x8eB\x8Bo\x8ou\x8un\x8nd\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*40 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8co\x8om\x8mp\x8pu\x8ut\x8te\x8eB\x8Bo\x8ou\x8un\x8nd\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
41 ····*·computeBound(ZZ)41 ····*·computeBound(ZZ)
42 ····*·computeBound(ZZ,ZZ,ZZ)42 ····*·computeBound(ZZ,ZZ,ZZ)
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./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
    
Offset 87, 19 lines modifiedOffset 87, 19 lines modified
  
87 o2·=·(-1,·5)87 o2·=·(-1,·5)
  
88 o2·:·Sequence</code></pre>88 o2·:·Sequence</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·h=degenerateK3BettiTables(a,b,e)91 <td>··············<pre><code·class="language-macaulay2">i3·:·h=degenerateK3BettiTables(a,b,e)
92 ·--·.00189157s·elapsed 
93 ·--·.00468804s·elapsed 
94 ·--·.0195169s·elapsed92 ·--·.00237356s·elapsed
95 ·--·.00721757s·elapsed93 ·--·.0052173s·elapsed
 94 ·--·.0205611s·elapsed
96 ·--·.00284809s·elapsed95 ·--·.0084417s·elapsed
 96 ·--·.00336082s·elapsed
  
97 ···························0··1···2···3···4···5···6···7··8·997 ···························0··1···2···3···4···5···6···7··8·9
98 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}98 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
99 ························0:·1··.···.···.···.···.···.···.··.·.99 ························0:·1··.···.···.···.···.···.···.··.·.
100 ························1:·.·36·160·315·288···.···.···.··.·.100 ························1:·.·36·160·315·288···.···.···.··.·.
101 ························2:·.··.···.···.···.·288·315·160·36·.101 ························2:·.··.···.···.···.·288·315·160·36·.
102 ························3:·.··.···.···.···.···.···.···.··.·1102 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 129, 15 lines modifiedOffset 129, 15 lines modified
  
129 o4·=·{0,·2,·3,·5}129 o4·=·{0,·2,·3,·5}
  
130 o4·:·List</code></pre>130 o4·:·List</code></pre>
131 </td>··········</tr>131 </td>··········</tr>
132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)133 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)
134 ·--·.424093s·elapsed134 ·--·.366147s·elapsed
  
135 ············0··1···2···3···4···5···6···7··8·9135 ············0··1···2···3···4···5···6···7··8·9
136 o5·=·total:·1·36·167·370·476·476·370·167·36·1136 o5·=·total:·1·36·167·370·476·476·370·167·36·1
137 ·········0:·1··.···.···.···.···.···.···.··.·.137 ·········0:·1··.···.···.···.···.···.···.··.·.
138 ·········1:·.·36·160·322·336·140··48···7··.·.138 ·········1:·.·36·160·322·336·140··48···7··.·.
139 ·········2:·.··.···7··48·140·336·322·160·36·.139 ·········2:·.··.···7··48·140·336·322·160·36·.
140 ·········3:·.··.···.···.···.···.···.···.··.·1140 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 165, 19 lines modifiedOffset 165, 19 lines modified
  
165 o7·=·(-1,·25)165 o7·=·(-1,·25)
  
166 o7·:·Sequence</code></pre>166 o7·:·Sequence</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i8·:·h=degenerateK3BettiTables(a,b,e)169 <td>··············<pre><code·class="language-macaulay2">i8·:·h=degenerateK3BettiTables(a,b,e)
170 ·--·.00413622s·elapsed 
171 ·--·.00480755s·elapsed 
172 ·--·.0205215s·elapsed170 ·--·.00212253s·elapsed
 171 ·--·.00539577s·elapsed
 172 ·--·.032081s·elapsed
173 ·--·.012116s·elapsed173 ·--·.0155421s·elapsed
174 ·--·.00333302s·elapsed174 ·--·.00382758s·elapsed
  
175 ···························0··1···2···3···4···5···6···7··8·9175 ···························0··1···2···3···4···5···6···7··8·9
176 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}176 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}
177 ························0:·1··.···.···.···.···.···.···.··.·.177 ························0:·1··.···.···.···.···.···.···.··.·.
178 ························1:·.·36·160·315·288···.···.···.··.·.178 ························1:·.·36·160·315·288···.···.···.··.·.
179 ························2:·.··.···.···.···.·288·315·160·36·.179 ························2:·.··.···.···.···.·288·315·160·36·.
180 ························3:·.··.···.···.···.···.···.···.··.·1180 ························3:·.··.···.···.···.···.···.···.··.·1
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Offset 28, 19 lines modifiedOffset 28, 19 lines modified
28 o1·:·Sequence28 o1·:·Sequence
29 i2·:·e=(-1,5)29 i2·:·e=(-1,5)
  
30 o2·=·(-1,·5)30 o2·=·(-1,·5)
  
31 o2·:·Sequence31 o2·:·Sequence
32 i3·:·h=degenerateK3BettiTables(a,b,e)32 i3·:·h=degenerateK3BettiTables(a,b,e)
33 ·--·.00189157s·elapsed 
34 ·--·.00468804s·elapsed 
35 ·--·.0195169s·elapsed33 ·--·.00237356s·elapsed
36 ·--·.00721757s·elapsed34 ·--·.0052173s·elapsed
 35 ·--·.0205611s·elapsed
37 ·--·.00284809s·elapsed36 ·--·.0084417s·elapsed
 37 ·--·.00336082s·elapsed
  
38 ···························0··1···2···3···4···5···6···7··8·938 ···························0··1···2···3···4···5···6···7··8·9
39 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}39 o3·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1}
40 ························0:·1··.···.···.···.···.···.···.··.·.40 ························0:·1··.···.···.···.···.···.···.··.·.
41 ························1:·.·36·160·315·288···.···.···.··.·.41 ························1:·.·36·160·315·288···.···.···.··.·.
42 ························2:·.··.···.···.···.·288·315·160·36·.42 ························2:·.··.···.···.···.·288·315·160·36·.
43 ························3:·.··.···.···.···.···.···.···.··.·143 ························3:·.··.···.···.···.···.···.···.··.·1
Offset 66, 15 lines modifiedOffset 66, 15 lines modified
66 o3·:·HashTable66 o3·:·HashTable
67 i4·:·keys·h67 i4·:·keys·h
  
68 o4·=·{0,·2,·3,·5}68 o4·=·{0,·2,·3,·5}
  
69 o4·:·List69 o4·:·List
70 i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)70 i5·:·elapsedTime·T=·minimalBetti·degenerateK3(a,b,e,Characteristic=>5)
71 ·--·.424093s·elapsed71 ·--·.366147s·elapsed
  
72 ············0··1···2···3···4···5···6···7··8·972 ············0··1···2···3···4···5···6···7··8·9
73 o5·=·total:·1·36·167·370·476·476·370·167·36·173 o5·=·total:·1·36·167·370·476·476·370·167·36·1
74 ·········0:·1··.···.···.···.···.···.···.··.·.74 ·········0:·1··.···.···.···.···.···.···.··.·.
75 ·········1:·.·36·160·322·336·140··48···7··.·.75 ·········1:·.·36·160·322·336·140··48···7··.·.
76 ·········2:·.··.···7··48·140·336·322·160·36·.76 ·········2:·.··.···7··48·140·336·322·160·36·.
77 ·········3:·.··.···.···.···.···.···.···.··.·177 ·········3:·.··.···.···.···.···.···.···.··.·1
Offset 95, 19 lines modifiedOffset 95, 19 lines modified
95 these·mistakes.95 these·mistakes.
96 i7·:·e=(-1,5^2)96 i7·:·e=(-1,5^2)
  
97 o7·=·(-1,·25)97 o7·=·(-1,·25)
  
98 o7·:·Sequence98 o7·:·Sequence
99 i8·:·h=degenerateK3BettiTables(a,b,e)99 i8·:·h=degenerateK3BettiTables(a,b,e)
100 ·--·.00413622s·elapsed 
101 ·--·.00480755s·elapsed 
102 ·--·.0205215s·elapsed100 ·--·.00212253s·elapsed
 101 ·--·.00539577s·elapsed
 102 ·--·.032081s·elapsed
103 ·--·.012116s·elapsed103 ·--·.0155421s·elapsed
104 ·--·.00333302s·elapsed104 ·--·.00382758s·elapsed
  
105 ···························0··1···2···3···4···5···6···7··8·9105 ···························0··1···2···3···4···5···6···7··8·9
106 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}106 o8·=·HashTable{0·=>·total:·1·36·160·315·288·288·315·160·36·1·····}
107 ························0:·1··.···.···.···.···.···.···.··.·.107 ························0:·1··.···.···.···.···.···.···.··.·.
108 ························1:·.·36·160·315·288···.···.···.··.·.108 ························1:·.·36·160·315·288···.···.···.··.·.
109 ························2:·.··.···.···.···.·288·315·160·36·.109 ························2:·.··.···.···.···.·288·315·160·36·.
110 ························3:·.··.···.···.···.···.···.···.··.·1110 ························3:·.··.···.···.···.···.···.···.··.·1
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./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
    
Offset 76, 33 lines modifiedOffset 76, 33 lines modified
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·a=477 <td>··············<pre><code·class="language-macaulay2">i1·:·a=4
  
78 o1·=·4</code></pre>78 o1·=·4</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·(d1,d2)=resonanceDet(a)81 <td>··············<pre><code·class="language-macaulay2">i2·:·(d1,d2)=resonanceDet(a)
82 ·--·.0474775s·elapsed82 ·--·.0141837s·elapsed
 83 ·--·.000035988s·elapsed
 84 ·--·.000093776s·elapsed
 85 ·--·.000089919s·elapsed
 86 ·--·.0000907s·elapsed
 87 ·--·.000113693s·elapsed
 88 ·--·.000100879s·elapsed
83 ·--·.000030627s·elapsed89 ·--·.000030677s·elapsed
84 ·--·.000069194s·elapsed90 ·--·.000092614s·elapsed
85 ·--·.000050365s·elapsed91 ·--·.000155772s·elapsed
86 ·--·.000058698s·elapsed 
87 ·--·.000063466s·elapsed 
88 ·--·.000199841s·elapsed92 ·--·.000128591s·elapsed
89 ·--·.000022634s·elapsed93 ·--·.000079249s·elapsed
90 ·--·.0526269s·elapsed 
91 ·--·.000108815s·elapsed 
92 ·--·.000124086s·elapsed 
93 ·--·.000068743s·elapsed 
94 ·--·.000058549s·elapsed 
95 ·--·.000059089s·elapsed94 ·--·.000097002s·elapsed
96 ·--·.000044588s·elapsed 
97 ·--·.00765186s·elapsed 
98 ·--·.000029114s·elapsed95 ·--·.000265427s·elapsed
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 99 ·--·.000265718s·elapsed
 100 ·--·.000091552s·elapsed
101 (number·of·blocks=·,·18)101 (number·of·blocks=·,·18)
102 (size·of·the·matrices,·Tally{1·=>·4})102 (size·of·the·matrices,·Tally{1·=>·4})
103 ·····························2·=>·6103 ·····························2·=>·6
104 ·····························3·=>·2104 ·····························3·=>·2
105 ·····························4·=>·6105 ·····························4·=>·6
106 ·······0·1106 ·······0·1
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20 grading.·Viewed·as·a·resolution·over·QQ(e_1,e_2),·this·resolution·is·non-20 grading.·Viewed·as·a·resolution·over·QQ(e_1,e_2),·this·resolution·is·non-
21 minimal·and·carries·further·gradings.·We·decompose·the·crucial·map·of·the·a-th21 minimal·and·carries·further·gradings.·We·decompose·the·crucial·map·of·the·a-th
22 strand·into·blocks,·compute·their·determinants,·and·factor·the·product.22 strand·into·blocks,·compute·their·determinants,·and·factor·the·product.
23 i1·:·a=423 i1·:·a=4
  
24 o1·=·424 o1·=·4
25 i2·:·(d1,d2)=resonanceDet(a)25 i2·:·(d1,d2)=resonanceDet(a)
26 ·--·.0474775s·elapsed26 ·--·.0141837s·elapsed
 27 ·--·.000035988s·elapsed
 28 ·--·.000093776s·elapsed
 29 ·--·.000089919s·elapsed
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31 ·--·.000063466s·elapsed 
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34 ·--·.0526269s·elapsed 
35 ·--·.000108815s·elapsed 
36 ·--·.000124086s·elapsed 
37 ·--·.000068743s·elapsed 
38 ·--·.000058549s·elapsed 
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41 ·--·.00765186s·elapsed 
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44 ·--·.000017897s·elapsed41 ·--·.000074299s·elapsed
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 43 ·--·.000265718s·elapsed
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46 (size·of·the·matrices,·Tally{1·=>·4})46 (size·of·the·matrices,·Tally{1·=>·4})
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49 ·····························4·=>·649 ·····························4·=>·6
50 ·······0·150 ·······0·1
51 total:·1·151 total:·1·1
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Offset 7, 55 lines modifiedOffset 7, 55 lines modified
  
7 i2·:·m·=·syz·m1;7 i2·:·m·=·syz·m1;
  
8 ··············50·······478 ··············50·······47
9 o2·:·Matrix·ZZ···<--·ZZ9 o2·:·Matrix·ZZ···<--·ZZ
  
10 i3·:·time·LLL·m;10 i3·:·time·LLL·m;
11 ·--·used·0.00799502s·(cpu);·0.00708641s·(thread);·0s·(gc)11 ·--·used·0.0067243s·(cpu);·0.00710246s·(thread);·0s·(gc)
  
12 ··············50·······4712 ··············50·······47
13 o3·:·Matrix·ZZ···<--·ZZ13 o3·:·Matrix·ZZ···<--·ZZ
  
14 i4·:·time·LLL(m,·Strategy=>CohenEngine);14 i4·:·time·LLL(m,·Strategy=>CohenEngine);
15 ·--·used·0.0215451s·(cpu);·0.0227067s·(thread);·0s·(gc)15 ·--·used·0.0257578s·(cpu);·0.026925s·(thread);·0s·(gc)
  
16 ··············50·······4716 ··············50·······47
17 o4·:·Matrix·ZZ···<--·ZZ17 o4·:·Matrix·ZZ···<--·ZZ
  
18 i5·:·time·LLL(m,·Strategy=>CohenTopLevel);18 i5·:·time·LLL(m,·Strategy=>CohenTopLevel);
19 ·--·used·0.0782916s·(cpu);·0.0805048s·(thread);·0s·(gc)19 ·--·used·0.102686s·(cpu);·0.105617s·(thread);·0s·(gc)
  
20 ··············50·······4720 ··············50·······47
21 o5·:·Matrix·ZZ···<--·ZZ21 o5·:·Matrix·ZZ···<--·ZZ
  
22 i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});22 i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});
23 ·--·used·0.00955947s·(cpu);·0.0102083s·(thread);·0s·(gc)23 ·--·used·0.0088902s·(cpu);·0.0112675s·(thread);·0s·(gc)
  
24 ··············50·······4724 ··············50·······47
25 o6·:·Matrix·ZZ···<--·ZZ25 o6·:·Matrix·ZZ···<--·ZZ
  
26 i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});26 i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});
27 ·--·used·0.0505885s·(cpu);·0.0509241s·(thread);·0s·(gc)27 ·--·used·0.0606047s·(cpu);·0.0635356s·(thread);·0s·(gc)
  
28 ··············50·······4728 ··············50·······47
29 o7·:·Matrix·ZZ···<--·ZZ29 o7·:·Matrix·ZZ···<--·ZZ
  
30 i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});30 i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});
31 ·--·used·0.0499709s·(cpu);·0.0530298s·(thread);·0s·(gc)31 ·--·used·0.0705691s·(cpu);·0.0708845s·(thread);·0s·(gc)
  
32 ··············50·······4732 ··············50·······47
33 o8·:·Matrix·ZZ···<--·ZZ33 o8·:·Matrix·ZZ···<--·ZZ
  
34 i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});34 i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});
35 ·--·used·0.243689s·(cpu);·0.245412s·(thread);·0s·(gc)35 ·--·used·0.41175s·(cpu);·0.413445s·(thread);·0s·(gc)
  
36 ··············50·······4736 ··············50·······47
37 o9·:·Matrix·ZZ···<--·ZZ37 o9·:·Matrix·ZZ···<--·ZZ
  
38 i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});38 i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});
39 ·--·used·0.123572s·(cpu);·0.123882s·(thread);·0s·(gc)39 ·--·used·0.197135s·(cpu);·0.198522s·(thread);·0s·(gc)
  
40 ···············50·······4740 ···············50·······47
41 o10·:·Matrix·ZZ···<--·ZZ41 o10·:·Matrix·ZZ···<--·ZZ
  
42 i11·:·42 i11·:·
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./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
    
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162 <td>··············<pre><code·class="language-macaulay2">i2·:·m·=·syz·m1;162 <td>··············<pre><code·class="language-macaulay2">i2·:·m·=·syz·m1;
  
163 ··············50·······47163 ··············50·······47
164 o2·:·Matrix·ZZ···&lt;--·ZZ</code></pre>164 o2·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
165 </td>··········</tr>165 </td>··········</tr>
166 ··········<tr>166 ··········<tr>
167 <td>··············<pre><code·class="language-macaulay2">i3·:·time·LLL·m;167 <td>··············<pre><code·class="language-macaulay2">i3·:·time·LLL·m;
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169 ··············50·······47169 ··············50·······47
170 o3·:·Matrix·ZZ···&lt;--·ZZ</code></pre>170 o3·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
171 </td>··········</tr>171 </td>··········</tr>
172 ··········<tr>172 ··········<tr>
173 <td>··············<pre><code·class="language-macaulay2">i4·:·time·LLL(m,·Strategy=>CohenEngine);173 <td>··············<pre><code·class="language-macaulay2">i4·:·time·LLL(m,·Strategy=>CohenEngine);
174 ·--·used·0.0215451s·(cpu);·0.0227067s·(thread);·0s·(gc)174 ·--·used·0.0257578s·(cpu);·0.026925s·(thread);·0s·(gc)
  
175 ··············50·······47175 ··············50·······47
176 o4·:·Matrix·ZZ···&lt;--·ZZ</code></pre>176 o4·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
177 </td>··········</tr>177 </td>··········</tr>
178 ··········<tr>178 ··········<tr>
179 <td>··············<pre><code·class="language-macaulay2">i5·:·time·LLL(m,·Strategy=>CohenTopLevel);179 <td>··············<pre><code·class="language-macaulay2">i5·:·time·LLL(m,·Strategy=>CohenTopLevel);
180 ·--·used·0.0782916s·(cpu);·0.0805048s·(thread);·0s·(gc)180 ·--·used·0.102686s·(cpu);·0.105617s·(thread);·0s·(gc)
  
181 ··············50·······47181 ··············50·······47
182 o5·:·Matrix·ZZ···&lt;--·ZZ</code></pre>182 o5·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
183 </td>··········</tr>183 </td>··········</tr>
184 ··········<tr>184 ··········<tr>
185 <td>··············<pre><code·class="language-macaulay2">i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});185 <td>··············<pre><code·class="language-macaulay2">i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});
186 ·--·used·0.00955947s·(cpu);·0.0102083s·(thread);·0s·(gc)186 ·--·used·0.0088902s·(cpu);·0.0112675s·(thread);·0s·(gc)
  
187 ··············50·······47187 ··············50·······47
188 o6·:·Matrix·ZZ···&lt;--·ZZ</code></pre>188 o6·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ··········<tr>190 ··········<tr>
191 <td>··············<pre><code·class="language-macaulay2">i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});191 <td>··············<pre><code·class="language-macaulay2">i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});
192 ·--·used·0.0505885s·(cpu);·0.0509241s·(thread);·0s·(gc)192 ·--·used·0.0606047s·(cpu);·0.0635356s·(thread);·0s·(gc)
  
193 ··············50·······47193 ··············50·······47
194 o7·:·Matrix·ZZ···&lt;--·ZZ</code></pre>194 o7·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
195 </td>··········</tr>195 </td>··········</tr>
196 ··········<tr>196 ··········<tr>
197 <td>··············<pre><code·class="language-macaulay2">i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});197 <td>··············<pre><code·class="language-macaulay2">i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});
198 ·--·used·0.0499709s·(cpu);·0.0530298s·(thread);·0s·(gc)198 ·--·used·0.0705691s·(cpu);·0.0708845s·(thread);·0s·(gc)
  
199 ··············50·······47199 ··············50·······47
200 o8·:·Matrix·ZZ···&lt;--·ZZ</code></pre>200 o8·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
201 </td>··········</tr>201 </td>··········</tr>
202 ··········<tr>202 ··········<tr>
203 <td>··············<pre><code·class="language-macaulay2">i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});203 <td>··············<pre><code·class="language-macaulay2">i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});
204 ·--·used·0.243689s·(cpu);·0.245412s·(thread);·0s·(gc)204 ·--·used·0.41175s·(cpu);·0.413445s·(thread);·0s·(gc)
  
205 ··············50·······47205 ··············50·······47
206 o9·:·Matrix·ZZ···&lt;--·ZZ</code></pre>206 o9·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
207 </td>··········</tr>207 </td>··········</tr>
208 ··········<tr>208 ··········<tr>
209 <td>··············<pre><code·class="language-macaulay2">i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});209 <td>··············<pre><code·class="language-macaulay2">i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});
210 ·--·used·0.123572s·(cpu);·0.123882s·(thread);·0s·(gc)210 ·--·used·0.197135s·(cpu);·0.198522s·(thread);·0s·(gc)
  
211 ···············50·······47211 ···············50·······47
212 o10·:·Matrix·ZZ···&lt;--·ZZ</code></pre>212 o10·:·Matrix·ZZ···&lt;--·ZZ</code></pre>
213 </td>··········</tr>213 </td>··········</tr>
214 ········</table>214 ········</table>
215 ······</div>215 ······</div>
216 ······<div>216 ······<div>
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118 i2·:·m·=·syz·m1;118 i2·:·m·=·syz·m1;
  
119 ··············50·······47119 ··············50·······47
120 o2·:·Matrix·ZZ···<--·ZZ120 o2·:·Matrix·ZZ···<--·ZZ
121 i3·:·time·LLL·m;121 i3·:·time·LLL·m;
122 ·--·used·0.00799502s·(cpu);·0.00708641s·(thread);·0s·(gc)122 ·--·used·0.0067243s·(cpu);·0.00710246s·(thread);·0s·(gc)
  
123 ··············50·······47123 ··············50·······47
124 o3·:·Matrix·ZZ···<--·ZZ124 o3·:·Matrix·ZZ···<--·ZZ
125 i4·:·time·LLL(m,·Strategy=>CohenEngine);125 i4·:·time·LLL(m,·Strategy=>CohenEngine);
126 ·--·used·0.0215451s·(cpu);·0.0227067s·(thread);·0s·(gc)126 ·--·used·0.0257578s·(cpu);·0.026925s·(thread);·0s·(gc)
  
127 ··············50·······47127 ··············50·······47
128 o4·:·Matrix·ZZ···<--·ZZ128 o4·:·Matrix·ZZ···<--·ZZ
129 i5·:·time·LLL(m,·Strategy=>CohenTopLevel);129 i5·:·time·LLL(m,·Strategy=>CohenTopLevel);
130 ·--·used·0.0782916s·(cpu);·0.0805048s·(thread);·0s·(gc)130 ·--·used·0.102686s·(cpu);·0.105617s·(thread);·0s·(gc)
  
131 ··············50·······47131 ··············50·······47
132 o5·:·Matrix·ZZ···<--·ZZ132 o5·:·Matrix·ZZ···<--·ZZ
133 i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});133 i6·:·time·LLL(m,·Strategy=>{Givens,RealFP});
134 ·--·used·0.00955947s·(cpu);·0.0102083s·(thread);·0s·(gc)134 ·--·used·0.0088902s·(cpu);·0.0112675s·(thread);·0s·(gc)
  
135 ··············50·······47135 ··············50·······47
136 o6·:·Matrix·ZZ···<--·ZZ136 o6·:·Matrix·ZZ···<--·ZZ
137 i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});137 i7·:·time·LLL(m,·Strategy=>{Givens,RealQP});
138 ·--·used·0.0505885s·(cpu);·0.0509241s·(thread);·0s·(gc)138 ·--·used·0.0606047s·(cpu);·0.0635356s·(thread);·0s·(gc)
  
139 ··············50·······47139 ··············50·······47
140 o7·:·Matrix·ZZ···<--·ZZ140 o7·:·Matrix·ZZ···<--·ZZ
141 i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});141 i8·:·time·LLL(m,·Strategy=>{Givens,RealXD});
142 ·--·used·0.0499709s·(cpu);·0.0530298s·(thread);·0s·(gc)142 ·--·used·0.0705691s·(cpu);·0.0708845s·(thread);·0s·(gc)
  
143 ··············50·······47143 ··············50·······47
144 o8·:·Matrix·ZZ···<--·ZZ144 o8·:·Matrix·ZZ···<--·ZZ
145 i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});145 i9·:·time·LLL(m,·Strategy=>{Givens,RealRR});
146 ·--·used·0.243689s·(cpu);·0.245412s·(thread);·0s·(gc)146 ·--·used·0.41175s·(cpu);·0.413445s·(thread);·0s·(gc)
  
147 ··············50·······47147 ··············50·······47
148 o9·:·Matrix·ZZ···<--·ZZ148 o9·:·Matrix·ZZ···<--·ZZ
149 i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});149 i10·:·time·LLL(m,·Strategy=>{BKZ,Givens,RealQP});
150 ·--·used·0.123572s·(cpu);·0.123882s·(thread);·0s·(gc)150 ·--·used·0.197135s·(cpu);·0.198522s·(thread);·0s·(gc)
  
151 ···············50·······47151 ···············50·······47
152 o10·:·Matrix·ZZ···<--·ZZ152 o10·:·Matrix·ZZ···<--·ZZ
153 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*153 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
154 ····*·Default·value:·_\x8N_\x8T_\x8L154 ····*·Default·value:·_\x8N_\x8T_\x8L
155 ····*·Function:·_\x8L_\x8L_\x8L·--·compute·an·LLL·basis155 ····*·Function:·_\x8L_\x8L_\x8L·--·compute·an·LLL·basis
156 ····*·Option·key:·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y·--·an·optional·argument156 ····*·Option·key:·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y·--·an·optional·argument
⊟
757 B
./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=287 #:len=28
8 YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ==8 YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ==
9 #:len=2899 #:len=289
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh
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569 B
./usr/share/doc/Macaulay2/LatticePolytopes/example-output/_are__Isomorphic.out
    
Offset 16, 14 lines modifiedOffset 16, 14 lines modified
  
16 ··············3·······816 ··············3·······8
17 o4·:·Matrix·ZZ··<--·ZZ17 o4·:·Matrix·ZZ··<--·ZZ
  
18 i5·:·P·=·convexHull(M);18 i5·:·P·=·convexHull(M);
  
19 i6·:·time·areIsomorphic(P,P);19 i6·:·time·areIsomorphic(P,P);
20 ·--·used·4.99971s·(cpu);·0.909027s·(thread);·0s·(gc)20 ·--·used·5.46099s·(cpu);·1.05153s·(thread);·0s·(gc)
  
21 i7·:·time·areIsomorphic(P,P,smoothTest=>false);21 i7·:·time·areIsomorphic(P,P,smoothTest=>false);
22 ·--·used·2.78477s·(cpu);·0.522481s·(thread);·0s·(gc)22 ·--·used·2.98649s·(cpu);·0.587501s·(thread);·0s·(gc)
  
23 i8·:·23 i8·:·
24 ·····24 ·····
⊟
1.89 KB
./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
    
Offset 114, 19 lines modifiedOffset 114, 19 lines modified
114 o4·:·Matrix·ZZ··&lt;--·ZZ</code></pre>114 o4·:·Matrix·ZZ··&lt;--·ZZ</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ··········<tr>116 ··········<tr>
117 <td>··············<pre><code·class="language-macaulay2">i5·:·P·=·convexHull(M);</code></pre>117 <td>··············<pre><code·class="language-macaulay2">i5·:·P·=·convexHull(M);</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i6·:·time·areIsomorphic(P,P);120 <td>··············<pre><code·class="language-macaulay2">i6·:·time·areIsomorphic(P,P);
121 ·--·used·4.99971s·(cpu);·0.909027s·(thread);·0s·(gc)</code></pre>121 ·--·used·5.46099s·(cpu);·1.05153s·(thread);·0s·(gc)</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i7·:·time·areIsomorphic(P,P,smoothTest=>false);124 <td>··············<pre><code·class="language-macaulay2">i7·:·time·areIsomorphic(P,P,smoothTest=>false);
125 ·--·used·2.78477s·(cpu);·0.522481s·(thread);·0s·(gc)</code></pre>125 ·--·used·2.98649s·(cpu);·0.587501s·(thread);·0s·(gc)</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ········</table>127 ········</table>
128 ······</div>128 ······</div>
129 ······<div·class="waystouse">129 ······<div·class="waystouse">
130 ········<h2>Ways·to·use·<span·class="tt">areIsomorphic</span>:</h2>130 ········<h2>Ways·to·use·<span·class="tt">areIsomorphic</span>:</h2>
131 ········<ul>131 ········<ul>
132 ··········<li>132 ··········<li>
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880 B
html2text {}
    
Offset 36, 15 lines modifiedOffset 36, 15 lines modified
36 ·····|·0·0·1·0·1·0·1·1·|36 ·····|·0·0·1·0·1·0·1·1·|
37 ·····|·0·0·0·1·0·1·1·1·|37 ·····|·0·0·0·1·0·1·1·1·|
  
38 ··············3·······838 ··············3·······8
39 o4·:·Matrix·ZZ··<--·ZZ39 o4·:·Matrix·ZZ··<--·ZZ
40 i5·:·P·=·convexHull(M);40 i5·:·P·=·convexHull(M);
41 i6·:·time·areIsomorphic(P,P);41 i6·:·time·areIsomorphic(P,P);
42 ·--·used·4.99971s·(cpu);·0.909027s·(thread);·0s·(gc)42 ·--·used·5.46099s·(cpu);·1.05153s·(thread);·0s·(gc)
43 i7·:·time·areIsomorphic(P,P,smoothTest=>false);43 i7·:·time·areIsomorphic(P,P,smoothTest=>false);
44 ·--·used·2.78477s·(cpu);·0.522481s·(thread);·0s·(gc)44 ·--·used·2.98649s·(cpu);·0.587501s·(thread);·0s·(gc)
45 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·a\x8ar\x8re\x8eI\x8Is\x8so\x8om\x8mo\x8or\x8rp\x8ph\x8hi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*45 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·a\x8ar\x8re\x8eI\x8Is\x8so\x8om\x8mo\x8or\x8rp\x8ph\x8hi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*
46 ····*·areIsomorphic(Matrix,Matrix)46 ····*·areIsomorphic(Matrix,Matrix)
47 ····*·areIsomorphic(Polyhedron,Polyhedron)47 ····*·areIsomorphic(Polyhedron,Polyhedron)
48 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*48 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
49 The·object·_\x8a_\x8r_\x8e_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8c·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.49 The·object·_\x8a_\x8r_\x8e_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8c·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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714 B
./usr/share/doc/Macaulay2/LexIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=97 #:len=9
8 TGV4SWRlYWxz8 TGV4SWRlYWxz
9 #:len=4709 #:len=470
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp
11 dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV411 dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV4
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712 B
./usr/share/doc/Macaulay2/Licenses/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 TGljZW5zZXM=8 TGljZW5zZXM=
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733 B
./usr/share/doc/Macaulay2/LieTypes/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
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752 B
./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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558 B
./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out
    
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29 i5·:·findRegion({{0,0},{4,4}},M,f)29 i5·:·findRegion({{0,0},{4,4}},M,f)
  
30 o5·=·{{1,·2},·{3,·1}}30 o5·=·{{1,·2},·{3,·1}}
  
31 o5·:·List31 o5·:·List
  
32 i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)32 i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)
33 ·--·.0950887s·elapsed33 ·--·.168008s·elapsed
  
34 o6·=·{{1,·2},·{3,·1}}34 o6·=·{{1,·2},·{3,·1}}
  
35 o6·:·List35 o6·:·List
  
36 i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})36 i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
37 ·--·.0105146s·elapsed37 ·--·.0243888s·elapsed
  
38 o7·=·{{1,·2},·{3,·1}}38 o7·=·{{1,·2},·{3,·1}}
  
39 o7·:·List39 o7·:·List
  
40 i8·:·40 i8·:·
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585 B
./usr/share/doc/Macaulay2/LinearTruncations/example-output/_linear__Truncations__Bound.out
    
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30 i5·:·apply(L,·d·->·isLinearComplex·res·prune·truncate(d,M))30 i5·:·apply(L,·d·->·isLinearComplex·res·prune·truncate(d,M))
  
31 o5·=·{true,·true}31 o5·=·{true,·true}
  
32 o5·:·List32 o5·:·List
  
33 i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)33 i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)
34 ·--·48.264s·elapsed34 ·--·55.7514s·elapsed
  
35 o6·=·{{4,·3,·3},·{4,·4,·2}}35 o6·=·{{4,·3,·3},·{4,·4,·2}}
  
36 o6·:·List36 o6·:·List
  
37 i7·:·elapsedTime·linearTruncationsBound·M37 i7·:·elapsedTime·linearTruncationsBound·M
38 ·--·.020348s·elapsed38 ·--·.0202847s·elapsed
  
39 o7·=·{{4,·3,·3},·{4,·4,·2}}39 o7·=·{{4,·3,·3},·{4,·4,·2}}
  
40 o7·:·List40 o7·:·List
  
41 i8·:·41 i8·:·
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./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html
    
Offset 117, 23 lines modifiedOffset 117, 23 lines modified
117 ········</table>117 ········</table>
118 ········<div>118 ········<div>
119 ··········<p>If·some·degrees·<span·class="tt">d</span>·are·known·to·satisfy·<span·class="tt">f(d,M)</span>,·then·they·can·be·specified·using·the·option·<span·class="tt">Inner</span>·in·order·to·expedite·the·computation.··Similarly,·degrees·not·above·those·given·in·<span·class="tt">Outer</span>·will·be·assumed·not·to·satisfy·<span·class="tt">f(d,M)</span>.·If·<span·class="tt">f</span>·takes·options·these·can·also·be·given·to·<span·class="tt">findRegion</span>.</p>119 ··········<p>If·some·degrees·<span·class="tt">d</span>·are·known·to·satisfy·<span·class="tt">f(d,M)</span>,·then·they·can·be·specified·using·the·option·<span·class="tt">Inner</span>·in·order·to·expedite·the·computation.··Similarly,·degrees·not·above·those·given·in·<span·class="tt">Outer</span>·will·be·assumed·not·to·satisfy·<span·class="tt">f(d,M)</span>.·If·<span·class="tt">f</span>·takes·options·these·can·also·be·given·to·<span·class="tt">findRegion</span>.</p>
120 ········</div>120 ········</div>
121 ········<table·class="examples">121 ········<table·class="examples">
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)123 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)
124 ·--·.0950887s·elapsed124 ·--·.168008s·elapsed
  
125 o6·=·{{1,·2},·{3,·1}}125 o6·=·{{1,·2},·{3,·1}}
  
126 o6·:·List</code></pre>126 o6·:·List</code></pre>
127 </td>··········</tr>127 </td>··········</tr>
128 ··········<tr>128 ··········<tr>
129 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})129 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{{1,1}})
130 ·--·.0105146s·elapsed130 ·--·.0243888s·elapsed
  
131 o7·=·{{1,·2},·{3,·1}}131 o7·=·{{1,·2},·{3,·1}}
  
132 o7·:·List</code></pre>132 o7·:·List</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ········</table>134 ········</table>
135 ······</div>135 ······</div>
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Offset 49, 22 lines modifiedOffset 49, 22 lines modified
  
49 o5·:·List49 o5·:·List
50 If·some·degrees·d·are·known·to·satisfy·f(d,M),·then·they·can·be·specified·using50 If·some·degrees·d·are·known·to·satisfy·f(d,M),·then·they·can·be·specified·using
51 the·option·Inner·in·order·to·expedite·the·computation.·Similarly,·degrees·not51 the·option·Inner·in·order·to·expedite·the·computation.·Similarly,·degrees·not
52 above·those·given·in·Outer·will·be·assumed·not·to·satisfy·f(d,M).·If·f·takes52 above·those·given·in·Outer·will·be·assumed·not·to·satisfy·f(d,M).·If·f·takes
53 options·these·can·also·be·given·to·findRegion.53 options·these·can·also·be·given·to·findRegion.
54 i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)54 i6·:·elapsedTime·findRegion({{0,0},{4,4}},M,f)
55 ·--·.0950887s·elapsed55 ·--·.168008s·elapsed
  
56 o6·=·{{1,·2},·{3,·1}}56 o6·=·{{1,·2},·{3,·1}}
  
57 o6·:·List57 o6·:·List
58 i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{58 i7·:·elapsedTime·findRegion({{0,0},{4,4}},M,f,Inner=>{{1,2},{3,1}},Outer=>{
59 {1,1}})59 {1,1}})
60 ·--·.0105146s·elapsed60 ·--·.0243888s·elapsed
  
61 o7·=·{{1,·2},·{3,·1}}61 o7·=·{{1,·2},·{3,·1}}
  
62 o7·:·List62 o7·:·List
63 *\x8**\x8**\x8**\x8**\x8*·C\x8Co\x8on\x8nt\x8tr\x8ri\x8ib\x8bu\x8ut\x8to\x8or\x8rs\x8s·*\x8**\x8**\x8**\x8**\x8*63 *\x8**\x8**\x8**\x8**\x8*·C\x8Co\x8on\x8nt\x8tr\x8ri\x8ib\x8bu\x8ut\x8to\x8or\x8rs\x8s·*\x8**\x8**\x8**\x8**\x8*
64 Mahrud·Sayrafi·contributed·to·the·code·for·this·function.64 Mahrud·Sayrafi·contributed·to·the·code·for·this·function.
65 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*65 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html
    
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112 ········</table>112 ········</table>
113 ········<div>113 ········<div>
114 ··········<p>The·output·is·a·list·of·the·minimal·multidegrees·$d$·such·that·the·sum·of·the·positive·coordinates·of·$b-d$·is·at·most·$i$·for·all·degrees·$b$·appearing·in·the·<span·class="tt">i</span>-th·step·of·the·resolution·of·$M$.</p>114 ··········<p>The·output·is·a·list·of·the·minimal·multidegrees·$d$·such·that·the·sum·of·the·positive·coordinates·of·$b-d$·is·at·most·$i$·for·all·degrees·$b$·appearing·in·the·<span·class="tt">i</span>-th·step·of·the·resolution·of·$M$.</p>
115 ········</div>115 ········</div>
116 ········<table·class="examples">116 ········<table·class="examples">
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)118 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)
119 ·--·48.264s·elapsed119 ·--·55.7514s·elapsed
  
120 o6·=·{{4,·3,·3},·{4,·4,·2}}120 o6·=·{{4,·3,·3},·{4,·4,·2}}
  
121 o6·:·List</code></pre>121 o6·:·List</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·linearTruncationsBound·M124 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·linearTruncationsBound·M
125 ·--·.020348s·elapsed125 ·--·.0202847s·elapsed
  
126 o7·=·{{4,·3,·3},·{4,·4,·2}}126 o7·=·{{4,·3,·3},·{4,·4,·2}}
  
127 o7·:·List</code></pre>127 o7·:·List</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ········</table>129 ········</table>
130 ······</div>130 ······</div>
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49 o5·=·{true,·true}49 o5·=·{true,·true}
  
50 o5·:·List50 o5·:·List
51 The·output·is·a·list·of·the·minimal·multidegrees·$d$·such·that·the·sum·of·the51 The·output·is·a·list·of·the·minimal·multidegrees·$d$·such·that·the·sum·of·the
52 positive·coordinates·of·$b-d$·is·at·most·$i$·for·all·degrees·$b$·appearing·in52 positive·coordinates·of·$b-d$·is·at·most·$i$·for·all·degrees·$b$·appearing·in
53 the·i-th·step·of·the·resolution·of·$M$.53 the·i-th·step·of·the·resolution·of·$M$.
54 i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)54 i6·:·elapsedTime·linearTruncations({{2,2,2},{4,4,4}},·M)
55 ·--·48.264s·elapsed55 ·--·55.7514s·elapsed
  
56 o6·=·{{4,·3,·3},·{4,·4,·2}}56 o6·=·{{4,·3,·3},·{4,·4,·2}}
  
57 o6·:·List57 o6·:·List
58 i7·:·elapsedTime·linearTruncationsBound·M58 i7·:·elapsedTime·linearTruncationsBound·M
59 ·--·.020348s·elapsed59 ·--·.0202847s·elapsed
  
60 o7·=·{{4,·3,·3},·{4,·4,·2}}60 o7·=·{{4,·3,·3},·{4,·4,·2}}
  
61 o7·:·List61 o7·:·List
62 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*62 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
63 In·general·linearTruncationsBound·will·not·find·the·minimal·degrees·where·$M$63 In·general·linearTruncationsBound·will·not·find·the·minimal·degrees·where·$M$
64 has·a·linear·resolution·but·will·be·faster·than·repeatedly·truncating·$M$.64 has·a·linear·resolution·but·will·be·faster·than·repeatedly·truncating·$M$.
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./usr/share/doc/Macaulay2/LocalRings/example-output/_hilbert__Samuel__Function.out
    
Offset 15, 15 lines modifiedOffset 15, 15 lines modified
  
15 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|15 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|
  
16 ······························116 ······························1
17 o4·:·RP-module,·quotient·of·RP17 o4·:·RP-module,·quotient·of·RP
  
18 i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)18 i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)
19 ·--·.173722s·elapsed19 ·--·.186818s·elapsed
  
20 o5·=·{1,·3,·6,·7,·6,·3,·1}20 o5·=·{1,·3,·6,·7,·6,·3,·1}
  
21 o5·:·List21 o5·:·List
  
22 i6·:·oo//sum22 i6·:·oo//sum
  
Offset 44, 21 lines modifiedOffset 44, 21 lines modified
  
44 ··············2···344 ··············2···3
45 o10·=·ideal·(x·,·y·)45 o10·=·ideal·(x·,·y·)
  
46 o10·:·Ideal·of·RP46 o10·:·Ideal·of·RP
  
47 i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds47 i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds
48 ·--·.0105186s·elapsed48 ·--·.0255763s·elapsed
  
49 o11·=·{1,·2,·3,·4,·5,·6}49 o11·=·{1,·2,·3,·4,·5,·6}
  
50 o11·:·List50 o11·:·List
  
51 i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds51 i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds
52 ·--·.317027s·elapsed52 ·--·.400211s·elapsed
  
53 o12·=·{6,·12,·18,·24,·30,·36}53 o12·=·{6,·12,·18,·24,·30,·36}
  
54 o12·:·List54 o12·:·List
  
55 i13·:·55 i13·:·
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Offset 107, 15 lines modifiedOffset 107, 15 lines modified
107 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|107 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|
  
108 ······························1108 ······························1
109 o4·:·RP-module,·quotient·of·RP</code></pre>109 o4·:·RP-module,·quotient·of·RP</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)112 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)
113 ·--·.173722s·elapsed113 ·--·.186818s·elapsed
  
114 o5·=·{1,·3,·6,·7,·6,·3,·1}114 o5·=·{1,·3,·6,·7,·6,·3,·1}
  
115 o5·:·List</code></pre>115 o5·:·List</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i6·:·oo//sum118 <td>··············<pre><code·class="language-macaulay2">i6·:·oo//sum
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147 ··············2···3147 ··············2···3
148 o10·=·ideal·(x·,·y·)148 o10·=·ideal·(x·,·y·)
  
149 o10·:·Ideal·of·RP</code></pre>149 o10·:·Ideal·of·RP</code></pre>
150 </td>··········</tr>150 </td>··········</tr>
151 ··········<tr>151 ··········<tr>
152 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds152 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds
153 ·--·.0105186s·elapsed153 ·--·.0255763s·elapsed
  
154 o11·=·{1,·2,·3,·4,·5,·6}154 o11·=·{1,·2,·3,·4,·5,·6}
  
155 o11·:·List</code></pre>155 o11·:·List</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds158 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds
159 ·--·.317027s·elapsed159 ·--·.400211s·elapsed
  
160 o12·=·{6,·12,·18,·24,·30,·36}160 o12·=·{6,·12,·18,·24,·30,·36}
  
161 o12·:·List</code></pre>161 o12·:·List</code></pre>
162 </td>··········</tr>162 </td>··········</tr>
163 ········</table>163 ········</table>
164 ······</div>164 ······</div>
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42 i4·:·M·=·RP^1/I42 i4·:·M·=·RP^1/I
  
43 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|43 o4·=·cokernel·|·x5+y3+z3·y5+x3+z3·z5+x3+y3·|
  
44 ······························144 ······························1
45 o4·:·RP-module,·quotient·of·RP45 o4·:·RP-module,·quotient·of·RP
46 i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)46 i5·:·elapsedTime·hilbertSamuelFunction(M,·0,·6)
47 ·--·.173722s·elapsed47 ·--·.186818s·elapsed
  
48 o5·=·{1,·3,·6,·7,·6,·3,·1}48 o5·=·{1,·3,·6,·7,·6,·3,·1}
  
49 o5·:·List49 o5·:·List
50 i6·:·oo//sum50 i6·:·oo//sum
  
51 o6·=·2751 o6·=·27
Offset 66, 21 lines modifiedOffset 66, 21 lines modified
66 i10·:·q·=·ideal"x2,y3"66 i10·:·q·=·ideal"x2,y3"
  
67 ··············2···367 ··············2···3
68 o10·=·ideal·(x·,·y·)68 o10·=·ideal·(x·,·y·)
  
69 o10·:·Ideal·of·RP69 o10·:·Ideal·of·RP
70 i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds70 i11·:·elapsedTime·hilbertSamuelFunction(N,·0,·5)·--·n+1·--·0.02·seconds
71 ·--·.0105186s·elapsed71 ·--·.0255763s·elapsed
  
72 o11·=·{1,·2,·3,·4,·5,·6}72 o11·=·{1,·2,·3,·4,·5,·6}
  
73 o11·:·List73 o11·:·List
74 i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds74 i12·:·elapsedTime·hilbertSamuelFunction(q,·N,·0,·5)·--·6(n+1)·--·0.32·seconds
75 ·--·.317027s·elapsed75 ·--·.400211s·elapsed
  
76 o12·=·{6,·12,·18,·24,·30,·36}76 o12·=·{6,·12,·18,·24,·30,·36}
  
77 o12·:·List77 o12·:·List
78 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*78 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
79 Hilbert-Samuel·function·with·respect·to·a·parameter·ideal·other·than·the79 Hilbert-Samuel·function·with·respect·to·a·parameter·ideal·other·than·the
80 maximal·ideal·can·be·slower.80 maximal·ideal·can·be·slower.
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763 B
./usr/share/doc/Macaulay2/MCMApproximations/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=317 #:len=31
8 Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ==8 Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ==
9 #:len=2939 #:len=293
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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⊟
5.78 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=247 #:len=24
8 dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp8 dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp
9 #:len=2979 #:len=297
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300 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
    
Offset 5, 12 lines modifiedOffset 5, 12 lines modified
5 i2·:·f5 i2·:·f
  
6 o2·=·10737418246 o2·=·1073741824
  
7 i3·:·(c·=·Command·"date";)7 i3·:·(c·=·Command·"date";)
  
8 i4·:·c8 i4·:·c
9 Tue·Jun··3·02:41:20·-12·20259 Tue·Jul··7·13:13:38·+14·2026
  
10 o4·=·010 o4·=·0
  
11 i5·:·11 i5·:·
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481 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Database.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·95790764644464592961 --·-*-·M2-comint·-*-·hash:·9579076464446459296
  
2 i1·:·filename·=·temporaryFileName·()·|·".dbm"2 i1·:·filename·=·temporaryFileName·()·|·".dbm"
  
3 o1·=·/tmp/M2-3695705-0/0.dbm3 o1·=·/tmp/M2-2676632-0/0.dbm
  
4 i2·:·x·=·openDatabaseOut·filename4 i2·:·x·=·openDatabaseOut·filename
  
5 o2·=·/tmp/M2-3695705-0/0.dbm5 o2·=·/tmp/M2-2676632-0/0.dbm
  
6 o2·:·Database6 o2·:·Database
  
7 i3·:·x#"first"·=·"hi·there"7 i3·:·x#"first"·=·"hi·there"
  
8 o3·=·hi·there8 o3·=·hi·there
  
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1.25 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Fast__Nonminimal.out
    
Offset 9, 25 lines modifiedOffset 9, 25 lines modified
9 i2·:·S·=·ring·I9 i2·:·S·=·ring·I
  
10 o2·=·S10 o2·=·S
  
11 o2·:·PolynomialRing11 o2·:·PolynomialRing
  
12 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)12 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
13 ·--·1.80458s·elapsed13 ·--·2.4765s·elapsed
  
14 ······1······35······241······841······1781······2464······2294······1432······576······135······1414 ······1······35······241······841······1781······2464······2294······1432······576······135······14
15 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S···<--·015 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S···<--·0
16 ·········································································································16 ·········································································································
17 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······1117 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······11
  
18 o3·:·ChainComplex18 o3·:·ChainComplex
  
19 i4·:·elapsedTime·C1·=·res·ideal(I_*)19 i4·:·elapsedTime·C1·=·res·ideal(I_*)
20 ·--·.974698s·elapsed20 ·--·1.11323s·elapsed
  
21 ······1······35······140······385······819······1080······819······385······140······35······121 ······1······35······140······385······819······1080······819······385······140······35······1
22 o4·=·S··<--·S···<--·S····<--·S····<--·S····<--·S·····<--·S····<--·S····<--·S····<--·S···<--·S··<--·022 o4·=·S··<--·S···<--·S····<--·S····<--·S····<--·S·····<--·S····<--·S····<--·S····<--·S···<--·S··<--·0
23 ····································································································23 ····································································································
24 ·····0······1·······2········3········4········5·········6········7········8········9·······10·····1124 ·····0······1·······2········3········4········5·········6········7········8········9·······10·····11
  
25 o4·:·ChainComplex25 o4·:·ChainComplex
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1.03 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___File_sp_lt_lt_sp__Thing.out
    
Offset 12, 19 lines modifiedOffset 12, 19 lines modified
  
12 o2·=·stdio12 o2·=·stdio
  
13 o2·:·File13 o2·:·File
  
14 i3·:·fn·=·temporaryFileName()14 i3·:·fn·=·temporaryFileName()
  
15 o3·=·/tmp/M2-3696118-0/015 o3·=·/tmp/M2-2676839-0/0
  
16 i4·:·fn·<<·"hi·there"·<<·endl·<<·close16 i4·:·fn·<<·"hi·there"·<<·endl·<<·close
  
17 o4·=·/tmp/M2-3696118-0/017 o4·=·/tmp/M2-2676839-0/0
  
18 o4·:·File18 o4·:·File
  
19 i5·:·get·fn19 i5·:·get·fn
  
20 o5·=·hi·there20 o5·=·hi·there
  
Offset 49, 27 lines modifiedOffset 49, 27 lines modified
49 x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x·+·149 x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x·+·1
50 o8·=·stdio50 o8·=·stdio
  
51 o8·:·File51 o8·:·File
  
52 i9·:·fn·<<·f·<<·close52 i9·:·fn·<<·f·<<·close
  
53 o9·=·/tmp/M2-3696118-0/053 o9·=·/tmp/M2-2676839-0/0
  
54 o9·:·File54 o9·:·File
  
55 i10·:·get·fn55 i10·:·get·fn
  
56 o10·=··10······9······8·······7·······6·······5·······4·······3······256 o10·=··10······9······8·······7·······6·······5·······4·······3······2
57 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x57 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x
58 ······+·158 ······+·1
  
59 i11·:·fn·<<·toExternalString·f·<<·close59 i11·:·fn·<<·toExternalString·f·<<·close
  
60 o11·=·/tmp/M2-3696118-0/060 o11·=·/tmp/M2-2676839-0/0
  
61 o11·:·File61 o11·:·File
  
62 i12·:·get·fn62 i12·:·get·fn
  
63 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+63 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
64 ······164 ······1
⊟
668 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___G__Cstats.out
    
Offset 1, 17 lines modifiedOffset 1, 17 lines modified
1 --·-*-·M2-comint·-*-·hash:·17318994284947214871 --·-*-·M2-comint·-*-·hash:·1731899428494721487
  
2 i1·:·s·=·GCstats()2 i1·:·s·=·GCstats()
  
3 o1·=·HashTable{"bytesAlloc"·=>·15433937402········}3 o1·=·HashTable{"bytesAlloc"·=>·15347371322········}
4 ···············"GC_free_space_divisor"·=>·34 ···············"GC_free_space_divisor"·=>·3
5 ···············"GC_LARGE_ALLOC_WARN_INTERVAL"·=>·15 ···············"GC_LARGE_ALLOC_WARN_INTERVAL"·=>·1
6 ···············"gcCpuTimeSecs"·=>·06 ···············"gcCpuTimeSecs"·=>·0
7 ···············"heapSize"·=>·2190827527 ···············"heapSize"·=>·219082752
8 ···············"numGCs"·=>·8308 ···············"numGCs"·=>·826
9 ···············"numGCThreads"·=>·129 ···············"numGCThreads"·=>·12
  
10 o1·:·HashTable10 o1·:·HashTable
  
11 i2·:·s#"heapSize"11 i2·:·s#"heapSize"
  
12 o2·=·21908275212 o2·=·219082752
⊟
853 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Minimal__Generators.out
    
Offset 40, 26 lines modifiedOffset 40, 26 lines modified
40 o6·:·PolynomialRing40 o6·:·PolynomialRing
  
41 i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));41 i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));
  
42 o7·:·Ideal·of·R42 o7·:·Ideal·of·R
  
43 i8·:·time·gens·gb·I;43 i8·:·time·gens·gb·I;
44 ·--·used·0.0200335s·(cpu);·0.0200315s·(thread);·0s·(gc)44 ·--·used·0.0350028s·(cpu);·0.0348658s·(thread);·0s·(gc)
  
45 ·············1······42845 ·············1······428
46 o8·:·Matrix·R··<--·R46 o8·:·Matrix·R··<--·R
  
47 i9·:·time·J1·=·saturate(I);47 i9·:·time·J1·=·saturate(I);
48 ·--·used·0.50658s·(cpu);·0.160597s·(thread);·0s·(gc)48 ·--·used·0.459704s·(cpu);·0.23252s·(thread);·0s·(gc)
  
49 o9·:·Ideal·of·R49 o9·:·Ideal·of·R
  
50 i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);50 i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);
51 ·--·used·9.307e-05s·(cpu);·9.2579e-05s·(thread);·0s·(gc)51 ·--·used·0.000134432s·(cpu);·0.000134502s·(thread);·0s·(gc)
  
52 o10·:·Ideal·of·R52 o10·:·Ideal·of·R
  
53 i11·:·numgens·J53 i11·:·numgens·J
  
54 o11·=·754 o11·=·7
  
⊟
657 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
    
Offset 3, 13 lines modifiedOffset 3, 13 lines modified
3 i1·:·M·=·random(RR^200,·RR^200);3 i1·:·M·=·random(RR^200,·RR^200);
  
4 ················200·········2004 ················200·········200
5 o1·:·Matrix·RR······<--·RR5 o1·:·Matrix·RR······<--·RR
6 ··············53··········536 ··············53··········53
  
7 i2·:·time·SVD(M);7 i2·:·time·SVD(M);
8 ·--·used·0.0289738s·(cpu);·0.0289739s·(thread);·0s·(gc)8 ·--·used·0.0372969s·(cpu);·0.0372156s·(thread);·0s·(gc)
  
9 i3·:·time·SVD(M,·DivideConquer=>true);9 i3·:·time·SVD(M,·DivideConquer=>true);
10 ·--·used·0.0270797s·(cpu);·0.0270974s·(thread);·0s·(gc)10 ·--·used·0.0357002s·(cpu);·0.0357113s·(thread);·0s·(gc)
  
11 i4·:·11 i4·:·
⊟
686 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
    
Offset 351, 15 lines modifiedOffset 351, 15 lines modified
351 ···············|·b·e·h·k·n·q·|351 ···············|·b·e·h·k·n·q·|
352 ···············|·c·f·i·l·o·r·|352 ···············|·c·f·i·l·o·r·|
  
353 ·····························3353 ·····························3
354 o58·:·R-module,·quotient·of·R354 o58·:·R-module,·quotient·of·R
  
355 i59·:·time·C·=·resolution·M355 i59·:·time·C·=·resolution·M
356 ·--·used·0.00148457s·(cpu);·0.00147841s·(thread);·0s·(gc)356 ·--·used·0.00300443s·(cpu);·0.00299295s·(thread);·0s·(gc)
  
357 ·······3······6······15······18······6357 ·······3······6······15······18······6
358 o59·=·R··<--·R··<--·R···<--·R···<--·R··<--·0358 o59·=·R··<--·R··<--·R···<--·R···<--·R··<--·0
359 ············································359 ············································
360 ······0······1······2·······3·······4······5360 ······0······1······2·······3·······4······5
  
361 o59·:·ChainComplex361 o59·:·ChainComplex
⊟
322 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
    
Offset 1, 9 lines modifiedOffset 1, 9 lines modified
1 --·-*-·M2-comint·-*-·hash:·13303793594201 --·-*-·M2-comint·-*-·hash:·1330379359420
  
2 i1·:·benchmark·"sqrt·2p100000"2 i1·:·benchmark·"sqrt·2p100000"
  
3 o1·=·.00033905422200956893 o1·=·.0003060271523195235
  
4 o1·:·RR·(of·precision·53)4 o1·:·RR·(of·precision·53)
  
5 i2·:·5 i2·:·
⊟
803 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_betti_lp..._cm__Minimize_eq_gt..._rp.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i2·:·S·=·ring·I9 i2·:·S·=·ring·I
  
10 o2·=·S10 o2·=·S
  
11 o2·:·PolynomialRing11 o2·:·PolynomialRing
  
12 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)12 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
13 ·--·1.86611s·elapsed13 ·--·2.48419s·elapsed
  
14 ······1······35······241······841······1781······2464······2294······1432······576······135······1414 ······1······35······241······841······1781······2464······2294······1432······576······135······14
15 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S···<--·015 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S···<--·0
16 ·········································································································16 ·········································································································
17 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······1117 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······11
  
18 o3·:·ChainComplex18 o3·:·ChainComplex
⊟
654 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
    
Offset 18, 29 lines modifiedOffset 18, 29 lines modified
  
18 o4·=·<<task,·running>>18 o4·=·<<task,·running>>
  
19 o4·:·Task19 o4·:·Task
  
20 i5·:·n20 i5·:·n
  
21 o5·=·93953821 o5·=·939547
  
22 i6·:·sleep·122 i6·:·sleep·1
  
23 o6·=·023 o6·=·0
  
24 i7·:·t24 i7·:·t
  
25 o7·=·<<task,·running>>25 o7·=·<<task,·running>>
  
26 o7·:·Task26 o7·:·Task
  
27 i8·:·n27 i8·:·n
  
28 o8·=·191211628 o8·=·1911978
  
29 i9·:·isReady·t29 i9·:·isReady·t
  
30 o9·=·false30 o9·=·false
  
31 i10·:·cancelTask·t31 i10·:·cancelTask·t
  
Offset 53, 22 lines modifiedOffset 53, 22 lines modified
  
53 o12·=·<<task,·canceled>>53 o12·=·<<task,·canceled>>
  
54 o12·:·Task54 o12·:·Task
  
55 i13·:·n55 i13·:·n
  
56 o13·=·191226656 o13·=·1912172
  
57 i14·:·sleep·157 i14·:·sleep·1
  
58 o14·=·058 o14·=·0
  
59 i15·:·n59 i15·:·n
  
60 o15·=·191226660 o15·=·1912172
  
61 i16·:·isReady·t61 i16·:·isReady·t
  
62 o16·=·false62 o16·=·false
  
63 i17·:·63 i17·:·
⊟
562 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out
    
Offset 1, 19 lines modifiedOffset 1, 19 lines modified
1 --·-*-·M2-comint·-*-·hash:·85355102461401752781 --·-*-·M2-comint·-*-·hash:·8535510246140175278
  
2 i1·:·dir·=·temporaryFileName()2 i1·:·dir·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3680876-0/03 o1·=·/tmp/M2-2656981-0/0
  
4 i2·:·makeDirectory·dir4 i2·:·makeDirectory·dir
  
5 o2·=·/tmp/M2-3680876-0/05 o2·=·/tmp/M2-2656981-0/0
  
6 i3·:·changeDirectory·dir6 i3·:·changeDirectory·dir
  
7 o3·=·/tmp/M2-3680876-0/0/7 o3·=·/tmp/M2-2656981-0/0/
  
8 i4·:·currentDirectory()8 i4·:·currentDirectory()
  
9 o4·=·/tmp/M2-3680876-0/0/9 o4·=·/tmp/M2-2656981-0/0/
  
10 i5·:·10 i5·:·
⊟
990 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
    
Offset 1, 27 lines modifiedOffset 1, 27 lines modified
1 --·-*-·M2-comint·-*-·hash:·109865180196083357191 --·-*-·M2-comint·-*-·hash:·10986518019608335719
  
2 i1·:·run·"uname·-a"2 i1·:·run·"uname·-a"
3 Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.1.140-1·(2025-05-22)·x86_64·GNU/Linux3 Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
  
4 o1·=·04 o1·=·0
  
5 i2·:·"!grep·a"·<<·"·ba·\n·bc·\n·ad·\n·ef·\n"·<<·close5 i2·:·"!grep·a"·<<·"·ba·\n·bc·\n·ad·\n·ef·\n"·<<·close
6 ·ba·6 ·ba·
7 ·ad·7 ·ad·
  
8 o2·=·!grep·a8 o2·=·!grep·a
  
9 o2·:·File9 o2·:·File
  
10 i3·:·peek·get·"!uname·-a"10 i3·:·peek·get·"!uname·-a"
  
11 o3·=·"Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian11 o3·=·"Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC
12 ·····"12 ·····"
13 ·····6.1.140-1·(2025-05-22)·x86_64·GNU/Linux13 ·····Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
  
14 i4·:·f·=·openInOut·"!egrep·'^in'"14 i4·:·f·=·openInOut·"!egrep·'^in'"
  
15 o4·=·!egrep·'^in'15 o4·=·!egrep·'^in'
  
16 o4·:·File16 o4·:·File
  
⊟
1.32 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out
    
Offset 126, 15 lines modifiedOffset 126, 15 lines modified
  
126 ················ZZ126 ················ZZ
127 o23·:·Ideal·of·----[x..z,·w]127 o23·:·Ideal·of·----[x..z,·w]
128 ···············1277128 ···············1277
  
129 i24·:·gb·I129 i24·:·gb·I
  
130 ···--·registering·gb·5·at·0x7f651ca08700130 ···--·registering·gb·5·at·0x7f9c78d31700
  
131 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4131 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4
132 ···--·number·of·monomials················=·8132 ···--·number·of·monomials················=·8
133 ···--·#reduction·steps·=·2133 ···--·#reduction·steps·=·2
134 ···--·#spairs·done·=·6134 ···--·#spairs·done·=·6
135 ···--·ncalls·=·0135 ···--·ncalls·=·0
136 ···--·nloop·=·0136 ···--·nloop·=·0
Offset 177, 15 lines modifiedOffset 177, 15 lines modified
  
177 i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});177 i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});
  
178 ··············1······4178 ··············1······4
179 o32·:·Matrix·R··<--·R179 o32·:·Matrix·R··<--·R
  
180 i33·:·time·betti·gb·f180 i33·:·time·betti·gb·f
181 ·--·used·0.131994s·(cpu);·0.131734s·(thread);·0s·(gc)181 ·--·used·0.271829s·(cpu);·0.271966s·(thread);·0s·(gc)
  
182 ·············0··1182 ·············0··1
183 o33·=·total:·1·53183 o33·=·total:·1·53
184 ··········0:·1··.184 ··········0:·1··.
185 ··········1:·.··.185 ··········1:·.··.
186 ··········2:·.··2186 ··········2:·.··2
187 ··········3:·.··1187 ··········3:·.··1
Offset 208, 15 lines modifiedOffset 208, 15 lines modified
  
208 ············3····5·····8·····9····12·····14····17208 ············3····5·····8·····9····12·····14····17
209 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T209 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T
  
210 o35·:·ZZ[T]210 o35·:·ZZ[T]
  
211 i36·:·time·betti·gb·f211 i36·:·time·betti·gb·f
212 ·--·used·0.000224087s·(cpu);·0.00225795s·(thread);·0s·(gc)212 ·--·used·0.00399783s·(cpu);·0.00357706s·(thread);·0s·(gc)
  
213 ·············0··1213 ·············0··1
214 o36·=·total:·1·53214 o36·=·total:·1·53
215 ··········0:·1··.215 ··········0:·1··.
216 ··········1:·.··.216 ··········1:·.··.
217 ··········2:·.··2217 ··········2:·.··2
218 ··········3:·.··1218 ··········3:·.··1
⊟
867 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spresolutions.out
    
Offset 36, 16 lines modifiedOffset 36, 16 lines modified
36 ··········<<·res·M·<<·endl·<<·endl;36 ··········<<·res·M·<<·endl·<<·endl;
37 ··········break;37 ··········break;
38 ··········)·else·(38 ··········)·else·(
39 ··········<<·"--·computation·interrupted"·<<·endl;39 ··········<<·"--·computation·interrupted"·<<·endl;
40 ··········status·M.cache.resolution;40 ··········status·M.cache.resolution;
41 ··········<<·"--·continuing·the·computation"·<<·endl;41 ··········<<·"--·continuing·the·computation"·<<·endl;
42 ··········))42 ··········))
 43 ·--·used·1.3894s·(cpu);·0.998677s·(thread);·0s·(gc)
43 ·--·used·1.14265s·(cpu);·1.00262s·(thread);·0s·(gc)44 ·--·used·0.147466s·(cpu);·0.150413s·(thread);·0s·(gc)
44 ·--·used·0.541325s·(cpu);·0.464256s·(thread);·0s·(gc) 
45 --·computation·started:·45 --·computation·started:·
46 --·computation·interrupted46 --·computation·interrupted
47 --·continuing·the·computation47 --·continuing·the·computation
48 --·computation·complete48 --·computation·complete
49 ·4······11······89······122······4049 ·4······11······89······122······40
50 R··<--·R···<--·R···<--·R····<--·R···<--·050 R··<--·R···<--·R···<--·R····<--·R···<--·0
51 ·········································51 ·········································
⊟
3.04 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__Directory_lp__String_cm__String_rp.out
    
Offset 1, 76 lines modifiedOffset 1, 76 lines modified
1 --·-*-·M2-comint·-*-·hash:·114227932945643102731 --·-*-·M2-comint·-*-·hash:·11422793294564310273
  
2 i1·:·src·=·temporaryFileName()·|·"/"2 i1·:·src·=·temporaryFileName()·|·"/"
  
3 o1·=·/tmp/M2-3689060-0/0/3 o1·=·/tmp/M2-2666599-0/0/
  
4 i2·:·dst·=·temporaryFileName()·|·"/"4 i2·:·dst·=·temporaryFileName()·|·"/"
  
5 o2·=·/tmp/M2-3689060-0/1/5 o2·=·/tmp/M2-2666599-0/1/
  
6 i3·:·makeDirectory·(src|"a/")6 i3·:·makeDirectory·(src|"a/")
  
7 o3·=·/tmp/M2-3689060-0/0/a/7 o3·=·/tmp/M2-2666599-0/0/a/
  
8 i4·:·makeDirectory·(src|"b/")8 i4·:·makeDirectory·(src|"b/")
  
9 o4·=·/tmp/M2-3689060-0/0/b/9 o4·=·/tmp/M2-2666599-0/0/b/
  
10 i5·:·makeDirectory·(src|"b/c/")10 i5·:·makeDirectory·(src|"b/c/")
  
11 o5·=·/tmp/M2-3689060-0/0/b/c/11 o5·=·/tmp/M2-2666599-0/0/b/c/
  
12 i6·:·src|"a/f"·<<·"hi·there"·<<·close12 i6·:·src|"a/f"·<<·"hi·there"·<<·close
  
13 o6·=·/tmp/M2-3689060-0/0/a/f13 o6·=·/tmp/M2-2666599-0/0/a/f
  
14 o6·:·File14 o6·:·File
  
15 i7·:·src|"a/g"·<<·"hi·there"·<<·close15 i7·:·src|"a/g"·<<·"hi·there"·<<·close
  
16 o7·=·/tmp/M2-3689060-0/0/a/g16 o7·=·/tmp/M2-2666599-0/0/a/g
  
17 o7·:·File17 o7·:·File
  
18 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close18 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close
  
19 o8·=·/tmp/M2-3689060-0/0/b/c/g19 o8·=·/tmp/M2-2666599-0/0/b/c/g
  
20 o8·:·File20 o8·:·File
  
21 i9·:·stack·findFiles·src21 i9·:·stack·findFiles·src
  
22 o9·=·/tmp/M2-3689060-0/0/22 o9·=·/tmp/M2-2666599-0/0/
23 ·····/tmp/M2-3689060-0/0/b/23 ·····/tmp/M2-2666599-0/0/b/
24 ·····/tmp/M2-3689060-0/0/b/c/24 ·····/tmp/M2-2666599-0/0/b/c/
25 ·····/tmp/M2-3689060-0/0/b/c/g25 ·····/tmp/M2-2666599-0/0/b/c/g
26 ·····/tmp/M2-3689060-0/0/a/26 ·····/tmp/M2-2666599-0/0/a/
27 ·····/tmp/M2-3689060-0/0/a/f 
28 ·····/tmp/M2-3689060-0/0/a/g27 ·····/tmp/M2-2666599-0/0/a/g
 28 ·····/tmp/M2-2666599-0/0/a/f
  
29 i10·:·copyDirectory(src,dst,Verbose=>true)29 i10·:·copyDirectory(src,dst,Verbose=>true)
30 ·--·copying:·/tmp/M2-3689060-0/0/b/c/g·->·/tmp/M2-3689060-0/1/b/c/g30 ·--·copying:·/tmp/M2-2666599-0/0/b/c/g·->·/tmp/M2-2666599-0/1/b/c/g
31 ·--·copying:·/tmp/M2-3689060-0/0/a/f·->·/tmp/M2-3689060-0/1/a/f 
32 ·--·copying:·/tmp/M2-3689060-0/0/a/g·->·/tmp/M2-3689060-0/1/a/g31 ·--·copying:·/tmp/M2-2666599-0/0/a/g·->·/tmp/M2-2666599-0/1/a/g
 32 ·--·copying:·/tmp/M2-2666599-0/0/a/f·->·/tmp/M2-2666599-0/1/a/f
  
33 i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)33 i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)
34 ·--·skipping:·/tmp/M2-3689060-0/0/b/c/g·not·newer·than·/tmp/M2-3689060-0/1/b/c/g34 ·--·skipping:·/tmp/M2-2666599-0/0/b/c/g·not·newer·than·/tmp/M2-2666599-0/1/b/c/g
35 ·--·skipping:·/tmp/M2-3689060-0/0/a/f·not·newer·than·/tmp/M2-3689060-0/1/a/f 
36 ·--·skipping:·/tmp/M2-3689060-0/0/a/g·not·newer·than·/tmp/M2-3689060-0/1/a/g35 ·--·skipping:·/tmp/M2-2666599-0/0/a/g·not·newer·than·/tmp/M2-2666599-0/1/a/g
 36 ·--·skipping:·/tmp/M2-2666599-0/0/a/f·not·newer·than·/tmp/M2-2666599-0/1/a/f
  
37 i12·:·stack·findFiles·dst37 i12·:·stack·findFiles·dst
  
38 o12·=·/tmp/M2-3689060-0/1/38 o12·=·/tmp/M2-2666599-0/1/
39 ······/tmp/M2-3689060-0/1/b/39 ······/tmp/M2-2666599-0/1/b/
40 ······/tmp/M2-3689060-0/1/b/c/40 ······/tmp/M2-2666599-0/1/b/c/
41 ······/tmp/M2-3689060-0/1/b/c/g41 ······/tmp/M2-2666599-0/1/b/c/g
42 ······/tmp/M2-3689060-0/1/a/42 ······/tmp/M2-2666599-0/1/a/
43 ······/tmp/M2-3689060-0/1/a/f 
44 ······/tmp/M2-3689060-0/1/a/g43 ······/tmp/M2-2666599-0/1/a/g
 44 ······/tmp/M2-2666599-0/1/a/f
  
45 i13·:·get·(dst|"b/c/g")45 i13·:·get·(dst|"b/c/g")
  
46 o13·=·ho·there46 o13·=·ho·there
  
47 i14·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d47 i14·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d
  
⊟
1.25 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out
    
Offset 1, 41 lines modifiedOffset 1, 41 lines modified
1 --·-*-·M2-comint·-*-·hash:·115394754201557751101 --·-*-·M2-comint·-*-·hash:·11539475420155775110
  
2 i1·:·src·=·temporaryFileName()2 i1·:·src·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3683335-0/03 o1·=·/tmp/M2-2663535-0/0
  
4 i2·:·dst·=·temporaryFileName()4 i2·:·dst·=·temporaryFileName()
  
5 o2·=·/tmp/M2-3683335-0/15 o2·=·/tmp/M2-2663535-0/1
  
6 i3·:·src·<<·"hi·there"·<<·close6 i3·:·src·<<·"hi·there"·<<·close
  
7 o3·=·/tmp/M2-3683335-0/07 o3·=·/tmp/M2-2663535-0/0
  
8 o3·:·File8 o3·:·File
  
9 i4·:·copyFile(src,dst,Verbose=>true)9 i4·:·copyFile(src,dst,Verbose=>true)
10 ·--·copying:·/tmp/M2-3683335-0/0·->·/tmp/M2-3683335-0/110 ·--·copying:·/tmp/M2-2663535-0/0·->·/tmp/M2-2663535-0/1
  
11 i5·:·get·dst11 i5·:·get·dst
  
12 o5·=·hi·there12 o5·=·hi·there
  
13 i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)13 i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
14 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/114 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1
  
15 i7·:·src·<<·"ho·there"·<<·close15 i7·:·src·<<·"ho·there"·<<·close
  
16 o7·=·/tmp/M2-3683335-0/016 o7·=·/tmp/M2-2663535-0/0
  
17 o7·:·File17 o7·:·File
  
18 i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)18 i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
19 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/119 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1
  
20 i9·:·get·dst20 i9·:·get·dst
  
21 o9·=·hi·there21 o9·=·hi·there
  
22 i10·:·removeFile·src22 i10·:·removeFile·src
  
⊟
559 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out
    
Offset 1, 23 lines modifiedOffset 1, 23 lines modified
1 --·-*-·M2-comint·-*-·hash:·155081537832322324531 --·-*-·M2-comint·-*-·hash:·15508153783232232453
  
2 i1·:·t1·=·cpuTime()2 i1·:·t1·=·cpuTime()
  
3 o1·=·171.4596511183 o1·=·258.3199765700001
  
4 o1·:·RR·(of·precision·53)4 o1·:·RR·(of·precision·53)
  
5 i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;5 i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;
  
6 i3·:·t2·=·cpuTime()6 i3·:·t2·=·cpuTime()
  
7 o3·=·172.2719530127 o3·=·259.871625954
  
8 o3·:·RR·(of·precision·53)8 o3·:·RR·(of·precision·53)
  
9 i4·:·t2-t19 i4·:·t2-t1
  
10 o4·=·.812301894000000910 o4·=·1.551649383999973
  
11 o4·:·RR·(of·precision·53)11 o4·:·RR·(of·precision·53)
  
12 i5·:·12 i5·:·
⊟
611 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Time.out
    
Offset 1, 24 lines modifiedOffset 1, 24 lines modified
1 --·-*-·M2-comint·-*-·hash:·36608394761079672591 --·-*-·M2-comint·-*-·hash:·3660839476107967259
  
2 i1·:·currentTime()2 i1·:·currentTime()
  
3 o1·=·17489617273 o1·=·1783379688
  
4 i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)4 i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)
  
5 o2·=·55.422390825324335 o2·=·56.51305259139096
  
6 o2·:·RR·(of·precision·53)6 o2·:·RR·(of·precision·53)
  
7 i3·:·12·*·(oo·-·floor·oo)7 i3·:·12·*·(oo·-·floor·oo)
  
8 o3·=·5.0686899038919368 o3·=·6.156631096691513
  
9 o3·:·RR·(of·precision·53)9 o3·:·RR·(of·precision·53)
  
10 i4·:·run·"date"10 i4·:·run·"date"
11 Tue·Jun··3·02:42:07·-12·202511 Tue·Jul··7·13:14:48·+14·2026
  
12 o4·=·012 o4·=·0
  
13 i5·:·13 i5·:·
⊟
390 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elapsed__Timing.out
    
Offset 1, 14 lines modifiedOffset 1, 14 lines modified
1 --·-*-·M2-comint·-*-·hash:·17311068032072987151 --·-*-·M2-comint·-*-·hash:·1731106803207298715
  
2 i1·:·elapsedTiming·sleep·12 i1·:·elapsedTiming·sleep·1
  
3 o1·=·03 o1·=·0
4 ·····--·1.00014·seconds4 ·····--·1.00019·seconds
  
5 o1·:·Time5 o1·:·Time
  
6 i2·:·peek·oo6 i2·:·peek·oo
  
7 o2·=·Time{1.00014,·0}7 o2·=·Time{1.00019,·0}
  
8 i3·:·8 i3·:·
⊟
4.01 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_elimination_spof_spvariables.out
    
Offset 6, 15 lines modifiedOffset 6, 15 lines modified
  
6 ···············3···················3·····2···············36 ···············3···················3·····2···············3
7 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)7 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)
  
8 o2·:·Ideal·of·R8 o2·:·Ideal·of·R
  
9 i3·:·time·leadTerm·gens·gb·I9 i3·:·time·leadTerm·gens·gb·I
10 ·--·used·0.0945913s·(cpu);·0.0945912s·(thread);·0s·(gc)10 ·--·used·0.141724s·(cpu);·0.141723s·(thread);·0s·(gc)
  
11 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy411 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy4
12 ·····------------------------------------------------------------------------12 ·····------------------------------------------------------------------------
13 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z213 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z2
14 ·····------------------------------------------------------------------------14 ·····------------------------------------------------------------------------
15 ·····267076255345488270sy3z4·5256861933965245618410txyz615 ·····267076255345488270sy3z4·5256861933965245618410txyz6
16 ·····------------------------------------------------------------------------16 ·····------------------------------------------------------------------------
Offset 85, 15 lines modifiedOffset 85, 15 lines modified
  
85 ···············3···················3·····2···············385 ···············3···················3·····2···············3
86 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)86 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)
  
87 o7·:·Ideal·of·R87 o7·:·Ideal·of·R
  
88 i8·:·time·G·=·eliminate(I,{s,t})88 i8·:·time·G·=·eliminate(I,{s,t})
89 ·--·used·0.0997789s·(cpu);·0.0997815s·(thread);·0s·(gc)89 ·--·used·0.147031s·(cpu);·0.14662s·(thread);·0s·(gc)
  
90 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···90 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···
91 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+91 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
92 ·····------------------------------------------------------------------------92 ·····------------------------------------------------------------------------
93 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·93 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·
94 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z94 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
95 ·····------------------------------------------------------------------------95 ·····------------------------------------------------------------------------
Offset 154, 15 lines modifiedOffset 154, 15 lines modified
154 i10·:·R1·=·QQ[x,y,z,s,t,·Degrees=>{3,3,4,1,1}];154 i10·:·R1·=·QQ[x,y,z,s,t,·Degrees=>{3,3,4,1,1}];
  
155 i11·:·I1·=·substitute(I,R1);155 i11·:·I1·=·substitute(I,R1);
  
156 o11·:·Ideal·of·R1156 o11·:·Ideal·of·R1
  
157 i12·:·time·G·=·eliminate(I1,{s,t})157 i12·:·time·G·=·eliminate(I1,{s,t})
158 ·--·used·0.0325054s·(cpu);·0.0325089s·(thread);·0s·(gc)158 ·--·used·0.0468117s·(cpu);·0.0468164s·(thread);·0s·(gc)
  
159 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7··159 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7··
160 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-160 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-
161 ······-----------------------------------------------------------------------161 ······-----------------------------------------------------------------------
162 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2··162 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2··
163 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-163 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-
164 ······-----------------------------------------------------------------------164 ······-----------------------------------------------------------------------
Offset 228, 15 lines modifiedOffset 228, 15 lines modified
  
228 ···················3·············3·····2·········3228 ···················3·············3·····2·········3
229 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})229 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})
  
230 o16·:·RingMap·A·<--·B230 o16·:·RingMap·A·<--·B
  
231 i17·:·time·G·=·kernel·F231 i17·:·time·G·=·kernel·F
232 ·--·used·0.249418s·(cpu);·0.117309s·(thread);·0s·(gc)232 ·--·used·0.365832s·(cpu);·0.159494s·(thread);·0s·(gc)
  
233 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···233 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···
234 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+234 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
235 ······-----------------------------------------------------------------------235 ······-----------------------------------------------------------------------
236 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·236 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·
237 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z237 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
238 ······-----------------------------------------------------------------------238 ······-----------------------------------------------------------------------
Offset 297, 23 lines modifiedOffset 297, 23 lines modified
297 i19·:·use·ring·I297 i19·:·use·ring·I
  
298 o19·=·R298 o19·=·R
  
299 o19·:·PolynomialRing299 o19·:·PolynomialRing
  
300 i20·:·time·f1·=·resultant(I_0,I_2,s)300 i20·:·time·f1·=·resultant(I_0,I_2,s)
301 ·--·used·0.00127094s·(cpu);·0.00126649s·(thread);·0s·(gc)301 ·--·used·0.00182866s·(cpu);·0.00182897s·(thread);·0s·(gc)
  
302 ·········9····9······7····3302 ·········9····9······7····3
303 o20·=·x*t··-·t··-·z*t··-·z303 o20·=·x*t··-·t··-·z*t··-·z
  
304 o20·:·R304 o20·:·R
  
305 i21·:·time·f2·=·resultant(I_1,f1,t)305 i21·:·time·f2·=·resultant(I_1,f1,t)
306 ·--·used·0.0285206s·(cpu);·0.0285309s·(thread);·0s·(gc)306 ·--·used·0.0445255s·(cpu);·0.0445387s·(thread);·0s·(gc)
  
307 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2··307 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2··
308 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+308 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+
309 ······-----------------------------------------------------------------------309 ······-----------------------------------------------------------------------
310 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8···310 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8···
311 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+311 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+
312 ······-----------------------------------------------------------------------312 ······-----------------------------------------------------------------------
⊟
876 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_end__Package.out
    
Offset 59, 15 lines modifiedOffset 59, 15 lines modified
59 ····································Version·=>·0.059 ····································Version·=>·0.0
60 ·············package·prefix·=>·/usr/60 ·············package·prefix·=>·/usr/
61 ·············PackageIsLoaded·=>·true61 ·············PackageIsLoaded·=>·true
62 ·············pkgname·=>·Foo62 ·············pkgname·=>·Foo
63 ·············private·dictionary·=>·Foo#"private·dictionary"63 ·············private·dictionary·=>·Foo#"private·dictionary"
64 ·············processed·documentation·=>·MutableHashTable{}64 ·············processed·documentation·=>·MutableHashTable{}
65 ·············raw·documentation·=>·MutableHashTable{}65 ·············raw·documentation·=>·MutableHashTable{}
66 ·············source·directory·=>·/tmp/M2-3680509-0/88-rundir/66 ·············source·directory·=>·/tmp/M2-2656573-0/88-rundir/
67 ·············source·file·=>·stdio67 ·············source·file·=>·stdio
68 ·············test·inputs·=>·MutableList{}68 ·············test·inputs·=>·MutableList{}
  
69 i7·:·dictionaryPath69 i7·:·dictionaryPath
  
70 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,70 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,
71 ·····------------------------------------------------------------------------71 ·····------------------------------------------------------------------------
⊟
469 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out
    
Offset 1, 20 lines modifiedOffset 1, 20 lines modified
1 --·-*-·M2-comint·-*-·hash:·74750389365702248991 --·-*-·M2-comint·-*-·hash:·7475038936570224899
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3681031-0/03 o1·=·/tmp/M2-2657156-0/0
  
4 i2·:·fileExists·fn4 i2·:·fileExists·fn
  
5 o2·=·false5 o2·=·false
  
6 i3·:·fn·<<·"hi·there"·<<·close6 i3·:·fn·<<·"hi·there"·<<·close
  
7 o3·=·/tmp/M2-3681031-0/07 o3·=·/tmp/M2-2657156-0/0
  
8 o3·:·File8 o3·:·File
  
9 i4·:·fileExists·fn9 i4·:·fileExists·fn
  
10 o4·=·true10 o4·=·true
  
⊟
580 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out
    
Offset 1, 28 lines modifiedOffset 1, 28 lines modified
1 --·-*-·M2-comint·-*-·hash:·12166954471952379941 --·-*-·M2-comint·-*-·hash:·1216695447195237994
  
2 i1·:·f·=·temporaryFileName()·<<·"hi·there"2 i1·:·f·=·temporaryFileName()·<<·"hi·there"
  
3 o1·=·/tmp/M2-3696937-0/03 o1·=·/tmp/M2-2679084-0/0
  
4 o1·:·File4 o1·:·File
  
5 i2·:·fileLength·f5 i2·:·fileLength·f
  
6 o2·=·86 o2·=·8
  
7 i3·:·close·f7 i3·:·close·f
  
8 o3·=·/tmp/M2-3696937-0/08 o3·=·/tmp/M2-2679084-0/0
  
9 o3·:·File9 o3·:·File
  
10 i4·:·filename·=·toString·f10 i4·:·filename·=·toString·f
  
11 o4·=·/tmp/M2-3696937-0/011 o4·=·/tmp/M2-2679084-0/0
  
12 i5·:·fileLength·filename12 i5·:·fileLength·filename
  
13 o5·=·813 o5·=·8
  
14 i6·:·get·filename14 i6·:·get·filename
  
⊟
557 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out
    
Offset 1, 25 lines modifiedOffset 1, 25 lines modified
1 --·-*-·M2-comint·-*-·hash:·112021406211239936331 --·-*-·M2-comint·-*-·hash:·11202140621123993633
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3694014-0/03 o1·=·/tmp/M2-2675848-0/0
  
4 i2·:·f·=·fn·<<·"hi·there"4 i2·:·f·=·fn·<<·"hi·there"
  
5 o2·=·/tmp/M2-3694014-0/05 o2·=·/tmp/M2-2675848-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·fileMode·f7 i3·:·fileMode·f
  
8 o3·=·4208 o3·=·420
  
9 i4·:·close·f9 i4·:·close·f
  
10 o4·=·/tmp/M2-3694014-0/010 o4·=·/tmp/M2-2675848-0/0
  
11 o4·:·File11 o4·:·File
  
12 i5·:·removeFile·fn12 i5·:·removeFile·fn
  
13 i6·:·13 i6·:·
⊟
454 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·47825702021974645321 --·-*-·M2-comint·-*-·hash:·4782570202197464532
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3683492-0/03 o1·=·/tmp/M2-2663710-0/0
  
4 i2·:·fn·<<·"hi·there"·<<·close4 i2·:·fn·<<·"hi·there"·<<·close
  
5 o2·=·/tmp/M2-3683492-0/05 o2·=·/tmp/M2-2663710-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·fileMode·fn7 i3·:·fileMode·fn
  
8 o3·=·4208 o3·=·420
  
⊟
639 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·174738782678455754421 --·-*-·M2-comint·-*-·hash:·17473878267845575442
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3682085-0/03 o1·=·/tmp/M2-2662405-0/0
  
4 i2·:·f·=·fn·<<·"hi·there"4 i2·:·f·=·fn·<<·"hi·there"
  
5 o2·=·/tmp/M2-3682085-0/05 o2·=·/tmp/M2-2662405-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·m·=·7·+·7*8·+·7*647 i3·:·m·=·7·+·7*8·+·7*64
  
8 o3·=·5118 o3·=·511
  
Offset 18, 15 lines modifiedOffset 18, 15 lines modified
  
18 i5·:·fileMode·f18 i5·:·fileMode·f
  
19 o5·=·51119 o5·=·511
  
20 i6·:·close·f20 i6·:·close·f
  
21 o6·=·/tmp/M2-3682085-0/021 o6·=·/tmp/M2-2662405-0/0
  
22 o6·:·File22 o6·:·File
  
23 i7·:·fileMode·fn23 i7·:·fileMode·fn
  
24 o7·=·51124 o7·=·511
  
⊟
477 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·167727843907993347231 --·-*-·M2-comint·-*-·hash:·16772784390799334723
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3696540-0/03 o1·=·/tmp/M2-2677458-0/0
  
4 i2·:·fn·<<·"hi·there"·<<·close4 i2·:·fn·<<·"hi·there"·<<·close
  
5 o2·=·/tmp/M2-3696540-0/05 o2·=·/tmp/M2-2677458-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·m·=·fileMode·fn7 i3·:·m·=·fileMode·fn
  
8 o3·=·4208 o3·=·420
  
⊟
262 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out
    
Offset 1, 7 lines modifiedOffset 1, 7 lines modified
1 --·-*-·M2-comint·-*-·hash:·13313107110751 --·-*-·M2-comint·-*-·hash:·1331310711075
  
2 i1·:·currentTime()·-·fileTime·"."2 i1·:·currentTime()·-·fileTime·"."
  
3 o1·=·283 o1·=·41
  
4 i2·:·4 i2·:·
⊟
640 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.out
    
Offset 29, 15 lines modifiedOffset 29, 15 lines modified
29 ·····{4}·|·0·····x2-3··y3-1··|29 ·····{4}·|·0·····x2-3··y3-1··|
  
30 ·············3······330 ·············3······3
31 o6·:·Matrix·R··<--·R31 o6·:·Matrix·R··<--·R
  
32 i7·:·syz·f32 i7·:·syz·f
  
33 ···--·registering·gb·0·at·0x7fee57b4300033 ···--·registering·gb·0·at·0x7f493ae25000
  
34 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb·elements·=·334 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb·elements·=·3
35 ···--·number·of·monomials················=·935 ···--·number·of·monomials················=·9
36 ···--·#reduction·steps·=·636 ···--·#reduction·steps·=·6
37 ···--·#spairs·done·=·637 ···--·#spairs·done·=·6
38 ···--·ncalls·=·038 ···--·ncalls·=·0
39 ···--·nloop·=·039 ···--·nloop·=·0
⊟
295 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_get.out
    
Offset 10, 11 lines modifiedOffset 10, 11 lines modified
  
10 o2·=·hi·there10 o2·=·hi·there
  
11 i3·:·removeFile·"test-file"11 i3·:·removeFile·"test-file"
  
12 i4·:·get·"!date"12 i4·:·get·"!date"
  
13 o4·=·Tue·Jun··3·02:41:41·-12·202513 o4·=·Tue·Jul··7·13:14:10·+14·2026
  
  
14 i5·:·14 i5·:·
⊟
255 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_group__I__D.out
    
Offset 1, 7 lines modifiedOffset 1, 7 lines modified
1 --·-*-·M2-comint·-*-·hash:·13309756143231 --·-*-·M2-comint·-*-·hash:·1330975614323
  
2 i1·:·groupID()2 i1·:·groupID()
  
3 o1·=·35526593 o1·=·2549134
  
4 i2·:·4 i2·:·
⊟
426 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Directory.out
    
Offset 2, 19 lines modifiedOffset 2, 19 lines modified
  
2 i1·:·isDirectory·"."2 i1·:·isDirectory·"."
  
3 o1·=·true3 o1·=·true
  
4 i2·:·fn·=·temporaryFileName()4 i2·:·fn·=·temporaryFileName()
  
5 o2·=·/tmp/M2-3680745-0/05 o2·=·/tmp/M2-2656842-0/0
  
6 i3·:·fn·<<·"hi·there"·<<·close6 i3·:·fn·<<·"hi·there"·<<·close
  
7 o3·=·/tmp/M2-3680745-0/07 o3·=·/tmp/M2-2656842-0/0
  
8 o3·:·File8 o3·:·File
  
9 i4·:·isDirectory·fn9 i4·:·isDirectory·fn
  
10 o4·=·false10 o4·=·false
  
⊟
848 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
    
Offset 75, 15 lines modifiedOffset 75, 15 lines modified
75 o17·=·false75 o17·=·false
  
76 i18·:·isPrime(m*m*m1*m1*m2^6)76 i18·:·isPrime(m*m*m1*m1*m2^6)
  
77 o18·=·false77 o18·=·false
  
78 i19·:·elapsedTime·facs·=·factor(m*m1)78 i19·:·elapsedTime·facs·=·factor(m*m1)
79 ·--·6.7843s·elapsed79 ·--·4.57625s·elapsed
  
80 o19·=·1000000000000000000000000000057*100000000000000000001000000008380 o19·=·1000000000000000000000000000057*1000000000000000000010000000083
  
81 o19·:·Expression·of·class·Product81 o19·:·Expression·of·class·Product
  
82 i20·:·facs·=·facs//toList/toList82 i20·:·facs·=·facs//toList/toList
  
Offset 97, 17 lines modifiedOffset 97, 17 lines modified
  
97 i22·:·m3·=·nextPrime·(m^3)97 i22·:·m3·=·nextPrime·(m^3)
  
98 o22·=·1000000000000000000000000000171000000000000000000000000009747000000000098 o22·=·10000000000000000000000000001710000000000000000000000000097470000000000
99 ······0000000000000018561399 ······00000000000000185613
  
100 i23·:·elapsedTime·isPrime·m3100 i23·:·elapsedTime·isPrime·m3
101 ·--·.0452662s·elapsed101 ·--·.0643698s·elapsed
  
102 o23·=·true102 o23·=·true
  
103 i24·:·elapsedTime·isPseudoprime·m3103 i24·:·elapsedTime·isPseudoprime·m3
104 ·--·.000131339s·elapsed104 ·--·.000153268s·elapsed
  
105 o24·=·true105 o24·=·true
  
106 i25·:·106 i25·:·
⊟
446 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·47822052457580536291 --·-*-·M2-comint·-*-·hash:·4782205245758053629
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3697277-0/03 o1·=·/tmp/M2-2679154-0/0
  
4 i2·:·fn·<<·"hi·there"·<<·close4 i2·:·fn·<<·"hi·there"·<<·close
  
5 o2·=·/tmp/M2-3697277-0/05 o2·=·/tmp/M2-2679154-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·isRegularFile·fn7 i3·:·isRegularFile·fn
  
8 o3·=·true8 o3·=·true
  
⊟
546 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_make__Directory_lp__String_rp.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·51133721592045717461 --·-*-·M2-comint·-*-·hash:·5113372159204571746
  
2 i1·:·dir·=·temporaryFileName()2 i1·:·dir·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3681260-0/03 o1·=·/tmp/M2-2659965-0/0
  
4 i2·:·makeDirectory·(dir|"/a/b/c")4 i2·:·makeDirectory·(dir|"/a/b/c")
  
5 o2·=·/tmp/M2-3681260-0/0/a/b/c5 o2·=·/tmp/M2-2659965-0/0/a/b/c
  
6 i3·:·removeDirectory·(dir|"/a/b/c")6 i3·:·removeDirectory·(dir|"/a/b/c")
  
7 i4·:·removeDirectory·(dir|"/a/b")7 i4·:·removeDirectory·(dir|"/a/b")
  
8 i5·:·removeDirectory·(dir|"/a")8 i5·:·removeDirectory·(dir|"/a")
  
⊟
815 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_memoize.out
    
Offset 3, 31 lines modifiedOffset 3, 31 lines modified
3 i1·:·fib·=·n·->·if·n·<=·1·then·1·else·fib(n-1)·+·fib(n-2)3 i1·:·fib·=·n·->·if·n·<=·1·then·1·else·fib(n-1)·+·fib(n-2)
  
4 o1·=·fib4 o1·=·fib
  
5 o1·:·FunctionClosure5 o1·:·FunctionClosure
  
6 i2·:·time·fib·286 i2·:·time·fib·28
7 ·--·used·0.625112s·(cpu);·0.489916s·(thread);·0s·(gc)7 ·--·used·0.850079s·(cpu);·0.673465s·(thread);·0s·(gc)
  
8 o2·=·5142298 o2·=·514229
  
9 i3·:·fib·=·memoize·fib9 i3·:·fib·=·memoize·fib
  
10 o3·=·fib10 o3·=·fib
  
11 o3·:·FunctionClosure11 o3·:·FunctionClosure
  
12 i4·:·time·fib·2812 i4·:·time·fib·28
13 ·--·used·4.67e-05s·(cpu);·4.637e-05s·(thread);·0s·(gc)13 ·--·used·5.8079e-05s·(cpu);·5.7037e-05s·(thread);·0s·(gc)
  
14 o4·=·51422914 o4·=·514229
  
15 i5·:·time·fib·2815 i5·:·time·fib·28
16 ·--·used·1.402e-06s·(cpu);·1.352e-06s·(thread);·0s·(gc)16 ·--·used·3.397e-06s·(cpu);·2.885e-06s·(thread);·0s·(gc)
  
17 o5·=·51422917 o5·=·514229
  
18 i6·:·fib·=·memoize(·n·->·fib(n-1)·+·fib(n-2)·)18 i6·:·fib·=·memoize(·n·->·fib(n-1)·+·fib(n-2)·)
  
19 o6·=·fib19 o6·=·fib
  
⊟
4.06 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_methods.out
    
Offset 18, 20 lines modifiedOffset 18, 20 lines modified
18 ·····{13·=>·(poincare,·BettiTally)································}18 ·····{13·=>·(poincare,·BettiTally)································}
19 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}19 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}
20 ·····{15·=>·(degree,·BettiTally)··································}20 ·····{15·=>·(degree,·BettiTally)··································}
21 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}21 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}
22 ·····{17·=>·(^,·Ring,·BettiTally)·································}22 ·····{17·=>·(^,·Ring,·BettiTally)·································}
23 ·····{18·=>·(regularity,·BettiTally)······························}23 ·····{18·=>·(regularity,·BettiTally)······························}
24 ·····{19·=>·(mathML,·BettiTally)··································}24 ·····{19·=>·(mathML,·BettiTally)··································}
25 ·····{20·=>·(codim,·BettiTally)···································}25 ·····{20·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············}
26 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}26 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}
27 ·····{22·=>·(dual,·BettiTally)····································} 
28 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············} 
29 ·····{24·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}27 ·····{22·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}
30 ·····{25·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}28 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}
 29 ·····{24·=>·(dual,·BettiTally)····································}
 30 ·····{25·=>·(codim,·BettiTally)···································}
  
31 o1·:·NumberedVerticalList31 o1·:·NumberedVerticalList
  
32 i2·:·methods·resolution32 i2·:·methods·resolution
  
33 o2·=·{0·=>·(resolution,·Ideal)·}33 o2·=·{0·=>·(resolution,·Ideal)·}
34 ·····{1·=>·(resolution,·Module)}34 ·····{1·=>·(resolution,·Module)}
Offset 60, 20 lines modifiedOffset 60, 20 lines modified
60 ·····{1·=>·(++,·Module,·GradedModule)}60 ·····{1·=>·(++,·Module,·GradedModule)}
61 ·····{2·=>·(++,·Module,·Module)······}61 ·····{2·=>·(++,·Module,·Module)······}
  
62 o4·:·NumberedVerticalList62 o4·:·NumberedVerticalList
  
63 i5·:·methods(·Matrix,·Matrix·)63 i5·:·methods(·Matrix,·Matrix·)
  
64 o5·=·{0·=>·(diff',·Matrix,·Matrix)·······························}64 o5·=·{0·=>·(-,·Matrix,·Matrix)···································}
65 ·····{1·=>·(contract',·Matrix,·Matrix)···························} 
66 ·····{2·=>·(+,·Matrix,·Matrix)···································}65 ·····{1·=>·(+,·Matrix,·Matrix)···································}
 66 ·····{2·=>·(diff',·Matrix,·Matrix)·······························}
67 ·····{3·=>·(diff,·Matrix,·Matrix)································}67 ·····{3·=>·(diff,·Matrix,·Matrix)································}
68 ·····{4·=>·(contract,·Matrix,·Matrix)····························}68 ·····{4·=>·(contract',·Matrix,·Matrix)···························}
69 ·····{5·=>·(-,·Matrix,·Matrix)···································}69 ·····{5·=>·(contract,·Matrix,·Matrix)····························}
70 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}70 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}
71 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}71 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}
72 ·····{8·=>·(==,·Matrix,·Matrix)··································}72 ·····{8·=>·(==,·Matrix,·Matrix)··································}
73 ·····{9·=>·(*,·Matrix,·Matrix)···································}73 ·····{9·=>·(*,·Matrix,·Matrix)···································}
74 ·····{10·=>·(|,·Matrix,·Matrix)··································}74 ·····{10·=>·(|,·Matrix,·Matrix)··································}
75 ·····{11·=>·(||,·Matrix,·Matrix)·································}75 ·····{11·=>·(||,·Matrix,·Matrix)·································}
76 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}76 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}
Offset 84, 16 lines modifiedOffset 84, 16 lines modified
84 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}84 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}
85 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}85 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}
86 ·····{19·=>·(//,·Matrix,·Matrix)·································}86 ·····{19·=>·(//,·Matrix,·Matrix)·································}
87 ·····{20·=>·(\\,·Matrix,·Matrix)·································}87 ·····{20·=>·(\\,·Matrix,·Matrix)·································}
88 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}88 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}
89 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}89 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}
90 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}90 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}
91 ·····{24·=>·(%,·Matrix,·Matrix)··································} 
92 ·····{25·=>·(remainder,·Matrix,·Matrix)··························}91 ·····{24·=>·(remainder,·Matrix,·Matrix)··························}
 92 ·····{25·=>·(%,·Matrix,·Matrix)··································}
93 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}93 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}
94 ·····{27·=>·(solve,·Matrix,·Matrix)······························}94 ·····{27·=>·(solve,·Matrix,·Matrix)······························}
95 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}95 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}
96 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}96 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}
97 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}97 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}
98 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}98 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}
99 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}99 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}
⊟
1.61 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i2·:·S·=·ring·I9 i2·:·S·=·ring·I
  
10 o2·=·S10 o2·=·S
  
11 o2·:·PolynomialRing11 o2·:·PolynomialRing
  
12 i3·:·elapsedTime·C·=·minimalBetti·I12 i3·:·elapsedTime·C·=·minimalBetti·I
13 ·--·1.70586s·elapsed13 ·--·2.62963s·elapsed
  
14 ············0··1···2···3···4····5···6···7···8··9·1014 ············0··1···2···3···4····5···6···7···8··9·10
15 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··115 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··1
16 ·········0:·1··.···.···.···.····.···.···.···.··.··.16 ·········0:·1··.···.···.···.····.···.···.···.··.··.
17 ·········1:·.·35·140·189··84····.···.···.···.··.··.17 ·········1:·.·35·140·189··84····.···.···.···.··.··.
18 ·········2:·.··.···.·196·735·1080·735·196···.··.··.18 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
19 ·········3:·.··.···.···.···.····.··84·189·140·35··.19 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 26, 44 lines modifiedOffset 26, 44 lines modified
26 o3·:·BettiTally26 o3·:·BettiTally
  
27 i4·:·I·=·ideal·I_*;27 i4·:·I·=·ideal·I_*;
  
28 o4·:·Ideal·of·S28 o4·:·Ideal·of·S
  
29 i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)29 i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)
30 ·--·.77006s·elapsed30 ·--·.805845s·elapsed
  
31 ············0··1···2···3···4····5···6···731 ············0··1···2···3···4····5···6···7
32 o5·=·total:·1·35·140·385·819·1080·735·19632 o5·=·total:·1·35·140·385·819·1080·735·196
33 ·········0:·1··.···.···.···.····.···.···.33 ·········0:·1··.···.···.···.····.···.···.
34 ·········1:·.·35·140·189··84····.···.···.34 ·········1:·.·35·140·189··84····.···.···.
35 ·········2:·.··.···.·196·735·1080·735·19635 ·········2:·.··.···.·196·735·1080·735·196
  
36 o5·:·BettiTally36 o5·:·BettiTally
  
37 i6·:·I·=·ideal·I_*;37 i6·:·I·=·ideal·I_*;
  
38 o6·:·Ideal·of·S38 o6·:·Ideal·of·S
  
39 i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)39 i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)
40 ·--·.0289581s·elapsed40 ·--·.0301415s·elapsed
  
41 ············0··1···2···3··441 ············0··1···2···3··4
42 o7·=·total:·1·35·140·189·8442 o7·=·total:·1·35·140·189·84
43 ·········0:·1··.···.···.··.43 ·········0:·1··.···.···.··.
44 ·········1:·.·35·140·189·8444 ·········1:·.·35·140·189·84
  
45 o7·:·BettiTally45 o7·:·BettiTally
  
46 i8·:·I·=·ideal·I_*;46 i8·:·I·=·ideal·I_*;
  
47 o8·:·Ideal·of·S47 o8·:·Ideal·of·S
  
48 i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)48 i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)
49 ·--·1.20943s·elapsed49 ·--·1.52404s·elapsed
  
50 ············0··1···2···3···4····550 ············0··1···2···3···4····5
51 o9·=·total:·1·35·140·385·819·108051 o9·=·total:·1·35·140·385·819·1080
52 ·········0:·1··.···.···.···.····.52 ·········0:·1··.···.···.···.····.
53 ·········1:·.·35·140·189··84····.53 ·········1:·.·35·140·189··84····.
54 ·········2:·.··.···.·196·735·108054 ·········2:·.··.···.·196·735·1080
  
⊟
497 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out
    
Offset 1, 22 lines modifiedOffset 1, 22 lines modified
1 --·-*-·M2-comint·-*-·hash:·155552268095099331351 --·-*-·M2-comint·-*-·hash:·15555226809509933135
  
2 i1·:·p·=·temporaryFileName()·|·"/"2 i1·:·p·=·temporaryFileName()·|·"/"
  
3 o1·=·/tmp/M2-3681409-0/0/3 o1·=·/tmp/M2-2660565-0/0/
  
4 i2·:·mkdir·p4 i2·:·mkdir·p
  
5 i3·:·isDirectory·p5 i3·:·isDirectory·p
  
6 o3·=·true6 o3·=·true
  
7 i4·:·(fn·=·p·|·"foo")·<<·"hi·there"·<<·close7 i4·:·(fn·=·p·|·"foo")·<<·"hi·there"·<<·close
  
8 o4·=·/tmp/M2-3681409-0/0/foo8 o4·=·/tmp/M2-2660565-0/0/foo
  
9 o4·:·File9 o4·:·File
  
10 i5·:·get·fn10 i5·:·get·fn
  
11 o5·=·hi·there11 o5·=·hi·there
  
⊟
945 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_move__File_lp__String_cm__String_rp.out
    
Offset 1, 31 lines modifiedOffset 1, 31 lines modified
1 --·-*-·M2-comint·-*-·hash:·48579440424710932181 --·-*-·M2-comint·-*-·hash:·4857944042471093218
  
2 i1·:·src·=·temporaryFileName()2 i1·:·src·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3681093-0/03 o1·=·/tmp/M2-2657350-0/0
  
4 i2·:·dst·=·temporaryFileName()4 i2·:·dst·=·temporaryFileName()
  
5 o2·=·/tmp/M2-3681093-0/15 o2·=·/tmp/M2-2657350-0/1
  
6 i3·:·src·<<·"hi·there"·<<·close6 i3·:·src·<<·"hi·there"·<<·close
  
7 o3·=·/tmp/M2-3681093-0/07 o3·=·/tmp/M2-2657350-0/0
  
8 o3·:·File8 o3·:·File
  
9 i4·:·moveFile(src,dst,Verbose=>true)9 i4·:·moveFile(src,dst,Verbose=>true)
10 --moving:·/tmp/M2-3681093-0/0·->·/tmp/M2-3681093-0/110 --moving:·/tmp/M2-2657350-0/0·->·/tmp/M2-2657350-0/1
  
11 i5·:·get·dst11 i5·:·get·dst
  
12 o5·=·hi·there12 o5·=·hi·there
  
13 i6·:·bak·=·moveFile(dst,Verbose=>true)13 i6·:·bak·=·moveFile(dst,Verbose=>true)
14 --backup·file·created:·/tmp/M2-3681093-0/1.bak14 --backup·file·created:·/tmp/M2-2657350-0/1.bak
  
15 o6·=·/tmp/M2-3681093-0/1.bak15 o6·=·/tmp/M2-2657350-0/1.bak
  
16 i7·:·removeFile·bak16 i7·:·removeFile·bak
  
17 i8·:·17 i8·:·
⊟
296 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_nanosleep.out
    
Offset 1, 8 lines modifiedOffset 1, 8 lines modified
1 --·-*-·M2-comint·-*-·hash:·13311146124411 --·-*-·M2-comint·-*-·hash:·1331114612441
  
2 i1·:·elapsedTime·nanosleep·5000000002 i1·:·elapsedTime·nanosleep·500000000
3 ·--·.500137s·elapsed3 ·--·.50013s·elapsed
  
4 o1·=·04 o1·=·0
  
5 i2·:·5 i2·:·
⊟
636 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out
    
Offset 5, 18 lines modifiedOffset 5, 18 lines modified
5 o1·=·{1,·2,·6,·24,·120,·720,·5040,·40320,·362880,·3628800}5 o1·=·{1,·2,·6,·24,·120,·720,·5040,·40320,·362880,·3628800}
  
6 o1·:·List6 o1·:·List
  
7 i2·:·L·=·random·toList·(1..10000);7 i2·:·L·=·random·toList·(1..10000);
  
8 i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);8 i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);
9 ·--·.680556s·elapsed9 ·--·1.29299s·elapsed
  
10 i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);10 i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);
11 ·--·.180746s·elapsed11 ·--·.23955s·elapsed
  
12 i5·:·allowableThreads12 i5·:·allowableThreads
  
13 o5·=·513 o5·=·5
  
14 i6·:·allowableThreads·=·maxAllowableThreads14 i6·:·allowableThreads·=·maxAllowableThreads
  
⊟
4.29 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out
    
Offset 67, 15 lines modifiedOffset 67, 15 lines modified
67 i3·:·S·=·ring·I67 i3·:·S·=·ring·I
  
68 o3·=·S68 o3·=·S
  
69 o3·:·PolynomialRing69 o3·:·PolynomialRing
  
70 i4·:·elapsedTime·minimalBetti·I70 i4·:·elapsedTime·minimalBetti·I
71 ·--·2.09728s·elapsed71 ·--·3.10166s·elapsed
  
72 ············0··1···2···3···4····5···6···7···8··9·1072 ············0··1···2···3···4····5···6···7···8··9·10
73 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··173 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··1
74 ·········0:·1··.···.···.···.····.···.···.···.··.··.74 ·········0:·1··.···.···.···.····.···.···.···.··.··.
75 ·········1:·.·35·140·189··84····.···.···.···.··.··.75 ·········1:·.·35·140·189··84····.···.···.···.··.··.
76 ·········2:·.··.···.·196·735·1080·735·196···.··.··.76 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
77 ·········3:·.··.···.···.···.····.··84·189·140·35··.77 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 84, 15 lines modifiedOffset 84, 15 lines modified
84 o4·:·BettiTally84 o4·:·BettiTally
  
85 i5·:·I·=·ideal·I_*;85 i5·:·I·=·ideal·I_*;
  
86 o5·:·Ideal·of·S86 o5·:·Ideal·of·S
  
87 i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)87 i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)
88 ·--·2.08346s·elapsed88 ·--·2.44486s·elapsed
  
89 ············0··1···2···3···4····5···6···7···8··9·1089 ············0··1···2···3···4····5···6···7···8··9·10
90 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··190 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··1
91 ·········0:·1··.···.···.···.····.···.···.···.··.··.91 ·········0:·1··.···.···.···.····.···.···.···.··.··.
92 ·········1:·.·35·140·189··84····.···.···.···.··.··.92 ·········1:·.·35·140·189··84····.···.···.···.··.··.
93 ·········2:·.··.···.·196·735·1080·735·196···.··.··.93 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
94 ·········3:·.··.···.···.···.····.··84·189·140·35··.94 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 105, 15 lines modifiedOffset 105, 15 lines modified
105 o7·:·Ideal·of·S105 o7·:·Ideal·of·S
  
106 i8·:·numTBBThreads·=·1106 i8·:·numTBBThreads·=·1
  
107 o8·=·1107 o8·=·1
  
108 i9·:·elapsedTime·minimalBetti(I)108 i9·:·elapsedTime·minimalBetti(I)
109 ·--·1.93825s·elapsed109 ·--·2.1033s·elapsed
  
110 ············0··1···2···3···4····5···6···7···8··9·10110 ············0··1···2···3···4····5···6···7···8··9·10
111 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1111 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1
112 ·········0:·1··.···.···.···.····.···.···.···.··.··.112 ·········0:·1··.···.···.···.····.···.···.···.··.··.
113 ·········1:·.·35·140·189··84····.···.···.···.··.··.113 ·········1:·.·35·140·189··84····.···.···.···.··.··.
114 ·········2:·.··.···.·196·735·1080·735·196···.··.··.114 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
115 ·········3:·.··.···.···.···.····.··84·189·140·35··.115 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 132, 15 lines modifiedOffset 132, 15 lines modified
132 o11·=·0132 o11·=·0
  
133 i12·:·I·=·ideal·I_*;133 i12·:·I·=·ideal·I_*;
  
134 o12·:·Ideal·of·S134 o12·:·Ideal·of·S
  
135 i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)135 i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
136 ·--·1.94911s·elapsed136 ·--·4.44298s·elapsed
  
137 ·······1······35······241······841······1781······2464······2294······1432······576······135······14137 ·······1······35······241······841······1781······2464······2294······1432······576······135······14
138 o13·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S138 o13·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S
139 ··································································································139 ··································································································
140 ······0······1·······2········3········4·········5·········6·········7·········8········9········10140 ······0······1·······2········3········4·········5·········6·········7·········8········9········10
  
141 o13·:·Complex141 o13·:·Complex
Offset 150, 15 lines modifiedOffset 150, 15 lines modified
150 o14·=·1150 o14·=·1
  
151 i15·:·I·=·ideal·I_*;151 i15·:·I·=·ideal·I_*;
  
152 o15·:·Ideal·of·S152 o15·:·Ideal·of·S
  
153 i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)153 i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
154 ·--·1.96008s·elapsed154 ·--·2.75209s·elapsed
  
155 ·······1······35······241······841······1781······2464······2294······1432······576······135······14155 ·······1······35······241······841······1781······2464······2294······1432······576······135······14
156 o16·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S156 o16·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S····<--·S····<--·S
157 ··································································································157 ··································································································
158 ······0······1·······2········3········4·········5·········6·········7·········8········9········10158 ······0······1·······2········3········4·········5·········6·········7·········8········9········10
  
159 o16·:·Complex159 o16·:·Complex
Offset 174, 43 lines modifiedOffset 174, 43 lines modified
174 o18·:·PolynomialRing174 o18·:·PolynomialRing
  
175 i19·:·I·=·ideal·random(S^1,·S^{4:-5});175 i19·:·I·=·ideal·random(S^1,·S^{4:-5});
  
176 o19·:·Ideal·of·S176 o19·:·Ideal·of·S
  
177 i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");177 i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
178 ·--·3.15699s·elapsed178 ·--·4.27149s·elapsed
  
179 ··············1······108179 ··············1······108
180 o20·:·Matrix·S··<--·S180 o20·:·Matrix·S··<--·S
  
181 i21·:·numTBBThreads·=·1181 i21·:·numTBBThreads·=·1
  
182 o21·=·1182 o21·=·1
  
183 i22·:·I·=·ideal·I_*;183 i22·:·I·=·ideal·I_*;
  
184 o22·:·Ideal·of·S184 o22·:·Ideal·of·S
  
185 i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");185 i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
186 ·--·5.57081s·elapsed186 ·--·5.95563s·elapsed
  
187 ··············1······108187 ··············1······108
188 o23·:·Matrix·S··<--·S188 o23·:·Matrix·S··<--·S
  
189 i24·:·numTBBThreads·=·10189 i24·:·numTBBThreads·=·10
  
190 o24·=·10190 o24·=·10
  
191 i25·:·I·=·ideal·I_*;191 i25·:·I·=·ideal·I_*;
  
192 o25·:·Ideal·of·S192 o25·:·Ideal·of·S
  
193 i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");193 i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
194 ·--·2.66126s·elapsed194 ·--·3.85508s·elapsed
  
195 ··············1······108195 ··············1······108
196 o26·:·Matrix·S··<--·S196 o26·:·Matrix·S··<--·S
  
197 i27·:·needsPackage·"AssociativeAlgebras"197 i27·:·needsPackage·"AssociativeAlgebras"
  
198 o27·=·AssociativeAlgebras198 o27·=·AssociativeAlgebras
Offset 233, 15 lines modifiedOffset 233, 15 lines modified
233 o30·=·ideal·(5a··+·2b*c·+·3c*b,·3a*c·+·5b··+·2c*a,·2a*b·+·3b*a·+·5c·)233 o30·=·ideal·(5a··+·2b*c·+·3c*b,·3a*c·+·5b··+·2c*a,·2a*b·+·3b*a·+·5c·)
  
234 ················ZZ234 ················ZZ
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⊟
2.88 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out
    
Offset 146, 65 lines modifiedOffset 146, 65 lines modified
146 o26·:·ZZ[T]146 o26·:·ZZ[T]
  
147 i27·:·gbTrace·=·3147 i27·:·gbTrace·=·3
  
148 o27·=·3148 o27·=·3
  
149 i28·:·time·poincare·I149 i28·:·time·poincare·I
150 ·--·used·0.00272158s·(cpu);·8.723e-06s·(thread);·0s·(gc)150 ·--·used·0.00115029s·(cpu);·9.398e-06s·(thread);·0s·(gc)
  
151 ············3·····6····9151 ············3·····6····9
152 o28·=·1·-·3T··+·3T··-·T152 o28·=·1·-·3T··+·3T··-·T
  
153 o28·:·ZZ[T]153 o28·:·ZZ[T]
  
154 i29·:·time·gens·gb·I;154 i29·:·time·gens·gb·I;
  
155 ···--·registering·gb·19·at·0x7fd8ebd7ee00155 ···--·registering·gb·19·at·0x7f23f5491e00
  
156 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of·(nonminimal)·gb·elements·=·11156 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of·(nonminimal)·gb·elements·=·11
157 ···--·number·of·monomials················=·4186157 ···--·number·of·monomials················=·4186
158 ···--·#reduction·steps·=·38158 ···--·#reduction·steps·=·38
159 ···--·#spairs·done·=·11159 ···--·#spairs·done·=·11
160 ···--·ncalls·=·10160 ···--·ncalls·=·10
161 ···--·nloop·=·29161 ···--·nloop·=·29
162 ···--·nsaved·=·0162 ···--·nsaved·=·0
163 ···--··--·used·0.00926653s·(cpu);·0.00991467s·(thread);·0s·(gc)163 ···--··--·used·0.010838s·(cpu);·0.0117141s·(thread);·0s·(gc)
  
164 ··············1······11164 ··············1······11
165 o29·:·Matrix·R··<--·R165 o29·:·Matrix·R··<--·R
  
166 i30·:·R·=·QQ[a..d];166 i30·:·R·=·QQ[a..d];
  
167 i31·:·I·=·ideal·random(R^1,·R^{3:-3});167 i31·:·I·=·ideal·random(R^1,·R^{3:-3});
  
168 ···--·registering·gb·20·at·0x7fd8ebd7ec40168 ···--·registering·gb·20·at·0x7f23f5491c40
  
169 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0169 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0
170 ···--·number·of·monomials················=·0170 ···--·number·of·monomials················=·0
171 ···--·#reduction·steps·=·0171 ···--·#reduction·steps·=·0
172 ···--·#spairs·done·=·0172 ···--·#spairs·done·=·0
173 ···--·ncalls·=·0173 ···--·ncalls·=·0
174 ···--·nloop·=·0174 ···--·nloop·=·0
175 ···--·nsaved·=·0175 ···--·nsaved·=·0
176 ···--·176 ···--·
177 o31·:·Ideal·of·R177 o31·:·Ideal·of·R
  
178 i32·:·time·p·=·poincare·I178 i32·:·time·p·=·poincare·I
  
179 ···--·registering·gb·21·at·0x7fd8ebd7e8c0179 ···--·registering·gb·21·at·0x7f23f54918c0
  
180 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of·(nonminimal)·gb·elements·=·11180 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of·(nonminimal)·gb·elements·=·11
181 ···--·number·of·monomials················=·267181 ···--·number·of·monomials················=·267
182 ···--·#reduction·steps·=·236182 ···--·#reduction·steps·=·236
183 ···--·#spairs·done·=·30183 ···--·#spairs·done·=·30
184 ···--·ncalls·=·10184 ···--·ncalls·=·10
185 ···--·nloop·=·20185 ···--·nloop·=·20
186 ···--·nsaved·=·0186 ···--·nsaved·=·0
187 ···--··--·used·0.00320777s·(cpu);·0.00422687s·(thread);·0s·(gc)187 ···--··--·used·0.0040092s·(cpu);·0.00515655s·(thread);·0s·(gc)
  
188 ············3·····6····9188 ············3·····6····9
189 o32·=·1·-·3T··+·3T··-·T189 o32·=·1·-·3T··+·3T··-·T
  
190 o32·:·ZZ[T]190 o32·:·ZZ[T]
  
191 i33·:·S·=·QQ[a..d,·MonomialOrder·=>·Eliminate·2]191 i33·:·S·=·QQ[a..d,·MonomialOrder·=>·Eliminate·2]
Offset 254, 27 lines modifiedOffset 254, 27 lines modified
  
254 i36·:·gbTrace·=·3254 i36·:·gbTrace·=·3
  
255 o36·=·3255 o36·=·3
  
256 i37·:·time·gens·gb·J;256 i37·:·time·gens·gb·J;
  
257 ···--·registering·gb·22·at·0x7fd8ebd7e700257 ···--·registering·gb·22·at·0x7f23f5491700
  
258 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m258 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
259 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m259 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
260 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m260 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
261 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements·=·39261 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements·=·39
262 ···--·number·of·monomials················=·1051262 ···--·number·of·monomials················=·1051
263 ···--·#reduction·steps·=·284263 ···--·#reduction·steps·=·284
264 ···--·#spairs·done·=·53264 ···--·#spairs·done·=·53
265 ···--·ncalls·=·46265 ···--·ncalls·=·46
266 ···--·nloop·=·54266 ···--·nloop·=·54
267 ···--·nsaved·=·0267 ···--·nsaved·=·0
268 ···--··--·used·0.0359964s·(cpu);·0.0365404s·(thread);·0s·(gc)268 ···--··--·used·0.044002s·(cpu);·0.0450079s·(thread);·0s·(gc)
  
269 ··············1······39269 ··············1······39
270 o37·:·Matrix·S··<--·S270 o37·:·Matrix·S··<--·S
  
271 i38·:·selectInSubring(1,·gens·gb·J)271 i38·:·selectInSubring(1,·gens·gb·J)
  
272 o38·=·|·243873059890414515367459726418219472801881021280016638460434780718278272 o38·=·|·243873059890414515367459726418219472801881021280016638460434780718278
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261 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_process__I__D.out
    
Offset 1, 7 lines modifiedOffset 1, 7 lines modified
1 --·-*-·M2-comint·-*-·hash:·13305136305631 --·-*-·M2-comint·-*-·hash:·1330513630563
  
2 i1·:·processID()2 i1·:·processID()
  
3 o1·=·36805093 o1·=·2656573
  
4 i2·:·4 i2·:·
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1.59 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
    
Offset 10, 15 lines modifiedOffset 10, 15 lines modified
  
10 ······123····110····1310 ······123····110····13
11 o2·=·x····+·x····+·x···+·111 o2·=·x····+·x····+·x···+·1
  
12 o2·:·R12 o2·:·R
  
13 i3·:·time·factor·f13 i3·:·time·factor·f
14 ·--·used·0.00258722s·(cpu);·0.00258178s·(thread);·0s·(gc)14 ·--·used·0.00491573s·(cpu);·0.00490021s·(thread);·0s·(gc)
  
15 ··············2········2·······2·······2·······2·······4·····3······2············4·····3·····2············4·····3·····2············10·····8·····6·····4······2·······10·····8······6·····4······2·······10······8····6····4·····2·······10······8·····6·····4······2·······10······8·····6·····4·····2·······10······8·····6······4·····2·······10······8·····6·····4······2·······10······8·····6······4·····2·······10·····8····6····4······2·······10·····8······6·····4······215 ··············2········2·······2·······2·······2·······4·····3······2············4·····3·····2············4·····3·····2············10·····8·····6·····4······2·······10·····8······6·····4······2·······10······8····6····4·····2·······10······8·····6·····4······2·······10······8·····6·····4·····2·······10······8·····6······4·····2·······10······8·····6·····4······2·······10······8·····6······4·····2·······10·····8····6····4······2·······10·····8······6·····4······2
16 o3·=·(x·+·1)(x··-·15)(x··+·8)(x··+·4)(x··+·2)(x··+·1)(x··-·4x··+·11x··-·4x·+·1)(x··-·6x··-·2x··-·6x·+·1)(x··+·9x··-·4x··+·9x·+·1)(x···-·5x··-·8x··-·4x··-·13x··+·1)(x···-·9x··+·15x··-·2x··+·10x··+·1)(x···-·10x··-·x··-·x··+·9x··+·1)(x···-·11x··-·8x··-·4x··+·11x··+·1)(x···-·13x··-·4x··-·8x··-·5x··+·1)(x···+·13x··-·2x··+·15x··+·5x··+·1)(x···+·11x··-·4x··-·8x··-·11x··+·1)(x···+·10x··-·2x··+·15x··-·9x··+·1)(x···+·9x··-·x··-·x··-·10x··+·1)(x···+·5x··+·15x··-·2x··+·13x··+·1)16 o3·=·(x·+·1)(x··-·15)(x··+·8)(x··+·4)(x··+·2)(x··+·1)(x··-·4x··+·11x··-·4x·+·1)(x··-·6x··-·2x··-·6x·+·1)(x··+·9x··-·4x··+·9x·+·1)(x···-·5x··-·8x··-·4x··-·13x··+·1)(x···-·9x··+·15x··-·2x··+·10x··+·1)(x···-·10x··-·x··-·x··+·9x··+·1)(x···-·11x··-·8x··-·4x··+·11x··+·1)(x···-·13x··-·4x··-·8x··-·5x··+·1)(x···+·13x··-·2x··+·15x··+·5x··+·1)(x···+·11x··-·4x··-·8x··-·11x··+·1)(x···+·10x··-·2x··+·15x··-·9x··+·1)(x···+·9x··-·x··-·x··-·10x··+·1)(x···+·5x··+·15x··-·2x··+·13x··+·1)
  
17 o3·:·Expression·of·class·Product17 o3·:·Expression·of·class·Product
  
18 i4·:·g·=·()·->·factor·f18 i4·:·g·=·()·->·factor·f
Offset 38, 11 lines modifiedOffset 38, 11 lines modified
38 o6·=·h38 o6·=·h
  
39 o6·:·FunctionClosure39 o6·:·FunctionClosure
  
40 i7·:·for·i·to·10·do·(g();h();h())40 i7·:·for·i·to·10·do·(g();h();h())
  
41 i8·:·profileSummary41 i8·:·profileSummary
42 g:·11·times,·used·.0250758·seconds42 g:·11·times,·used·.0390346·seconds
43 h:·22·times,·used·.0509056·seconds43 h:·22·times,·used·.0774187·seconds
  
44 i9·:·44 i9·:·
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1.27 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out
    
Offset 13, 15 lines modifiedOffset 13, 15 lines modified
13 i5·:·codim·I,·degree·I13 i5·:·codim·I,·degree·I
  
14 o5·=·(2,·10)14 o5·=·(2,·10)
  
15 o5·:·Sequence15 o5·:·Sequence
  
16 i6·:·time·randomKRationalPoint(I)16 i6·:·time·randomKRationalPoint(I)
17 ·--·used·0.120879s·(cpu);·0.0600286s·(thread);·0s·(gc)17 ·--·used·0.138709s·(cpu);·0.0707743s·(thread);·0s·(gc)
  
18 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)18 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)
19 ·············2······3···1·····3···0·····319 ·············2······3···1·····3···0·····3
  
20 o6·:·Ideal·of·R20 o6·:·Ideal·of·R
  
21 i7·:·R=kk[x_0..x_5];21 i7·:·R=kk[x_0..x_5];
Offset 33, 15 lines modifiedOffset 33, 15 lines modified
33 i9·:·codim·I,·degree·I33 i9·:·codim·I,·degree·I
  
34 o9·=·(3,·10)34 o9·=·(3,·10)
  
35 o9·:·Sequence35 o9·:·Sequence
  
36 i10·:·time·randomKRationalPoint(I)36 i10·:·time·randomKRationalPoint(I)
37 ·--·used·0.285537s·(cpu);·0.166293s·(thread);·0s·(gc)37 ·--·used·0.331339s·(cpu);·0.194846s·(thread);·0s·(gc)
  
38 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)38 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)
39 ··············4······5···3······5···2·····5···1······5···0······539 ··············4······5···3······5···2·····5···1······5···0······5
  
40 o10·:·Ideal·of·R40 o10·:·Ideal·of·R
  
41 i11·:·p=10007,kk=ZZ/p,R=kk[x_0..x_2]41 i11·:·p=10007,kk=ZZ/p,R=kk[x_0..x_2]
Offset 58, 12 lines modifiedOffset 58, 12 lines modified
  
58 i14·:·I=ideal·random(n,R);58 i14·:·I=ideal·random(n,R);
  
59 o14·:·Ideal·of·R59 o14·:·Ideal·of·R
  
60 i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c->degree·c);60 i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c->degree·c);
61 ·····················#select(degs,d->d==1))),f->f>0))61 ·····················#select(degs,d->d==1))),f->f>0))
62 ·--·used·2.47859s·(cpu);·1.39746s·(thread);·0s·(gc)62 ·--·used·4.09656s·(cpu);·1.66143s·(thread);·0s·(gc)
  
63 o15·=·5863 o15·=·58
  
64 i16·:·64 i16·:·
⊟
642 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_read__Directory.out
    
Offset 1, 26 lines modifiedOffset 1, 26 lines modified
1 --·-*-·M2-comint·-*-·hash:·209107367040705141 --·-*-·M2-comint·-*-·hash:·20910736704070514
  
2 i1·:·dir·=·temporaryFileName()2 i1·:·dir·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3695513-0/03 o1·=·/tmp/M2-2676437-0/0
  
4 i2·:·makeDirectory·dir4 i2·:·makeDirectory·dir
  
5 o2·=·/tmp/M2-3695513-0/05 o2·=·/tmp/M2-2676437-0/0
  
6 i3·:·(fn·=·dir·|·"/"·|·"foo")·<<·"hi·there"·<<·close6 i3·:·(fn·=·dir·|·"/"·|·"foo")·<<·"hi·there"·<<·close
  
7 o3·=·/tmp/M2-3695513-0/0/foo7 o3·=·/tmp/M2-2676437-0/0/foo
  
8 o3·:·File8 o3·:·File
  
9 i4·:·readDirectory·dir9 i4·:·readDirectory·dir
  
10 o4·=·{..,·.,·foo}10 o4·=·{foo,·.,·..}
  
11 o4·:·List11 o4·:·List
  
12 i5·:·removeFile·fn12 i5·:·removeFile·fn
  
13 i6·:·removeDirectory·dir13 i6·:·removeDirectory·dir
  
⊟
836 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reading_spfiles.out
    
Offset 1, 16 lines modifiedOffset 1, 16 lines modified
1 --·-*-·M2-comint·-*-·hash:·135135551042009447961 --·-*-·M2-comint·-*-·hash:·13513555104200944796
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3685338-0/03 o1·=·/tmp/M2-2665365-0/0
  
4 i2·:·fn·<<·"z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"·<<·endl·<<·close4 i2·:·fn·<<·"z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"·<<·endl·<<·close
  
5 o2·=·/tmp/M2-3685338-0/05 o2·=·/tmp/M2-2665365-0/0
  
6 o2·:·File6 o2·:·File
  
7 i3·:·get·fn7 i3·:·get·fn
  
8 o3·=·z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^28 o3·=·z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
9 ·····+8*y^39 ·····+8*y^3
Offset 38, 15 lines modifiedOffset 38, 15 lines modified
  
38 o6·:·Expression·of·class·Product38 o6·:·Expression·of·class·Product
  
39 i7·:·fn·<<·"sample·=·2^10039 i7·:·fn·<<·"sample·=·2^100
40 ·····print·sample40 ·····print·sample
41 ·····"·<<·close41 ·····"·<<·close
  
42 o7·=·/tmp/M2-3685338-0/042 o7·=·/tmp/M2-2665365-0/0
  
43 o7·:·File43 o7·:·File
  
44 i8·:·get·fn44 i8·:·get·fn
  
45 o8·=·sample·=·2^10045 o8·=·sample·=·2^100
46 ·····print·sample46 ·····print·sample
⊟
351 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_readlink.out
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
1 --·-*-·M2-comint·-*-·hash:·44086396114787811301 --·-*-·M2-comint·-*-·hash:·4408639611478781130
  
2 i1·:·p·=·temporaryFileName·()2 i1·:·p·=·temporaryFileName·()
  
3 o1·=·/tmp/M2-3696255-0/03 o1·=·/tmp/M2-2677144-0/0
  
4 i2·:·symlinkFile·("foo",·p)4 i2·:·symlinkFile·("foo",·p)
  
5 i3·:·readlink·p5 i3·:·readlink·p
  
6 o3·=·foo6 o3·=·foo
  
⊟
868 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out
    
Offset 1, 39 lines modifiedOffset 1, 39 lines modified
1 --·-*-·M2-comint·-*-·hash:·3240723472132246561 --·-*-·M2-comint·-*-·hash:·324072347213224656
  
2 i1·:·realpath·"."2 i1·:·realpath·"."
  
3 o1·=·/tmp/M2-3680509-0/83-rundir/3 o1·=·/tmp/M2-2656573-0/83-rundir/
  
4 i2·:·p·=·temporaryFileName()4 i2·:·p·=·temporaryFileName()
  
5 o2·=·/tmp/M2-3696286-0/05 o2·=·/tmp/M2-2677183-0/0
  
6 i3·:·q·=·temporaryFileName()6 i3·:·q·=·temporaryFileName()
  
7 o3·=·/tmp/M2-3696286-0/17 o3·=·/tmp/M2-2677183-0/1
  
8 i4·:·symlinkFile(p,q)8 i4·:·symlinkFile(p,q)
  
9 i5·:·p·<<·close9 i5·:·p·<<·close
  
10 o5·=·/tmp/M2-3696286-0/010 o5·=·/tmp/M2-2677183-0/0
  
11 o5·:·File11 o5·:·File
  
12 i6·:·readlink·q12 i6·:·readlink·q
  
13 o6·=·/tmp/M2-3696286-0/013 o6·=·/tmp/M2-2677183-0/0
  
14 i7·:·realpath·q14 i7·:·realpath·q
  
15 o7·=·/tmp/M2-3696286-0/015 o7·=·/tmp/M2-2677183-0/0
  
16 i8·:·removeFile·p16 i8·:·removeFile·p
  
17 i9·:·removeFile·q17 i9·:·removeFile·q
  
18 i10·:·realpath·""18 i10·:·realpath·""
  
19 o10·=·/tmp/M2-3680509-0/83-rundir/19 o10·=·/tmp/M2-2656573-0/83-rundir/
  
20 i11·:·20 i11·:·
⊟
1.15 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
    
Offset 1, 15 lines modifiedOffset 1, 15 lines modified
1 --·-*-·M2-comint·-*-·hash:·17293843743726626931 --·-*-·M2-comint·-*-·hash:·1729384374372662693
  
2 i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·"--·finalizing·sequence·#"|i|"·--"))2 i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·"--·finalizing·sequence·#"|i|"·--"))
  
3 i2·:·collectGarbage()·3 i2·:·collectGarbage()·
4 --finalization:·(1)[5]:·--·finalizing·sequence·#6·-- 
5 --finalization:·(2)[0]:·--·finalizing·sequence·#1·-- 
6 --finalization:·(3)[3]:·--·finalizing·sequence·#4·--4 --finalization:·(1)[3]:·--·finalizing·sequence·#4·--
7 --finalization:·(4)[4]:·--·finalizing·sequence·#5·--5 --finalization:·(2)[4]:·--·finalizing·sequence·#5·--
 6 --finalization:·(3)[6]:·--·finalizing·sequence·#7·--
8 --finalization:·(5)[7]:·--·finalizing·sequence·#8·--7 --finalization:·(4)[7]:·--·finalizing·sequence·#8·--
 8 --finalization:·(5)[1]:·--·finalizing·sequence·#2·--
9 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--9 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--
10 --finalization:·(7)[1]:·--·finalizing·sequence·#2·--10 --finalization:·(7)[0]:·--·finalizing·sequence·#1·--
11 --finalization:·(8)[6]:·--·finalizing·sequence·#7·--11 --finalization:·(8)[5]:·--·finalizing·sequence·#6·--
  
12 i3·:·12 i3·:·
⊟
492 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
    
Offset 1, 19 lines modifiedOffset 1, 19 lines modified
1 --·-*-·M2-comint·-*-·hash:·85329803100970600891 --·-*-·M2-comint·-*-·hash:·8532980310097060089
  
2 i1·:·dir·=·temporaryFileName()2 i1·:·dir·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3681736-0/03 o1·=·/tmp/M2-2661573-0/0
  
4 i2·:·makeDirectory·dir4 i2·:·makeDirectory·dir
  
5 o2·=·/tmp/M2-3681736-0/05 o2·=·/tmp/M2-2661573-0/0
  
6 i3·:·readDirectory·dir6 i3·:·readDirectory·dir
  
7 o3·=·{..,·.}7 o3·=·{.,·..}
  
8 o3·:·List8 o3·:·List
  
9 i4·:·removeDirectory·dir9 i4·:·removeDirectory·dir
  
10 i5·:·10 i5·:·
⊟
376 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__Path.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·17314202321481493871 --·-*-·M2-comint·-*-·hash:·1731420232148149387
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3680619-0/03 o1·=·/tmp/M2-2656700-0/0
  
4 i2·:·rootPath·|·fn4 i2·:·rootPath·|·fn
  
5 o2·=·/tmp/M2-3680619-0/05 o2·=·/tmp/M2-2656700-0/0
  
6 i3·:·6 i3·:·
⊟
395 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_root__U__R__I.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·17314202315255729681 --·-*-·M2-comint·-*-·hash:·1731420231525572968
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3695159-0/03 o1·=·/tmp/M2-2676330-0/0
  
4 i2·:·rootURI·|·fn4 i2·:·rootURI·|·fn
  
5 o2·=·file:///tmp/M2-3695159-0/05 o2·=·file:///tmp/M2-2676330-0/0
  
6 i3·:·6 i3·:·
⊟
679 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_saving_sppolynomials_spand_spmatrices_spin_spfiles.out
    
Offset 25, 19 lines modifiedOffset 25, 19 lines modified
25 o4·=·image·|·x2·x2-y2·xyz7·|25 o4·=·image·|·x2·x2-y2·xyz7·|
  
26 ·····························126 ·····························1
27 o4·:·R-module,·submodule·of·R27 o4·:·R-module,·submodule·of·R
  
28 i5·:·f·=·temporaryFileName()28 i5·:·f·=·temporaryFileName()
  
29 o5·=·/tmp/M2-3692110-0/029 o5·=·/tmp/M2-2674564-0/0
  
30 i6·:·f·<<·toString·(p,m,M)·<<·close30 i6·:·f·<<·toString·(p,m,M)·<<·close
  
31 o6·=·/tmp/M2-3692110-0/031 o6·=·/tmp/M2-2674564-0/0
  
32 o6·:·File32 o6·:·File
  
33 i7·:·get·f33 i7·:·get·f
  
34 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,34 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,
35 ·····x^2-y^2,·x*y*z^7}},image·matrix·{{x^2,·x^2-y^2,·x*y*z^7}})35 ·····x^2-y^2,·x*y*z^7}},image·matrix·{{x^2,·x^2-y^2,·x*y*z^7}})
⊟
1.19 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out
    
Offset 191, 18 lines modifiedOffset 191, 18 lines modified
191 o25·=·40191 o25·=·40
  
192 i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;192 i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;
  
193 i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;193 i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;
  
194 i30·:·time·X·=·solve(A,B);194 i30·:·time·X·=·solve(A,B);
195 ·--·used·0.000122043s·(cpu);·0.000111128s·(thread);·0s·(gc)195 ·--·used·0.000701215s·(cpu);·0.00069291s·(thread);·0s·(gc)
  
196 i31·:·time·X·=·solve(A,B,·MaximalRank=>true);196 i31·:·time·X·=·solve(A,B,·MaximalRank=>true);
197 ·--·used·7.7687e-05s·(cpu);·7.7697e-05s·(thread);·0s·(gc)197 ·--·used·0.000144901s·(cpu);·0.000144972s·(thread);·0s·(gc)
  
198 i32·:·norm(A*X-B)198 i32·:·norm(A*X-B)
  
199 o32·=·5.111850690840453e-15199 o32·=·5.111850690840453e-15
  
200 o32·:·RR·(of·precision·53)200 o32·:·RR·(of·precision·53)
  
Offset 211, 18 lines modifiedOffset 211, 18 lines modified
211 o33·=·100211 o33·=·100
  
212 i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;212 i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;
  
213 i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;213 i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;
  
214 i38·:·time·X·=·solve(A,B);214 i38·:·time·X·=·solve(A,B);
215 ·--·used·0.124122s·(cpu);·0.124129s·(thread);·0s·(gc)215 ·--·used·0.184679s·(cpu);·0.18469s·(thread);·0s·(gc)
  
216 i39·:·time·X·=·solve(A,B,·MaximalRank=>true);216 i39·:·time·X·=·solve(A,B,·MaximalRank=>true);
217 ·--·used·0.262736s·(cpu);·0.134172s·(thread);·0s·(gc)217 ·--·used·0.183642s·(cpu);·0.183647s·(thread);·0s·(gc)
  
218 i40·:·norm(A*X-B)218 i40·:·norm(A*X-B)
  
219 o40·=·1.491578274689709814082355885932e-28219 o40·=·1.491578274689709814082355885932e-28
  
220 o40·:·RR·(of·precision·100)220 o40·:·RR·(of·precision·100)
  
⊟
2.01 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__Directory_lp__String_cm__String_rp.out
    
Offset 1, 60 lines modifiedOffset 1, 60 lines modified
1 --·-*-·M2-comint·-*-·hash:·29895135282135576911 --·-*-·M2-comint·-*-·hash:·2989513528213557691
  
2 i1·:·src·=·temporaryFileName()·|·"/"2 i1·:·src·=·temporaryFileName()·|·"/"
  
3 o1·=·/tmp/M2-3685802-0/0/3 o1·=·/tmp/M2-2665493-0/0/
  
4 i2·:·dst·=·temporaryFileName()·|·"/"4 i2·:·dst·=·temporaryFileName()·|·"/"
  
5 o2·=·/tmp/M2-3685802-0/1/5 o2·=·/tmp/M2-2665493-0/1/
  
6 i3·:·makeDirectory·(src|"a/")6 i3·:·makeDirectory·(src|"a/")
  
7 o3·=·/tmp/M2-3685802-0/0/a/7 o3·=·/tmp/M2-2665493-0/0/a/
  
8 i4·:·makeDirectory·(src|"b/")8 i4·:·makeDirectory·(src|"b/")
  
9 o4·=·/tmp/M2-3685802-0/0/b/9 o4·=·/tmp/M2-2665493-0/0/b/
  
10 i5·:·makeDirectory·(src|"b/c/")10 i5·:·makeDirectory·(src|"b/c/")
  
11 o5·=·/tmp/M2-3685802-0/0/b/c/11 o5·=·/tmp/M2-2665493-0/0/b/c/
  
12 i6·:·src|"a/f"·<<·"hi·there"·<<·close12 i6·:·src|"a/f"·<<·"hi·there"·<<·close
  
13 o6·=·/tmp/M2-3685802-0/0/a/f13 o6·=·/tmp/M2-2665493-0/0/a/f
  
14 o6·:·File14 o6·:·File
  
15 i7·:·src|"a/g"·<<·"hi·there"·<<·close15 i7·:·src|"a/g"·<<·"hi·there"·<<·close
  
16 o7·=·/tmp/M2-3685802-0/0/a/g16 o7·=·/tmp/M2-2665493-0/0/a/g
  
17 o7·:·File17 o7·:·File
  
18 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close18 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close
  
19 o8·=·/tmp/M2-3685802-0/0/b/c/g19 o8·=·/tmp/M2-2665493-0/0/b/c/g
  
20 o8·:·File20 o8·:·File
  
21 i9·:·symlinkDirectory(src,dst,Verbose=>true)21 i9·:·symlinkDirectory(src,dst,Verbose=>true)
22 --symlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g22 --symlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
23 --symlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f 
24 --symlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g23 --symlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
 24 --symlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f
  
25 i10·:·get·(dst|"b/c/g")25 i10·:·get·(dst|"b/c/g")
  
26 o10·=·ho·there26 o10·=·ho·there
  
27 i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)27 i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
28 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g28 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
29 --unsymlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f 
30 --unsymlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g29 --unsymlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
 30 --unsymlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f
  
31 i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d31 i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d
  
32 o12·=·rm32 o12·=·rm
  
33 o12·:·FunctionClosure33 o12·:·FunctionClosure
  
⊟
368 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_symlink__File.out
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
1 --·-*-·M2-comint·-*-·hash:·93438446729403065951 --·-*-·M2-comint·-*-·hash:·9343844672940306595
  
2 i1·:·fn·=·temporaryFileName()2 i1·:·fn·=·temporaryFileName()
  
3 o1·=·/tmp/M2-3689376-0/03 o1·=·/tmp/M2-2667794-0/0
  
4 i2·:·symlinkFile("qwert",·fn)4 i2·:·symlinkFile("qwert",·fn)
  
5 i3·:·fileExists·fn5 i3·:·fileExists·fn
  
6 o3·=·false6 o3·=·false
  
⊟
438 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_temporary__File__Name.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·17319265312913021061 --·-*-·M2-comint·-*-·hash:·1731926531291302106
  
2 i1·:·temporaryFileName·()·|·".tex"2 i1·:·temporaryFileName·()·|·".tex"
  
3 o1·=·/tmp/M2-3697216-0/0.tex3 o1·=·/tmp/M2-2679119-0/0.tex
  
4 i2·:·temporaryFileName·()·|·".html"4 i2·:·temporaryFileName·()·|·".html"
  
5 o2·=·/tmp/M2-3697216-0/1.html5 o2·=·/tmp/M2-2679119-0/1.html
  
6 i3·:·6 i3·:·
⊟
351 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
    
Offset 1, 8 lines modifiedOffset 1, 8 lines modified
1 --·-*-·M2-comint·-*-·hash:·13324355007231 --·-*-·M2-comint·-*-·hash:·1332435500723
  
2 i1·:·time·3^302 i1·:·time·3^30
3 ·--·used·1.7386e-05s·(cpu);·4.627e-06s·(thread);·0s·(gc)3 ·--·used·1.7433e-05s·(cpu);·4.648e-06s·(thread);·0s·(gc)
  
4 o1·=·2058911320946494 o1·=·205891132094649
  
5 i2·:·5 i2·:·
⊟
414 B
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_timing.out
    
Offset 1, 14 lines modifiedOffset 1, 14 lines modified
1 --·-*-·M2-comint·-*-·hash:·17309883004690986031 --·-*-·M2-comint·-*-·hash:·1730988300469098603
  
2 i1·:·timing·3^302 i1·:·timing·3^30
  
3 o1·=·2058911320946493 o1·=·205891132094649
4 ·····--·.00001293·seconds4 ·····--·.000019527·seconds
  
5 o1·:·Time5 o1·:·Time
  
6 i2·:·peek·oo6 i2·:·peek·oo
  
7 o2·=·Time{.00001293,·205891132094649}7 o2·=·Time{.000019527,·205891132094649}
  
8 i3·:·8 i3·:·
⊟
4.94 KB
./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
    
Offset 33, 15 lines modifiedOffset 33, 15 lines modified
33 ···············"memtailor·version"·=>·1.033 ···············"memtailor·version"·=>·1.0
34 ···············"mpfi·version"·=>·1.5.434 ···············"mpfi·version"·=>·1.5.4
35 ···············"mpfr·version"·=>·4.2.235 ···············"mpfr·version"·=>·4.2.2
36 ···············"mpsolve·version"·=>·3.2.136 ···············"mpsolve·version"·=>·3.2.1
37 ···············"mysql·version"·=>·not·present37 ···············"mysql·version"·=>·not·present
38 ···············"normaliz·version"·=>·3.10.438 ···············"normaliz·version"·=>·3.10.4
39 ···············"ntl·version"·=>·11.5.139 ···············"ntl·version"·=>·11.5.1
40 ···············"operating·system·release"·=>·6.1.0-37-cloud-amd6440 ···············"operating·system·release"·=>·6.12.22+bpo-amd64
41 ···············"operating·system"·=>·Linux41 ···············"operating·system"·=>·Linux
42 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43 ···············"pointer·size"·=>·843 ···············"pointer·size"·=>·8
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42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>CodimensionLimit·--·an·optional·argument</h1>44 ······<h1>CodimensionLimit·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
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42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>CoefficientRing·--·an·optional·argument</h1>44 ······<h1>CoefficientRing·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
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77 o2·=·1073741824</code></pre>77 o2·=·1073741824</code></pre>
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84 Tue·Jun··3·02:41:20·-12·202584 Tue·Jul··7·13:13:38·+14·2026
  
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21 i2·:·f21 i2·:·f
  
22 o2·=·107374182422 o2·=·1073741824
23 i3·:·(c·=·Command·"date";)23 i3·:·(c·=·Command·"date";)
24 i4·:·c24 i4·:·c
25 Tue·Jun··3·02:41:20·-12·202525 Tue·Jul··7·13:13:38·+14·2026
  
26 o4·=·026 o4·=·0
27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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6 *\x8**\x8**\x8**\x8**\x8**\x8*·D\x8Da\x8at\x8ta\x8ab\x8ba\x8as\x8se\x8e·-\x8--\x8-·t\x8th\x8he\x8e·c\x8cl\x8la\x8as\x8ss\x8s·o\x8of\x8f·a\x8al\x8ll\x8l·d\x8da\x8at\x8ta\x8ab\x8ba\x8as\x8se\x8e·f\x8fi\x8il\x8le\x8es\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*6 *\x8**\x8**\x8**\x8**\x8**\x8*·D\x8Da\x8at\x8ta\x8ab\x8ba\x8as\x8se\x8e·-\x8--\x8-·t\x8th\x8he\x8e·c\x8cl\x8la\x8as\x8ss\x8s·o\x8of\x8f·a\x8al\x8ll\x8l·d\x8da\x8at\x8ta\x8ab\x8ba\x8as\x8se\x8e·f\x8fi\x8il\x8le\x8es\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*
7 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*7 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
8 A·database·file·is·just·like·a·hash·table,·except·both·the·keys·and·values·have8 A·database·file·is·just·like·a·hash·table,·except·both·the·keys·and·values·have
9 to·be·strings.·In·this·example·we·create·a·database·file,·store·a·few·entries,9 to·be·strings.·In·this·example·we·create·a·database·file,·store·a·few·entries,
10 remove·an·entry·with·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e,·close·the·file,·and·then·remove·the·file.10 remove·an·entry·with·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e,·close·the·file,·and·then·remove·the·file.
11 i1·:·filename·=·temporaryFileName·()·|·".dbm"11 i1·:·filename·=·temporaryFileName·()·|·".dbm"
  
12 o1·=·/tmp/M2-3695705-0/0.dbm12 o1·=·/tmp/M2-2676632-0/0.dbm
13 i2·:·x·=·openDatabaseOut·filename13 i2·:·x·=·openDatabaseOut·filename
  
14 o2·=·/tmp/M2-3695705-0/0.dbm14 o2·=·/tmp/M2-2676632-0/0.dbm
  
15 o2·:·Database15 o2·:·Database
16 i3·:·x#"first"·=·"hi·there"16 i3·:·x#"first"·=·"hi·there"
  
17 o3·=·hi·there17 o3·=·hi·there
18 i4·:·x#"first"18 i4·:·x#"first"
  
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34 ····<div·id="buttons">34 ····<div·id="buttons">
35 ······<div>35 ······<div>
36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a></span>······</div>
37 ······<div·class="right">37 ······<div·class="right">
38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Variables.html">next</a>·|·<a·href="___Pair__Limit.html">previous</a>·|·<a·href="___Variables.html">forward</a>·|·<a·href="___Pair__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Variables.html">next</a>·|·<a·href="___Degree__Limit.html">previous</a>·|·<a·href="___Variables.html">forward</a>·|·<a·href="___Degree__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>DegreeGroup·--·an·optional·argument</h1>44 ······<h1>DegreeGroup·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
48 ······<div>48 ······<div>
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35 ······<div>35 ······<div>
36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a></span>······</div>
37 ······<div·class="right">37 ······<div·class="right">
38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Generic.html">next</a>·|·<a·href="___Strategy.html">previous</a>·|·<a·href="___Generic.html">forward</a>·|·<a·href="___Strategy.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Degree__Group.html">next</a>·|·<a·href="___Codimension__Limit.html">previous</a>·|·<a·href="___Degree__Group.html">forward</a>·|·<a·href="___Codimension__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>DegreeLimit·--·an·optional·argument</h1>44 ······<h1>DegreeLimit·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
48 ······<div>48 ······<div>
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35 ······<div>35 ······<div>
36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a></span>······</div>
37 ······<div·class="right">37 ······<div·class="right">
38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Verify.html">next</a>·|·<a·href="___Skew__Commutative.html">previous</a>·|·<a·href="___Verify.html">forward</a>·|·<a·href="___Skew__Commutative.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Generic.html">next</a>·|·<a·href="___Skew__Commutative.html">previous</a>·|·<a·href="___Generic.html">forward</a>·|·<a·href="___Skew__Commutative.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>DegreeMap·--·an·optional·argument</h1>44 ······<h1>DegreeMap·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
48 ······<div>48 ······<div>
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84 o2·=·S84 o2·=·S
  
85 o2·:·PolynomialRing</code></pre>85 o2·:·PolynomialRing</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)88 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
89 ·--·1.80458s·elapsed89 ·--·2.4765s·elapsed
  
90 ······1······35······241······841······1781······2464······2294······1432······576······135······1490 ······1······35······241······841······1781······2464······2294······1432······576······135······14
91 o3·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·091 o3·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·0
92 ·········································································································92 ·········································································································
93 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······1193 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······11
  
94 o3·:·ChainComplex</code></pre>94 o3·:·ChainComplex</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C1·=·res·ideal(I_*)97 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C1·=·res·ideal(I_*)
98 ·--·.974698s·elapsed98 ·--·1.11323s·elapsed
  
99 ······1······35······140······385······819······1080······819······385······140······35······199 ······1······35······140······385······819······1080······819······385······140······35······1
100 o4·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·S··&lt;--·0100 o4·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·S··&lt;--·0
101 ····································································································101 ····································································································
102 ·····0······1·······2········3········4········5·········6········7········8········9·······10·····11102 ·····0······1·······2········3········4········5·········6········7········8········9·······10·····11
  
103 o4·:·ChainComplex</code></pre>103 o4·:·ChainComplex</code></pre>
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30 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,630 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,6
31 i2·:·S·=·ring·I31 i2·:·S·=·ring·I
  
32 o2·=·S32 o2·=·S
  
33 o2·:·PolynomialRing33 o2·:·PolynomialRing
34 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)34 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
35 ·--·1.80458s·elapsed35 ·--·2.4765s·elapsed
  
36 ······1······35······241······841······1781······2464······2294······143236 ······1······35······241······841······1781······2464······2294······1432
37 576······135······1437 576······135······14
38 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S38 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S
39 <--·S····<--·S···<--·039 <--·S····<--·S···<--·0
  
  
40 ·····0······1·······2········3········4·········5·········6·········7·········840 ·····0······1·······2········3········4·········5·········6·········7·········8
41 9········10······1141 9········10······11
  
42 o3·:·ChainComplex42 o3·:·ChainComplex
43 i4·:·elapsedTime·C1·=·res·ideal(I_*)43 i4·:·elapsedTime·C1·=·res·ideal(I_*)
44 ·--·.974698s·elapsed44 ·--·1.11323s·elapsed
  
45 ······1······35······140······385······819······1080······819······385······14045 ······1······35······140······385······819······1080······819······385······140
46 35······146 35······1
47 o4·=·S··<--·S···<--·S····<--·S····<--·S····<--·S·····<--·S····<--·S····<--·S47 o4·=·S··<--·S···<--·S····<--·S····<--·S····<--·S·····<--·S····<--·S····<--·S
48 <--·S···<--·S··<--·048 <--·S···<--·S··<--·0
  
  
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97 o2·=·stdio97 o2·=·stdio
  
98 o2·:·File</code></pre>98 o2·:·File</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·=·temporaryFileName()101 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·=·temporaryFileName()
  
102 o3·=·/tmp/M2-3696118-0/0</code></pre>102 o3·=·/tmp/M2-2676839-0/0</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·endl·&lt;&lt;·close105 <td>··············<pre><code·class="language-macaulay2">i4·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·endl·&lt;&lt;·close
  
106 o4·=·/tmp/M2-3696118-0/0106 o4·=·/tmp/M2-2676839-0/0
  
107 o4·:·File</code></pre>107 o4·:·File</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i5·:·get·fn110 <td>··············<pre><code·class="language-macaulay2">i5·:·get·fn
  
111 o5·=·hi·there</code></pre>111 o5·=·hi·there</code></pre>
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139 o8·=·stdio139 o8·=·stdio
  
140 o8·:·File</code></pre>140 o8·:·File</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i9·:·fn·&lt;&lt;·f·&lt;&lt;·close143 <td>··············<pre><code·class="language-macaulay2">i9·:·fn·&lt;&lt;·f·&lt;&lt;·close
  
144 o9·=·/tmp/M2-3696118-0/0144 o9·=·/tmp/M2-2676839-0/0
  
145 o9·:·File</code></pre>145 o9·:·File</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i10·:·get·fn148 <td>··············<pre><code·class="language-macaulay2">i10·:·get·fn
  
149 o10·=··10······9······8·······7·······6·······5·······4·······3······2149 o10·=··10······9······8·······7·······6·······5·······4·······3······2
150 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x150 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x
151 ······+·1</code></pre>151 ······+·1</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i11·:·fn·&lt;&lt;·toExternalString·f·&lt;&lt;·close154 <td>··············<pre><code·class="language-macaulay2">i11·:·fn·&lt;&lt;·toExternalString·f·&lt;&lt;·close
  
155 o11·=·/tmp/M2-3696118-0/0155 o11·=·/tmp/M2-2676839-0/0
  
156 o11·:·File</code></pre>156 o11·:·File</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i12·:·get·fn159 <td>··············<pre><code·class="language-macaulay2">i12·:·get·fn
  
160 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+160 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
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38 --·ho·there·--38 --·ho·there·--
  
39 o2·=·stdio39 o2·=·stdio
  
40 o2·:·File40 o2·:·File
41 i3·:·fn·=·temporaryFileName()41 i3·:·fn·=·temporaryFileName()
  
42 o3·=·/tmp/M2-3696118-0/042 o3·=·/tmp/M2-2676839-0/0
43 i4·:·fn·<<·"hi·there"·<<·endl·<<·close43 i4·:·fn·<<·"hi·there"·<<·endl·<<·close
  
44 o4·=·/tmp/M2-3696118-0/044 o4·=·/tmp/M2-2676839-0/0
  
45 o4·:·File45 o4·:·File
46 i5·:·get·fn46 i5·:·get·fn
  
47 o5·=·hi·there47 o5·=·hi·there
48 i6·:·R·=·QQ[x]48 i6·:·R·=·QQ[x]
  
Offset 68, 25 lines modifiedOffset 68, 25 lines modified
68 ·10······9······8·······7·······6·······5·······4·······3······268 ·10······9······8·······7·······6·······5·······4·······3······2
69 x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x·+·169 x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x·+·1
70 o8·=·stdio70 o8·=·stdio
  
71 o8·:·File71 o8·:·File
72 i9·:·fn·<<·f·<<·close72 i9·:·fn·<<·f·<<·close
  
73 o9·=·/tmp/M2-3696118-0/073 o9·=·/tmp/M2-2676839-0/0
  
74 o9·:·File74 o9·:·File
75 i10·:·get·fn75 i10·:·get·fn
  
76 o10·=··10······9······8·······7·······6·······5·······4·······3······276 o10·=··10······9······8·······7·······6·······5·······4·······3······2
77 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x77 ······x···+·10x··+·45x··+·120x··+·210x··+·252x··+·210x··+·120x··+·45x··+·10x
78 ······+·178 ······+·1
79 i11·:·fn·<<·toExternalString·f·<<·close79 i11·:·fn·<<·toExternalString·f·<<·close
  
80 o11·=·/tmp/M2-3696118-0/080 o11·=·/tmp/M2-2676839-0/0
  
81 o11·:·File81 o11·:·File
82 i12·:·get·fn82 i12·:·get·fn
  
83 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+83 o12·=·x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
84 ······184 ······1
85 i13·:·value·get·fn85 i13·:·value·get·fn
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45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 ········<p>Macaulay2·uses·the·Hans·Boehm·<a·href="___G__C_spgarbage_spcollector.html">garbage·collector</a>·to·reclaim·unused·memory.··The·function·<span·class="tt">GCstats</span>·provides·information·about·its·status,·such·as·the·total·number·of·bytes·allocated,·the·current·heap·size,·the·number·of·garbage·collections·done,·the·number·of·threads·used·in·each·collection,·the·total·cpu·time·spent·in·garbage·collection,·etc.</p>47 ········<p>Macaulay2·uses·the·Hans·Boehm·<a·href="___G__C_spgarbage_spcollector.html">garbage·collector</a>·to·reclaim·unused·memory.··The·function·<span·class="tt">GCstats</span>·provides·information·about·its·status,·such·as·the·total·number·of·bytes·allocated,·the·current·heap·size,·the·number·of·garbage·collections·done,·the·number·of·threads·used·in·each·collection,·the·total·cpu·time·spent·in·garbage·collection,·etc.</p>
48 ········<table·class="examples">48 ········<table·class="examples">
49 ··········<tr>49 ··········<tr>
50 <td>··············<pre><code·class="language-macaulay2">i1·:·s·=·GCstats()50 <td>··············<pre><code·class="language-macaulay2">i1·:·s·=·GCstats()
  
51 o1·=·HashTable{&quot;bytesAlloc&quot;·=>·15433937402········}51 o1·=·HashTable{&quot;bytesAlloc&quot;·=>·15347371322········}
52 ···············&quot;GC_free_space_divisor&quot;·=>·352 ···············&quot;GC_free_space_divisor&quot;·=>·3
53 ···············&quot;GC_LARGE_ALLOC_WARN_INTERVAL&quot;·=>·153 ···············&quot;GC_LARGE_ALLOC_WARN_INTERVAL&quot;·=>·1
54 ···············&quot;gcCpuTimeSecs&quot;·=>·054 ···············&quot;gcCpuTimeSecs&quot;·=>·0
55 ···············&quot;heapSize&quot;·=>·21908275255 ···············&quot;heapSize&quot;·=>·219082752
56 ···············&quot;numGCs&quot;·=>·83056 ···············&quot;numGCs&quot;·=>·826
57 ···············&quot;numGCThreads&quot;·=>·1257 ···············&quot;numGCThreads&quot;·=>·12
  
58 o1·:·HashTable</code></pre>58 o1·:·HashTable</code></pre>
59 </td>··········</tr>59 </td>··········</tr>
60 ········</table>60 ········</table>
61 ········<p>The·value·returned·is·a·hash·table,·from·which·individual·bits·of·information·can·be·easily·extracted,·as·follows.</p>61 ········<p>The·value·returned·is·a·hash·table,·from·which·individual·bits·of·information·can·be·easily·extracted,·as·follows.</p>
62 ········<table·class="examples">62 ········<table·class="examples">
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8 Macaulay2·uses·the·Hans·Boehm·_\x8g_\x8a_\x8r_\x8b_\x8a_\x8g_\x8e_\x8·_\x8c_\x8o_\x8l_\x8l_\x8e_\x8c_\x8t_\x8o_\x8r·to·reclaim·unused·memory.·The8 Macaulay2·uses·the·Hans·Boehm·_\x8g_\x8a_\x8r_\x8b_\x8a_\x8g_\x8e_\x8·_\x8c_\x8o_\x8l_\x8l_\x8e_\x8c_\x8t_\x8o_\x8r·to·reclaim·unused·memory.·The
9 function·GCstats·provides·information·about·its·status,·such·as·the·total9 function·GCstats·provides·information·about·its·status,·such·as·the·total
10 number·of·bytes·allocated,·the·current·heap·size,·the·number·of·garbage10 number·of·bytes·allocated,·the·current·heap·size,·the·number·of·garbage
11 collections·done,·the·number·of·threads·used·in·each·collection,·the·total·cpu11 collections·done,·the·number·of·threads·used·in·each·collection,·the·total·cpu
12 time·spent·in·garbage·collection,·etc.12 time·spent·in·garbage·collection,·etc.
13 i1·:·s·=·GCstats()13 i1·:·s·=·GCstats()
  
14 o1·=·HashTable{"bytesAlloc"·=>·15433937402········}14 o1·=·HashTable{"bytesAlloc"·=>·15347371322········}
15 ···············"GC_free_space_divisor"·=>·315 ···············"GC_free_space_divisor"·=>·3
16 ···············"GC_LARGE_ALLOC_WARN_INTERVAL"·=>·116 ···············"GC_LARGE_ALLOC_WARN_INTERVAL"·=>·1
17 ···············"gcCpuTimeSecs"·=>·017 ···············"gcCpuTimeSecs"·=>·0
18 ···············"heapSize"·=>·21908275218 ···············"heapSize"·=>·219082752
19 ···············"numGCs"·=>·83019 ···············"numGCs"·=>·826
20 ···············"numGCThreads"·=>·1220 ···············"numGCThreads"·=>·12
  
21 o1·:·HashTable21 o1·:·HashTable
22 The·value·returned·is·a·hash·table,·from·which·individual·bits·of·information22 The·value·returned·is·a·hash·table,·from·which·individual·bits·of·information
23 can·be·easily·extracted,·as·follows.23 can·be·easily·extracted,·as·follows.
24 i2·:·s#"heapSize"24 i2·:·s#"heapSize"
  
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34 ····<div·id="buttons">34 ····<div·id="buttons">
35 ······<div>35 ······<div>
36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Generic.html">Generic</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Generic.html">Generic</a></span>······</div>
37 ······<div·class="right">37 ······<div·class="right">
38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Basis__Element__Limit.html">next</a>·|·<a·href="___Degree__Limit.html">previous</a>·|·<a·href="___Basis__Element__Limit.html">forward</a>·|·<a·href="___Degree__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Basis__Element__Limit.html">next</a>·|·<a·href="___Degree__Map.html">previous</a>·|·<a·href="___Basis__Element__Limit.html">forward</a>·|·<a·href="___Degree__Map.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>Generic·--·an·optional·argument</h1>44 ······<h1>Generic·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
48 ······<div>48 ······<div>
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106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));107 <td>··············<pre><code·class="language-macaulay2">i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));
  
108 o7·:·Ideal·of·R</code></pre>108 o7·:·Ideal·of·R</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i8·:·time·gens·gb·I;111 <td>··············<pre><code·class="language-macaulay2">i8·:·time·gens·gb·I;
112 ·--·used·0.0200335s·(cpu);·0.0200315s·(thread);·0s·(gc)112 ·--·used·0.0350028s·(cpu);·0.0348658s·(thread);·0s·(gc)
  
113 ·············1······428113 ·············1······428
114 o8·:·Matrix·R··&lt;--·R</code></pre>114 o8·:·Matrix·R··&lt;--·R</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ··········<tr>116 ··········<tr>
117 <td>··············<pre><code·class="language-macaulay2">i9·:·time·J1·=·saturate(I);117 <td>··············<pre><code·class="language-macaulay2">i9·:·time·J1·=·saturate(I);
118 ·--·used·0.50658s·(cpu);·0.160597s·(thread);·0s·(gc)118 ·--·used·0.459704s·(cpu);·0.23252s·(thread);·0s·(gc)
  
119 o9·:·Ideal·of·R</code></pre>119 o9·:·Ideal·of·R</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);122 <td>··············<pre><code·class="language-macaulay2">i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);
123 ·--·used·9.307e-05s·(cpu);·9.2579e-05s·(thread);·0s·(gc)123 ·--·used·0.000134432s·(cpu);·0.000134502s·(thread);·0s·(gc)
  
124 o10·:·Ideal·of·R</code></pre>124 o10·:·Ideal·of·R</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ··········<tr>126 ··········<tr>
127 <td>··············<pre><code·class="language-macaulay2">i11·:·numgens·J127 <td>··············<pre><code·class="language-macaulay2">i11·:·numgens·J
  
128 o11·=·7</code></pre>128 o11·=·7</code></pre>
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44 o6·=·R44 o6·=·R
  
45 o6·:·PolynomialRing45 o6·:·PolynomialRing
46 i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));46 i7·:·I·=·truncate(8,·monomialCurveIdeal(R,{1,4,5,9}));
  
47 o7·:·Ideal·of·R47 o7·:·Ideal·of·R
48 i8·:·time·gens·gb·I;48 i8·:·time·gens·gb·I;
49 ·--·used·0.0200335s·(cpu);·0.0200315s·(thread);·0s·(gc)49 ·--·used·0.0350028s·(cpu);·0.0348658s·(thread);·0s·(gc)
  
50 ·············1······42850 ·············1······428
51 o8·:·Matrix·R··<--·R51 o8·:·Matrix·R··<--·R
52 i9·:·time·J1·=·saturate(I);52 i9·:·time·J1·=·saturate(I);
53 ·--·used·0.50658s·(cpu);·0.160597s·(thread);·0s·(gc)53 ·--·used·0.459704s·(cpu);·0.23252s·(thread);·0s·(gc)
  
54 o9·:·Ideal·of·R54 o9·:·Ideal·of·R
55 i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);55 i10·:·time·J·=·saturate(I,·MinimalGenerators·=>·false);
56 ·--·used·9.307e-05s·(cpu);·9.2579e-05s·(thread);·0s·(gc)56 ·--·used·0.000134432s·(cpu);·0.000134502s·(thread);·0s·(gc)
  
57 o10·:·Ideal·of·R57 o10·:·Ideal·of·R
58 i11·:·numgens·J58 i11·:·numgens·J
  
59 o11·=·759 o11·=·7
60 i12·:·numgens·J160 i12·:·numgens·J1
  
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34 ····<div·id="buttons">34 ····<div·id="buttons">
35 ······<div>35 ······<div>
36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a></span>······</div>
37 ······<div·class="right">37 ······<div·class="right">
38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Degree__Group.html">next</a>·|·<a·href="___Basis__Element__Limit.html">previous</a>·|·<a·href="___Degree__Group.html">forward</a>·|·<a·href="___Basis__Element__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Verify.html">next</a>·|·<a·href="___Basis__Element__Limit.html">previous</a>·|·<a·href="___Verify.html">forward</a>·|·<a·href="___Basis__Element__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>PairLimit·--·an·optional·argument</h1>44 ······<h1>PairLimit·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
48 ······<div>48 ······<div>
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61 ················200·········20061 ················200·········200
62 o1·:·Matrix·RR······&lt;--·RR62 o1·:·Matrix·RR······&lt;--·RR
63 ··············53··········53</code></pre>63 ··············53··········53</code></pre>
64 </td>··········</tr>64 </td>··········</tr>
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i2·:·time·SVD(M);66 <td>··············<pre><code·class="language-macaulay2">i2·:·time·SVD(M);
67 ·--·used·0.0289738s·(cpu);·0.0289739s·(thread);·0s·(gc)</code></pre>67 ·--·used·0.0372969s·(cpu);·0.0372156s·(thread);·0s·(gc)</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ··········<tr>69 ··········<tr>
70 <td>··············<pre><code·class="language-macaulay2">i3·:·time·SVD(M,·DivideConquer=>true);70 <td>··············<pre><code·class="language-macaulay2">i3·:·time·SVD(M,·DivideConquer=>true);
71 ·--·used·0.0270797s·(cpu);·0.0270974s·(thread);·0s·(gc)</code></pre>71 ·--·used·0.0357002s·(cpu);·0.0357113s·(thread);·0s·(gc)</code></pre>
72 </td>··········</tr>72 </td>··········</tr>
73 ········</table>73 ········</table>
74 ······</div>74 ······</div>
75 ······<div>75 ······<div>
76 ········<h2>Further·information</h2>76 ········<h2>Further·information</h2>
77 ········<ul>77 ········<ul>
78 ··········<li>78 ··········<li>
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12 For·large·matrices,·this·algorithm·is·often·much·faster.12 For·large·matrices,·this·algorithm·is·often·much·faster.
13 i1·:·M·=·random(RR^200,·RR^200);13 i1·:·M·=·random(RR^200,·RR^200);
  
14 ················200·········20014 ················200·········200
15 o1·:·Matrix·RR······<--·RR15 o1·:·Matrix·RR······<--·RR
16 ··············53··········5316 ··············53··········53
17 i2·:·time·SVD(M);17 i2·:·time·SVD(M);
18 ·--·used·0.0289738s·(cpu);·0.0289739s·(thread);·0s·(gc)18 ·--·used·0.0372969s·(cpu);·0.0372156s·(thread);·0s·(gc)
19 i3·:·time·SVD(M,·DivideConquer=>true);19 i3·:·time·SVD(M,·DivideConquer=>true);
20 ·--·used·0.0270797s·(cpu);·0.0270974s·(thread);·0s·(gc)20 ·--·used·0.0357002s·(cpu);·0.0357113s·(thread);·0s·(gc)
21 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*21 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
22 ····*·Default·value:·_\x8t_\x8r_\x8u_\x8e22 ····*·Default·value:·_\x8t_\x8r_\x8u_\x8e
23 ····*·Function:·_\x8S_\x8V_\x8D·--·singular·value·decomposition·of·a·matrix23 ····*·Function:·_\x8S_\x8V_\x8D·--·singular·value·decomposition·of·a·matrix
24 ····*·Option·key:·_\x8D_\x8i_\x8v_\x8i_\x8d_\x8e_\x8C_\x8o_\x8n_\x8q_\x8u_\x8e_\x8r·--·an·optional·argument24 ····*·Option·key:·_\x8D_\x8i_\x8v_\x8i_\x8d_\x8e_\x8C_\x8o_\x8n_\x8q_\x8u_\x8e_\x8r·--·an·optional·argument
25 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·D\x8Di\x8iv\x8vi\x8id\x8de\x8eC\x8Co\x8on\x8nq\x8qu\x8ue\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·D\x8Di\x8iv\x8vi\x8id\x8de\x8eC\x8Co\x8on\x8nq\x8qu\x8ue\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
26 ····*·_\x8S_\x8V_\x8D_\x8(_\x8._\x8._\x8._\x8,_\x8D_\x8i_\x8v_\x8i_\x8d_\x8e_\x8C_\x8o_\x8n_\x8q_\x8u_\x8e_\x8r_\x8=_\x8>_\x8._\x8._\x8._\x8)·--·Use·the·lapack·divide·and·conquer·SVD26 ····*·_\x8S_\x8V_\x8D_\x8(_\x8._\x8._\x8._\x8,_\x8D_\x8i_\x8v_\x8i_\x8d_\x8e_\x8C_\x8o_\x8n_\x8q_\x8u_\x8e_\x8r_\x8=_\x8>_\x8._\x8._\x8._\x8)·--·Use·the·lapack·divide·and·conquer·SVD
27 ······algorithm27 ······algorithm
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36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Strategy.html">Strategy</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Strategy.html">Strategy</a></span>······</div>
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38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
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41 </form><a·href="___Degree__Limit.html">next</a>·|·<a·href="___Threads.html">previous</a>·|·<a·href="___Degree__Limit.html">forward</a>·|·<a·href="___Threads.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Codimension__Limit.html">next</a>·|·<a·href="___Verbosity.html">previous</a>·|·<a·href="___Codimension__Limit.html">forward</a>·|·<a·href="___Verbosity.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>Strategy·--·an·optional·argument</h1>44 ······<h1>Strategy·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
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38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">40 ··<input·type="hidden"·name="q"·value="site:macaulay2.com/doc">
41 </form><a·href="___Strategy.html">next</a>·|·<a·href="___Syzygies.html">previous</a>·|·<a·href="___Strategy.html">forward</a>·|·<a·href="___Syzygies.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Verbosity.html">next</a>·|·<a·href="___Syzygies.html">previous</a>·|·<a·href="___Verbosity.html">forward</a>·|·<a·href="___Syzygies.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>Threads·--·an·optional·argument</h1>44 ······<h1>Threads·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
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36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a></span>······</div>36 <a·href="https://macaulay2.com/">Macaulay2</a>·<span·id="version-select-container"></span>·»·<a·title="Macaulay2·documentation"·href="index.html">Documentation·</a>·<br><a·href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a>·»·<span><a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">symbols·used·as·the·name·or·value·of·an·optional·argument</a>·>·<a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a></span>······</div>
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39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
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42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>Verbosity·--·an·optional·argument</h1>44 ······<h1>Verbosity·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
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38 <form·method="get"·action="https://www.google.com/search">38 <form·method="get"·action="https://www.google.com/search">
39 ··<input·placeholder="Search"·type="text"·name="q"·value="">39 ··<input·placeholder="Search"·type="text"·name="q"·value="">
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41 </form><a·href="___Verbosity.html">next</a>·|·<a·href="___Degree__Map.html">previous</a>·|·<a·href="___Verbosity.html">forward</a>·|·<a·href="___Degree__Map.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>41 </form><a·href="___Coefficient__Ring.html">next</a>·|·<a·href="___Pair__Limit.html">previous</a>·|·<a·href="___Coefficient__Ring.html">forward</a>·|·<a·href="___Pair__Limit.html">backward</a>·|·<a·href="_symbols_spused_spas_spthe_spname_spor_spvalue_spof_span_spoptional_spargument.html">up</a>·|·<a·href="master.html">index</a>·|·<a·href="toc.html">toc</a>······</div>
42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>Verify·--·an·optional·argument</h1>44 ······<h1>Verify·--·an·optional·argument</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>47 A·symbol·used·as·the·name·of·an·optional·argument.······</div>
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554 ·····························3554 ·····························3
555 o58·:·R-module,·quotient·of·R</code></pre>555 o58·:·R-module,·quotient·of·R</code></pre>
556 </td>··········</tr>556 </td>··········</tr>
557 ········</table>557 ········</table>
558 We·may·use·<a·title="projective·resolution"·href="_resolution.html">resolution</a>·to·produce·a·projective·resolution·of·it,·and·<a·title="time·a·computation"·href="_time.html">time</a>·to·report·the·time·required.········<table·class="examples">558 We·may·use·<a·title="projective·resolution"·href="_resolution.html">resolution</a>·to·produce·a·projective·resolution·of·it,·and·<a·title="time·a·computation"·href="_time.html">time</a>·to·report·the·time·required.········<table·class="examples">
559 ··········<tr>559 ··········<tr>
560 <td>··············<pre><code·class="language-macaulay2">i59·:·time·C·=·resolution·M560 <td>··············<pre><code·class="language-macaulay2">i59·:·time·C·=·resolution·M
561 ·--·used·0.00148457s·(cpu);·0.00147841s·(thread);·0s·(gc)561 ·--·used·0.00300443s·(cpu);·0.00299295s·(thread);·0s·(gc)
  
562 ·······3······6······15······18······6562 ·······3······6······15······18······6
563 o59·=·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R··&lt;--·0563 o59·=·R··&lt;--·R··&lt;--·R···&lt;--·R···&lt;--·R··&lt;--·0
564 ············································564 ············································
565 ······0······1······2·······3·······4······5565 ······0······1······2·······3·······4······5
  
566 o59·:·ChainComplex</code></pre>566 o59·:·ChainComplex</code></pre>
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386 ···············|·c·f·i·l·o·r·|386 ···············|·c·f·i·l·o·r·|
  
387 ·····························3387 ·····························3
388 o58·:·R-module,·quotient·of·R388 o58·:·R-module,·quotient·of·R
389 We·may·use·_\x8r_\x8e_\x8s_\x8o_\x8l_\x8u_\x8t_\x8i_\x8o_\x8n·to·produce·a·projective·resolution·of·it,·and·_\x8t_\x8i_\x8m_\x8e·to389 We·may·use·_\x8r_\x8e_\x8s_\x8o_\x8l_\x8u_\x8t_\x8i_\x8o_\x8n·to·produce·a·projective·resolution·of·it,·and·_\x8t_\x8i_\x8m_\x8e·to
390 report·the·time·required.390 report·the·time·required.
391 i59·:·time·C·=·resolution·M391 i59·:·time·C·=·resolution·M
392 ·--·used·0.00148457s·(cpu);·0.00147841s·(thread);·0s·(gc)392 ·--·used·0.00300443s·(cpu);·0.00299295s·(thread);·0s·(gc)
  
393 ·······3······6······15······18······6393 ·······3······6······15······18······6
394 o59·=·R··<--·R··<--·R···<--·R···<--·R··<--·0394 o59·=·R··<--·R··<--·R···<--·R···<--·R··<--·0
  
395 ······0······1······2·······3·······4······5395 ······0······1······2·······3·······4······5
  
396 o59·:·ChainComplex396 o59·:·ChainComplex
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
    
Offset 67, 15 lines modifiedOffset 67, 15 lines modified
67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 Produces·an·accurate·timing·for·the·code·contained·in·the·string·<span·class="tt">s</span>.··The·value·returned·is·the·number·of·seconds.········<table·class="examples">70 Produces·an·accurate·timing·for·the·code·contained·in·the·string·<span·class="tt">s</span>.··The·value·returned·is·the·number·of·seconds.········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·benchmark·&quot;sqrt·2p100000&quot;72 <td>··············<pre><code·class="language-macaulay2">i1·:·benchmark·&quot;sqrt·2p100000&quot;
  
73 o1·=·.000339054222009568973 o1·=·.0003060271523195235
  
74 o1·:·RR·(of·precision·53)</code></pre>74 o1·:·RR·(of·precision·53)</code></pre>
75 </td>··········</tr>75 </td>··········</tr>
76 ········</table>76 ········</table>
77 The·snippet·of·code·provided·will·be·run·enough·times·to·register·meaningfully·on·the·clock,·and·the·garbage·collector·will·be·called·beforehand.······</div>77 The·snippet·of·code·provided·will·be·run·enough·times·to·register·meaningfully·on·the·clock,·and·the·garbage·collector·will·be·called·beforehand.······</div>
78 ······<div·class="waystouse">78 ······<div·class="waystouse">
79 ········<h2>For·the·programmer</h2>79 ········<h2>For·the·programmer</h2>
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13 ··········o·a·_\x8r_\x8e_\x8a_\x8l_\x8·_\x8n_\x8u_\x8m_\x8b_\x8e_\x8r,·the·number·of·seconds·it·takes·to·evaluate·the·code13 ··········o·a·_\x8r_\x8e_\x8a_\x8l_\x8·_\x8n_\x8u_\x8m_\x8b_\x8e_\x8r,·the·number·of·seconds·it·takes·to·evaluate·the·code
14 ············in·s14 ············in·s
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 Produces·an·accurate·timing·for·the·code·contained·in·the·string·s.·The·value16 Produces·an·accurate·timing·for·the·code·contained·in·the·string·s.·The·value
17 returned·is·the·number·of·seconds.17 returned·is·the·number·of·seconds.
18 i1·:·benchmark·"sqrt·2p100000"18 i1·:·benchmark·"sqrt·2p100000"
  
19 o1·=·.000339054222009568919 o1·=·.0003060271523195235
  
20 o1·:·RR·(of·precision·53)20 o1·:·RR·(of·precision·53)
21 The·snippet·of·code·provided·will·be·run·enough·times·to·register·meaningfully21 The·snippet·of·code·provided·will·be·run·enough·times·to·register·meaningfully
22 on·the·clock,·and·the·garbage·collector·will·be·called·beforehand.22 on·the·clock,·and·the·garbage·collector·will·be·called·beforehand.
23 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
24 The·object·_\x8b_\x8e_\x8n_\x8c_\x8h_\x8m_\x8a_\x8r_\x8k·is·a·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8c_\x8l_\x8o_\x8s_\x8u_\x8r_\x8e.24 The·object·_\x8b_\x8e_\x8n_\x8c_\x8h_\x8m_\x8a_\x8r_\x8k·is·a·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8c_\x8l_\x8o_\x8s_\x8u_\x8r_\x8e.
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
    
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83 o2·=·S83 o2·=·S
  
84 o2·:·PolynomialRing</code></pre>84 o2·:·PolynomialRing</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)87 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
88 ·--·1.86611s·elapsed88 ·--·2.48419s·elapsed
  
89 ······1······35······241······841······1781······2464······2294······1432······576······135······1489 ······1······35······241······841······1781······2464······2294······1432······576······135······14
90 o3·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·090 o3·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S···&lt;--·0
91 ·········································································································91 ·········································································································
92 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······1192 ·····0······1·······2········3········4·········5·········6·········7·········8········9········10······11
  
93 o3·:·ChainComplex</code></pre>93 o3·:·ChainComplex</code></pre>
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27 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,627 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,6
28 i2·:·S·=·ring·I28 i2·:·S·=·ring·I
  
29 o2·=·S29 o2·=·S
  
30 o2·:·PolynomialRing30 o2·:·PolynomialRing
31 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)31 i3·:·elapsedTime·C·=·res(I,·FastNonminimal·=>·true)
32 ·--·1.86611s·elapsed32 ·--·2.48419s·elapsed
  
33 ······1······35······241······841······1781······2464······2294······143233 ······1······35······241······841······1781······2464······2294······1432
34 576······135······1434 576······135······14
35 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S35 o3·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<--·S
36 <--·S····<--·S···<--·036 <--·S····<--·S···<--·0
  
  
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html
    
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95 o4·=·&lt;&lt;task,·running>>95 o4·=·&lt;&lt;task,·running>>
  
96 o4·:·Task</code></pre>96 o4·:·Task</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·n99 <td>··············<pre><code·class="language-macaulay2">i5·:·n
  
100 o5·=·939538</code></pre>100 o5·=·939547</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i6·:·sleep·1103 <td>··············<pre><code·class="language-macaulay2">i6·:·sleep·1
  
104 o6·=·0</code></pre>104 o6·=·0</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
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112 o7·=·&lt;&lt;task,·running>>112 o7·=·&lt;&lt;task,·running>>
  
113 o7·:·Task</code></pre>113 o7·:·Task</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i8·:·n116 <td>··············<pre><code·class="language-macaulay2">i8·:·n
  
117 o8·=·1912116</code></pre>117 o8·=·1911978</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i9·:·isReady·t120 <td>··············<pre><code·class="language-macaulay2">i9·:·isReady·t
  
121 o9·=·false</code></pre>121 o9·=·false</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
Offset 138, 25 lines modifiedOffset 138, 25 lines modified
138 o12·=·&lt;&lt;task,·canceled>>138 o12·=·&lt;&lt;task,·canceled>>
  
139 o12·:·Task</code></pre>139 o12·:·Task</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i13·:·n142 <td>··············<pre><code·class="language-macaulay2">i13·:·n
  
143 o13·=·1912266</code></pre>143 o13·=·1912172</code></pre>
144 </td>··········</tr>144 </td>··········</tr>
145 ··········<tr>145 ··········<tr>
146 <td>··············<pre><code·class="language-macaulay2">i14·:·sleep·1146 <td>··············<pre><code·class="language-macaulay2">i14·:·sleep·1
  
147 o14·=·0</code></pre>147 o14·=·0</code></pre>
148 </td>··········</tr>148 </td>··········</tr>
149 ··········<tr>149 ··········<tr>
150 <td>··············<pre><code·class="language-macaulay2">i15·:·n150 <td>··············<pre><code·class="language-macaulay2">i15·:·n
  
151 o15·=·1912266</code></pre>151 o15·=·1912172</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i16·:·isReady·t154 <td>··············<pre><code·class="language-macaulay2">i16·:·isReady·t
  
155 o16·=·false</code></pre>155 o16·=·false</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
157 ········</table>157 ········</table>
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29 i4·:·t29 i4·:·t
  
30 o4·=·<<task,·running>>30 o4·=·<<task,·running>>
  
31 o4·:·Task31 o4·:·Task
32 i5·:·n32 i5·:·n
  
33 o5·=·93953833 o5·=·939547
34 i6·:·sleep·134 i6·:·sleep·1
  
35 o6·=·035 o6·=·0
36 i7·:·t36 i7·:·t
  
37 o7·=·<<task,·running>>37 o7·=·<<task,·running>>
  
38 o7·:·Task38 o7·:·Task
39 i8·:·n39 i8·:·n
  
40 o8·=·191211640 o8·=·1911978
41 i9·:·isReady·t41 i9·:·isReady·t
  
42 o9·=·false42 o9·=·false
43 i10·:·cancelTask·t43 i10·:·cancelTask·t
44 i11·:·sleep·244 i11·:·sleep·2
45 stdio:2:25:(3):[1]:·error:·interrupted45 stdio:2:25:(3):[1]:·error:·interrupted
  
Offset 56, 19 lines modifiedOffset 56, 19 lines modified
56 i12·:·t56 i12·:·t
  
57 o12·=·<<task,·canceled>>57 o12·=·<<task,·canceled>>
  
58 o12·:·Task58 o12·:·Task
59 i13·:·n59 i13·:·n
  
60 o13·=·191226660 o13·=·1912172
61 i14·:·sleep·161 i14·:·sleep·1
  
62 o14·=·062 o14·=·0
63 i15·:·n63 i15·:·n
  
64 o15·=·191226664 o15·=·1912172
65 i16·:·isReady·t65 i16·:·isReady·t
  
66 o16·=·false66 o16·=·false
67 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*67 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
68 ····*·_\x8c_\x8a_\x8n_\x8c_\x8e_\x8l_\x8T_\x8a_\x8s_\x8k_\x8(_\x8T_\x8a_\x8s_\x8k_\x8)·--·stop·a·task68 ····*·_\x8c_\x8a_\x8n_\x8c_\x8e_\x8l_\x8T_\x8a_\x8s_\x8k_\x8(_\x8T_\x8a_\x8s_\x8k_\x8)·--·stop·a·task
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70 ········<div>70 ········<div>
71 ··········<p>Change·the·current·working·directory·to·<var>dir</var>.</p>71 ··········<p>Change·the·current·working·directory·to·<var>dir</var>.</p>
72 ········</div>72 ········</div>
73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()75 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()
  
76 o1·=·/tmp/M2-3680876-0/0</code></pre>76 o1·=·/tmp/M2-2656981-0/0</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir79 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir
  
80 o2·=·/tmp/M2-3680876-0/0</code></pre>80 o2·=·/tmp/M2-2656981-0/0</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·changeDirectory·dir83 <td>··············<pre><code·class="language-macaulay2">i3·:·changeDirectory·dir
  
84 o3·=·/tmp/M2-3680876-0/0/</code></pre>84 o3·=·/tmp/M2-2656981-0/0/</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·currentDirectory()87 <td>··············<pre><code·class="language-macaulay2">i4·:·currentDirectory()
  
88 o4·=·/tmp/M2-3680876-0/0/</code></pre>88 o4·=·/tmp/M2-2656981-0/0/</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ········</table>90 ········</table>
91 ········<div>91 ········<div>
92 ··········<p>If·<var>dir</var>·is·omitted,·then·the·current·working·directory·is·changed·to·the·user's·home·directory.</p>92 ··········<p>If·<var>dir</var>·is·omitted,·then·the·current·working·directory·is·changed·to·the·user's·home·directory.</p>
93 ········</div>93 ········</div>
94 ······</div>94 ······</div>
95 ······<div>95 ······<div>
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11 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,11 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,
12 ····*·Outputs:12 ····*·Outputs:
13 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·the·new·working·directory;13 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·the·new·working·directory;
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 Change·the·current·working·directory·to·dir.15 Change·the·current·working·directory·to·dir.
16 i1·:·dir·=·temporaryFileName()16 i1·:·dir·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3680876-0/017 o1·=·/tmp/M2-2656981-0/0
18 i2·:·makeDirectory·dir18 i2·:·makeDirectory·dir
  
19 o2·=·/tmp/M2-3680876-0/019 o2·=·/tmp/M2-2656981-0/0
20 i3·:·changeDirectory·dir20 i3·:·changeDirectory·dir
  
21 o3·=·/tmp/M2-3680876-0/0/21 o3·=·/tmp/M2-2656981-0/0/
22 i4·:·currentDirectory()22 i4·:·currentDirectory()
  
23 o4·=·/tmp/M2-3680876-0/0/23 o4·=·/tmp/M2-2656981-0/0/
24 If·dir·is·omitted,·then·the·current·working·directory·is·changed·to·the·user's24 If·dir·is·omitted,·then·the·current·working·directory·is·changed·to·the·user's
25 home·directory.25 home·directory.
26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·current·working·directory27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·current·working·directory
28 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*28 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
29 The·object·_\x8c_\x8h_\x8a_\x8n_\x8g_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.29 The·object·_\x8c_\x8h_\x8a_\x8n_\x8g_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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42 ····</div>42 ····</div>
43 <hr>····<div>43 <hr>····<div>
44 ······<h1>communicating·with·programs</h1>44 ······<h1>communicating·with·programs</h1>
45 ······<div>45 ······<div>
46 The·most·naive·way·to·interact·with·another·program·is·simply·to·run·it,·let·it·communicate·directly·with·the·user,·and·wait·for·it·to·finish.··This·is·done·with·the·<a·title="run·an·external·command"·href="_run.html">run</a>·command.········<table·class="examples">46 The·most·naive·way·to·interact·with·another·program·is·simply·to·run·it,·let·it·communicate·directly·with·the·user,·and·wait·for·it·to·finish.··This·is·done·with·the·<a·title="run·an·external·command"·href="_run.html">run</a>·command.········<table·class="examples">
47 ··········<tr>47 ··········<tr>
48 <td>··············<pre><code·class="language-macaulay2">i1·:·run·&quot;uname·-a&quot;48 <td>··············<pre><code·class="language-macaulay2">i1·:·run·&quot;uname·-a&quot;
49 Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.1.140-1·(2025-05-22)·x86_64·GNU/Linux49 Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
  
50 o1·=·0</code></pre>50 o1·=·0</code></pre>
51 </td>··········</tr>51 </td>··········</tr>
52 ········</table>52 ········</table>
53 To·run·a·program·and·provide·it·with·input,·one·way·is·use·the·operator·<a·title="a·binary·operator·(file·output,·...)"·href="__lt_lt.html">&lt;&lt;</a>,·with·a·file·name·whose·first·character·is·an·exclamation·point;·the·rest·of·the·file·name·will·be·taken·as·the·command·to·run,·as·in·the·following·example.········<table·class="examples">53 To·run·a·program·and·provide·it·with·input,·one·way·is·use·the·operator·<a·title="a·binary·operator·(file·output,·...)"·href="__lt_lt.html">&lt;&lt;</a>,·with·a·file·name·whose·first·character·is·an·exclamation·point;·the·rest·of·the·file·name·will·be·taken·as·the·command·to·run,·as·in·the·following·example.········<table·class="examples">
54 ··········<tr>54 ··········<tr>
55 <td>··············<pre><code·class="language-macaulay2">i2·:·&quot;!grep·a&quot;·&lt;&lt;·&quot;·ba·\n·bc·\n·ad·\n·ef·\n&quot;·&lt;&lt;·close55 <td>··············<pre><code·class="language-macaulay2">i2·:·&quot;!grep·a&quot;·&lt;&lt;·&quot;·ba·\n·bc·\n·ad·\n·ef·\n&quot;·&lt;&lt;·close
Offset 62, 17 lines modifiedOffset 62, 17 lines modified
62 o2·:·File</code></pre>62 o2·:·File</code></pre>
63 </td>··········</tr>63 </td>··········</tr>
64 ········</table>64 ········</table>
65 More·often,·one·wants·to·write·Macaulay2·code·to·obtain·and·manipulate·the·output·from·the·other·program.··If·the·program·requires·no·input·data,·then·we·can·use·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>·with·a·file·name·whose·first·character·is·an·exclamation·point.··In·the·following·example,·we·also·peek·at·the·string·to·see·whether·it·includes·a·newline·character.········<table·class="examples">65 More·often,·one·wants·to·write·Macaulay2·code·to·obtain·and·manipulate·the·output·from·the·other·program.··If·the·program·requires·no·input·data,·then·we·can·use·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>·with·a·file·name·whose·first·character·is·an·exclamation·point.··In·the·following·example,·we·also·peek·at·the·string·to·see·whether·it·includes·a·newline·character.········<table·class="examples">
66 ··········<tr>66 ··········<tr>
67 <td>··············<pre><code·class="language-macaulay2">i3·:·peek·get·&quot;!uname·-a&quot;67 <td>··············<pre><code·class="language-macaulay2">i3·:·peek·get·&quot;!uname·-a&quot;
  
68 o3·=·&quot;Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian68 o3·=·&quot;Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC
69 ·····&quot;69 ·····&quot;
70 ·····6.1.140-1·(2025-05-22)·x86_64·GNU/Linux</code></pre>70 ·····Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux</code></pre>
71 </td>··········</tr>71 </td>··········</tr>
72 ········</table>72 ········</table>
73 Bidirectional·communication·with·a·program·is·also·possible.··We·use·<a·title="open·an·input·output·file"·href="_open__In__Out.html">openInOut</a>·to·create·a·file·that·serves·as·a·bidirectional·connection·to·a·program.··That·file·is·called·an·input·output·file.··In·this·example·we·open·a·connection·to·the·unix·utility·<span·class="tt">egrep</span>·and·use·it·to·locate·the·symbol·names·in·Macaulay2·that·begin·with·<span·class="tt">in</span>.········<table·class="examples">73 Bidirectional·communication·with·a·program·is·also·possible.··We·use·<a·title="open·an·input·output·file"·href="_open__In__Out.html">openInOut</a>·to·create·a·file·that·serves·as·a·bidirectional·connection·to·a·program.··That·file·is·called·an·input·output·file.··In·this·example·we·open·a·connection·to·the·unix·utility·<span·class="tt">egrep</span>·and·use·it·to·locate·the·symbol·names·in·Macaulay2·that·begin·with·<span·class="tt">in</span>.········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i4·:·f·=·openInOut·&quot;!egrep·'^in'&quot;75 <td>··············<pre><code·class="language-macaulay2">i4·:·f·=·openInOut·&quot;!egrep·'^in'&quot;
  
76 o4·=·!egrep·'^in'76 o4·=·!egrep·'^in'
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4 _\x8n_\x8e_\x8x_\x8t·|·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·|·_\x8f_\x8o_\x8r_\x8w_\x8a_\x8r_\x8d·|·_\x8b_\x8a_\x8c_\x8k_\x8w_\x8a_\x8r_\x8d·|·_\x8u_\x8p·|·_\x8i_\x8n_\x8d_\x8e_\x8x·|·_\x8t_\x8o_\x8c4 _\x8n_\x8e_\x8x_\x8t·|·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·|·_\x8f_\x8o_\x8r_\x8w_\x8a_\x8r_\x8d·|·_\x8b_\x8a_\x8c_\x8k_\x8w_\x8a_\x8r_\x8d·|·_\x8u_\x8p·|·_\x8i_\x8n_\x8d_\x8e_\x8x·|·_\x8t_\x8o_\x8c
5 ===============================================================================5 ===============================================================================
6 *\x8**\x8**\x8**\x8**\x8**\x8*·c\x8co\x8om\x8mm\x8mu\x8un\x8ni\x8ic\x8ca\x8at\x8ti\x8in\x8ng\x8g·w\x8wi\x8it\x8th\x8h·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8ms\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*6 *\x8**\x8**\x8**\x8**\x8**\x8*·c\x8co\x8om\x8mm\x8mu\x8un\x8ni\x8ic\x8ca\x8at\x8ti\x8in\x8ng\x8g·w\x8wi\x8it\x8th\x8h·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8ms\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*
7 The·most·naive·way·to·interact·with·another·program·is·simply·to·run·it,·let·it7 The·most·naive·way·to·interact·with·another·program·is·simply·to·run·it,·let·it
8 communicate·directly·with·the·user,·and·wait·for·it·to·finish.·This·is·done8 communicate·directly·with·the·user,·and·wait·for·it·to·finish.·This·is·done
9 with·the·_\x8r_\x8u_\x8n·command.9 with·the·_\x8r_\x8u_\x8n·command.
10 i1·:·run·"uname·-a"10 i1·:·run·"uname·-a"
11 Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian·6.1.140- 
12 1·(2025-05-22)·x86_64·GNU/Linux11 Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian
 12 6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
  
13 o1·=·013 o1·=·0
14 To·run·a·program·and·provide·it·with·input,·one·way·is·use·the·operator·_\x8<_\x8<,14 To·run·a·program·and·provide·it·with·input,·one·way·is·use·the·operator·_\x8<_\x8<,
15 with·a·file·name·whose·first·character·is·an·exclamation·point;·the·rest·of·the15 with·a·file·name·whose·first·character·is·an·exclamation·point;·the·rest·of·the
16 file·name·will·be·taken·as·the·command·to·run,·as·in·the·following·example.16 file·name·will·be·taken·as·the·command·to·run,·as·in·the·following·example.
17 i2·:·"!grep·a"·<<·"·ba·\n·bc·\n·ad·\n·ef·\n"·<<·close17 i2·:·"!grep·a"·<<·"·ba·\n·bc·\n·ad·\n·ef·\n"·<<·close
18 ·ba18 ·ba
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25 More·often,·one·wants·to·write·Macaulay2·code·to·obtain·and·manipulate·the25 More·often,·one·wants·to·write·Macaulay2·code·to·obtain·and·manipulate·the
26 output·from·the·other·program.·If·the·program·requires·no·input·data,·then·we26 output·from·the·other·program.·If·the·program·requires·no·input·data,·then·we
27 can·use·_\x8g_\x8e_\x8t·with·a·file·name·whose·first·character·is·an·exclamation·point.·In27 can·use·_\x8g_\x8e_\x8t·with·a·file·name·whose·first·character·is·an·exclamation·point.·In
28 the·following·example,·we·also·peek·at·the·string·to·see·whether·it·includes·a28 the·following·example,·we·also·peek·at·the·string·to·see·whether·it·includes·a
29 newline·character.29 newline·character.
30 i3·:·peek·get·"!uname·-a"30 i3·:·peek·get·"!uname·-a"
  
31 o3·=·"Linux·infom01-amd64·6.1.0-37-cloud-amd64·#1·SMP·PREEMPT_DYNAMIC·Debian31 o3·=·"Linux·i-capture-the-hostname·6.12.22+bpo-amd64·#1·SMP·PREEMPT_DYNAMIC
32 ·····"32 ·····"
33 ·····6.1.140-1·(2025-05-22)·x86_64·GNU/Linux33 ·····Debian·6.12.22-1~bpo12+1·(2025-04-25)·x86_64·GNU/Linux
34 Bidirectional·communication·with·a·program·is·also·possible.·We·use·_\x8o_\x8p_\x8e_\x8n_\x8I_\x8n_\x8O_\x8u_\x8t34 Bidirectional·communication·with·a·program·is·also·possible.·We·use·_\x8o_\x8p_\x8e_\x8n_\x8I_\x8n_\x8O_\x8u_\x8t
35 to·create·a·file·that·serves·as·a·bidirectional·connection·to·a·program.·That35 to·create·a·file·that·serves·as·a·bidirectional·connection·to·a·program.·That
36 file·is·called·an·input·output·file.·In·this·example·we·open·a·connection·to36 file·is·called·an·input·output·file.·In·this·example·we·open·a·connection·to
37 the·unix·utility·egrep·and·use·it·to·locate·the·symbol·names·in·Macaulay2·that37 the·unix·utility·egrep·and·use·it·to·locate·the·symbol·names·in·Macaulay2·that
38 begin·with·in.38 begin·with·in.
39 i4·:·f·=·openInOut·"!egrep·'^in'"39 i4·:·f·=·openInOut·"!egrep·'^in'"
  
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215 ················ZZ215 ················ZZ
216 o23·:·Ideal·of·----[x..z,·w]216 o23·:·Ideal·of·----[x..z,·w]
217 ···············1277</code></pre>217 ···············1277</code></pre>
218 </td>··········</tr>218 </td>··········</tr>
219 ··········<tr>219 ··········<tr>
220 <td>··············<pre><code·class="language-macaulay2">i24·:·gb·I220 <td>··············<pre><code·class="language-macaulay2">i24·:·gb·I
  
221 ···--·registering·gb·5·at·0x7f651ca08700221 ···--·registering·gb·5·at·0x7f9c78d31700
  
222 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4222 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4
223 ···--·number·of·monomials················=·8223 ···--·number·of·monomials················=·8
224 ···--·#reduction·steps·=·2224 ···--·#reduction·steps·=·2
225 ···--·#spairs·done·=·6225 ···--·#spairs·done·=·6
226 ···--·ncalls·=·0226 ···--·ncalls·=·0
227 ···--·nloop·=·0227 ···--·nloop·=·0
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301 <td>··············<pre><code·class="language-macaulay2">i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});301 <td>··············<pre><code·class="language-macaulay2">i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});
  
302 ··············1······4302 ··············1······4
303 o32·:·Matrix·R··&lt;--·R</code></pre>303 o32·:·Matrix·R··&lt;--·R</code></pre>
304 </td>··········</tr>304 </td>··········</tr>
305 ··········<tr>305 ··········<tr>
306 <td>··············<pre><code·class="language-macaulay2">i33·:·time·betti·gb·f306 <td>··············<pre><code·class="language-macaulay2">i33·:·time·betti·gb·f
307 ·--·used·0.131994s·(cpu);·0.131734s·(thread);·0s·(gc)307 ·--·used·0.271829s·(cpu);·0.271966s·(thread);·0s·(gc)
  
308 ·············0··1308 ·············0··1
309 o33·=·total:·1·53309 o33·=·total:·1·53
310 ··········0:·1··.310 ··········0:·1··.
311 ··········1:·.··.311 ··········1:·.··.
312 ··········2:·.··2312 ··········2:·.··2
313 ··········3:·.··1313 ··········3:·.··1
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339 ············3····5·····8·····9····12·····14····17339 ············3····5·····8·····9····12·····14····17
340 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T340 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T
  
341 o35·:·ZZ[T]</code></pre>341 o35·:·ZZ[T]</code></pre>
342 </td>··········</tr>342 </td>··········</tr>
343 ··········<tr>343 ··········<tr>
344 <td>··············<pre><code·class="language-macaulay2">i36·:·time·betti·gb·f344 <td>··············<pre><code·class="language-macaulay2">i36·:·time·betti·gb·f
345 ·--·used·0.000224087s·(cpu);·0.00225795s·(thread);·0s·(gc)345 ·--·used·0.00399783s·(cpu);·0.00357706s·(thread);·0s·(gc)
  
346 ·············0··1346 ·············0··1
347 o36·=·total:·1·53347 o36·=·total:·1·53
348 ··········0:·1··.348 ··········0:·1··.
349 ··········1:·.··.349 ··········1:·.··.
350 ··········2:·.··2350 ··········2:·.··2
351 ··········3:·.··1351 ··········3:·.··1
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139 o23·=·ideal·(x*y·-·z·,·y··-·w·)139 o23·=·ideal·(x*y·-·z·,·y··-·w·)
  
140 ················ZZ140 ················ZZ
141 o23·:·Ideal·of·----[x..z,·w]141 o23·:·Ideal·of·----[x..z,·w]
142 ···············1277142 ···············1277
143 i24·:·gb·I143 i24·:·gb·I
  
144 ···--·registering·gb·5·at·0x7f651ca08700144 ···--·registering·gb·5·at·0x7f9c78d31700
  
145 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4145 ···--·[gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber·of·(nonminimal)·gb·elements·=·4
146 ···--·number·of·monomials················=·8146 ···--·number·of·monomials················=·8
147 ···--·#reduction·steps·=·2147 ···--·#reduction·steps·=·2
148 ···--·#spairs·done·=·6148 ···--·#spairs·done·=·6
149 ···--·ncalls·=·0149 ···--·ncalls·=·0
150 ···--·nloop·=·0150 ···--·nloop·=·0
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212 o31·:·ZZ[T]212 o31·:·ZZ[T]
213 i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});213 i32·:·f·=·random(R^1,R^{-3,-3,-5,-6});
  
214 ··············1······4214 ··············1······4
215 o32·:·Matrix·R··<--·R215 o32·:·Matrix·R··<--·R
216 i33·:·time·betti·gb·f216 i33·:·time·betti·gb·f
217 ·--·used·0.131994s·(cpu);·0.131734s·(thread);·0s·(gc)217 ·--·used·0.271829s·(cpu);·0.271966s·(thread);·0s·(gc)
  
218 ·············0··1218 ·············0··1
219 o33·=·total:·1·53219 o33·=·total:·1·53
220 ··········0:·1··.220 ··········0:·1··.
221 ··········1:·.··.221 ··········1:·.··.
222 ··········2:·.··2222 ··········2:·.··2
223 ··········3:·.··1223 ··········3:·.··1
Offset 244, 15 lines modifiedOffset 244, 15 lines modified
244 i35·:·poincare·cokernel·f·=·(1-T^3)*(1-T^3)*(1-T^5)*(1-T^6)·--·cache·poincare244 i35·:·poincare·cokernel·f·=·(1-T^3)*(1-T^3)*(1-T^5)*(1-T^6)·--·cache·poincare
  
245 ············3····5·····8·····9····12·····14····17245 ············3····5·····8·····9····12·····14····17
246 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T246 o35·=·1·-·2T··-·T··+·2T··+·2T··-·T···-·2T···+·T
  
247 o35·:·ZZ[T]247 o35·:·ZZ[T]
248 i36·:·time·betti·gb·f248 i36·:·time·betti·gb·f
249 ·--·used·0.000224087s·(cpu);·0.00225795s·(thread);·0s·(gc)249 ·--·used·0.00399783s·(cpu);·0.00357706s·(thread);·0s·(gc)
  
250 ·············0··1250 ·············0··1
251 o36·=·total:·1·53251 o36·=·total:·1·53
252 ··········0:·1··.252 ··········0:·1··.
253 ··········1:·.··.253 ··········1:·.··.
254 ··········2:·.··2254 ··········2:·.··2
255 ··········3:·.··1255 ··········3:·.··1
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_spresolutions.html
    
Offset 92, 16 lines modifiedOffset 92, 16 lines modified
92 ··········&lt;&lt;·res·M·&lt;&lt;·endl·&lt;&lt;·endl;92 ··········&lt;&lt;·res·M·&lt;&lt;·endl·&lt;&lt;·endl;
93 ··········break;93 ··········break;
94 ··········)·else·(94 ··········)·else·(
95 ··········&lt;&lt;·&quot;--·computation·interrupted&quot;·&lt;&lt;·endl;95 ··········&lt;&lt;·&quot;--·computation·interrupted&quot;·&lt;&lt;·endl;
96 ··········status·M.cache.resolution;96 ··········status·M.cache.resolution;
97 ··········&lt;&lt;·&quot;--·continuing·the·computation&quot;·&lt;&lt;·endl;97 ··········&lt;&lt;·&quot;--·continuing·the·computation&quot;·&lt;&lt;·endl;
98 ··········))98 ··········))
 99 ·--·used·1.3894s·(cpu);·0.998677s·(thread);·0s·(gc)
99 ·--·used·1.14265s·(cpu);·1.00262s·(thread);·0s·(gc)100 ·--·used·0.147466s·(cpu);·0.150413s·(thread);·0s·(gc)
100 ·--·used·0.541325s·(cpu);·0.464256s·(thread);·0s·(gc) 
101 --·computation·started:·101 --·computation·started:·
102 --·computation·interrupted102 --·computation·interrupted
103 --·continuing·the·computation103 --·continuing·the·computation
104 --·computation·complete104 --·computation·complete
105 ·4······11······89······122······40105 ·4······11······89······122······40
106 R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R···&lt;--·0106 R··&lt;--·R···&lt;--·R···&lt;--·R····&lt;--·R···&lt;--·0
107 ·········································107 ·········································
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Offset 50, 16 lines modifiedOffset 50, 16 lines modified
50 ··········<<·res·M·<<·endl·<<·endl;50 ··········<<·res·M·<<·endl·<<·endl;
51 ··········break;51 ··········break;
52 ··········)·else·(52 ··········)·else·(
53 ··········<<·"--·computation·interrupted"·<<·endl;53 ··········<<·"--·computation·interrupted"·<<·endl;
54 ··········status·M.cache.resolution;54 ··········status·M.cache.resolution;
55 ··········<<·"--·continuing·the·computation"·<<·endl;55 ··········<<·"--·continuing·the·computation"·<<·endl;
56 ··········))56 ··········))
 57 ·--·used·1.3894s·(cpu);·0.998677s·(thread);·0s·(gc)
57 ·--·used·1.14265s·(cpu);·1.00262s·(thread);·0s·(gc)58 ·--·used·0.147466s·(cpu);·0.150413s·(thread);·0s·(gc)
58 ·--·used·0.541325s·(cpu);·0.464256s·(thread);·0s·(gc) 
59 --·computation·started:59 --·computation·started:
60 --·computation·interrupted60 --·computation·interrupted
61 --·continuing·the·computation61 --·continuing·the·computation
62 --·computation·complete62 --·computation·complete
63 ·4······11······89······122······4063 ·4······11······89······122······40
64 R··<--·R···<--·R···<--·R····<--·R···<--·064 R··<--·R···<--·R···<--·R····<--·R···<--·0
  
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
    
Offset 85, 90 lines modifiedOffset 85, 90 lines modified
85 ······</div>85 ······</div>
86 ······<div>86 ······<div>
87 ········<h2>Description</h2>87 ········<h2>Description</h2>
88 ········<table·class="examples">88 ········<table·class="examples">
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()·|·&quot;/&quot;90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()·|·&quot;/&quot;
  
91 o1·=·/tmp/M2-3689060-0/0/</code></pre>91 o1·=·/tmp/M2-2666599-0/0/</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()·|·&quot;/&quot;94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()·|·&quot;/&quot;
  
95 o2·=·/tmp/M2-3689060-0/1/</code></pre>95 o2·=·/tmp/M2-2666599-0/1/</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i3·:·makeDirectory·(src|&quot;a/&quot;)98 <td>··············<pre><code·class="language-macaulay2">i3·:·makeDirectory·(src|&quot;a/&quot;)
  
99 o3·=·/tmp/M2-3689060-0/0/a/</code></pre>99 o3·=·/tmp/M2-2666599-0/0/a/</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i4·:·makeDirectory·(src|&quot;b/&quot;)102 <td>··············<pre><code·class="language-macaulay2">i4·:·makeDirectory·(src|&quot;b/&quot;)
  
103 o4·=·/tmp/M2-3689060-0/0/b/</code></pre>103 o4·=·/tmp/M2-2666599-0/0/b/</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·makeDirectory·(src|&quot;b/c/&quot;)106 <td>··············<pre><code·class="language-macaulay2">i5·:·makeDirectory·(src|&quot;b/c/&quot;)
  
107 o5·=·/tmp/M2-3689060-0/0/b/c/</code></pre>107 o5·=·/tmp/M2-2666599-0/0/b/c/</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i6·:·src|&quot;a/f&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close110 <td>··············<pre><code·class="language-macaulay2">i6·:·src|&quot;a/f&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
111 o6·=·/tmp/M2-3689060-0/0/a/f111 o6·=·/tmp/M2-2666599-0/0/a/f
  
112 o6·:·File</code></pre>112 o6·:·File</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i7·:·src|&quot;a/g&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close115 <td>··············<pre><code·class="language-macaulay2">i7·:·src|&quot;a/g&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
116 o7·=·/tmp/M2-3689060-0/0/a/g116 o7·=·/tmp/M2-2666599-0/0/a/g
  
117 o7·:·File</code></pre>117 o7·:·File</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i8·:·src|&quot;b/c/g&quot;·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close120 <td>··············<pre><code·class="language-macaulay2">i8·:·src|&quot;b/c/g&quot;·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close
  
121 o8·=·/tmp/M2-3689060-0/0/b/c/g121 o8·=·/tmp/M2-2666599-0/0/b/c/g
  
122 o8·:·File</code></pre>122 o8·:·File</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i9·:·stack·findFiles·src125 <td>··············<pre><code·class="language-macaulay2">i9·:·stack·findFiles·src
  
126 o9·=·/tmp/M2-3689060-0/0/126 o9·=·/tmp/M2-2666599-0/0/
127 ·····/tmp/M2-3689060-0/0/b/127 ·····/tmp/M2-2666599-0/0/b/
128 ·····/tmp/M2-3689060-0/0/b/c/128 ·····/tmp/M2-2666599-0/0/b/c/
129 ·····/tmp/M2-3689060-0/0/b/c/g129 ·····/tmp/M2-2666599-0/0/b/c/g
130 ·····/tmp/M2-3689060-0/0/a/130 ·····/tmp/M2-2666599-0/0/a/
131 ·····/tmp/M2-3689060-0/0/a/f131 ·····/tmp/M2-2666599-0/0/a/g
132 ·····/tmp/M2-3689060-0/0/a/g</code></pre>132 ·····/tmp/M2-2666599-0/0/a/f</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i10·:·copyDirectory(src,dst,Verbose=>true)135 <td>··············<pre><code·class="language-macaulay2">i10·:·copyDirectory(src,dst,Verbose=>true)
136 ·--·copying:·/tmp/M2-3689060-0/0/b/c/g·->·/tmp/M2-3689060-0/1/b/c/g136 ·--·copying:·/tmp/M2-2666599-0/0/b/c/g·->·/tmp/M2-2666599-0/1/b/c/g
137 ·--·copying:·/tmp/M2-3689060-0/0/a/f·->·/tmp/M2-3689060-0/1/a/f137 ·--·copying:·/tmp/M2-2666599-0/0/a/g·->·/tmp/M2-2666599-0/1/a/g
138 ·--·copying:·/tmp/M2-3689060-0/0/a/g·->·/tmp/M2-3689060-0/1/a/g</code></pre>138 ·--·copying:·/tmp/M2-2666599-0/0/a/f·->·/tmp/M2-2666599-0/1/a/f</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)141 <td>··············<pre><code·class="language-macaulay2">i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)
142 ·--·skipping:·/tmp/M2-3689060-0/0/b/c/g·not·newer·than·/tmp/M2-3689060-0/1/b/c/g142 ·--·skipping:·/tmp/M2-2666599-0/0/b/c/g·not·newer·than·/tmp/M2-2666599-0/1/b/c/g
143 ·--·skipping:·/tmp/M2-3689060-0/0/a/f·not·newer·than·/tmp/M2-3689060-0/1/a/f143 ·--·skipping:·/tmp/M2-2666599-0/0/a/g·not·newer·than·/tmp/M2-2666599-0/1/a/g
144 ·--·skipping:·/tmp/M2-3689060-0/0/a/g·not·newer·than·/tmp/M2-3689060-0/1/a/g</code></pre>144 ·--·skipping:·/tmp/M2-2666599-0/0/a/f·not·newer·than·/tmp/M2-2666599-0/1/a/f</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i12·:·stack·findFiles·dst147 <td>··············<pre><code·class="language-macaulay2">i12·:·stack·findFiles·dst
  
148 o12·=·/tmp/M2-3689060-0/1/148 o12·=·/tmp/M2-2666599-0/1/
149 ······/tmp/M2-3689060-0/1/b/149 ······/tmp/M2-2666599-0/1/b/
150 ······/tmp/M2-3689060-0/1/b/c/150 ······/tmp/M2-2666599-0/1/b/c/
151 ······/tmp/M2-3689060-0/1/b/c/g151 ······/tmp/M2-2666599-0/1/b/c/g
152 ······/tmp/M2-3689060-0/1/a/152 ······/tmp/M2-2666599-0/1/a/
153 ······/tmp/M2-3689060-0/1/a/f153 ······/tmp/M2-2666599-0/1/a/g
154 ······/tmp/M2-3689060-0/1/a/g</code></pre>154 ······/tmp/M2-2666599-0/1/a/f</code></pre>
155 </td>··········</tr>155 </td>··········</tr>
156 ··········<tr>156 ··········<tr>
157 <td>··············<pre><code·class="language-macaulay2">i13·:·get·(dst|&quot;b/c/g&quot;)157 <td>··············<pre><code·class="language-macaulay2">i13·:·get·(dst|&quot;b/c/g&quot;)
  
158 o13·=·ho·there</code></pre>158 o13·=·ho·there</code></pre>
159 </td>··········</tr>159 </td>··········</tr>
160 ········</table>160 ········</table>
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Offset 26, 69 lines modifiedOffset 26, 69 lines modified
26 ············individual·file·operations26 ············individual·file·operations
27 ····*·Consequences:27 ····*·Consequences:
28 ··········o·a·copy·of·the·directory·tree·rooted·at·src·is·created,·rooted·at28 ··········o·a·copy·of·the·directory·tree·rooted·at·src·is·created,·rooted·at
29 ············dst29 ············dst
30 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
31 i1·:·src·=·temporaryFileName()·|·"/"31 i1·:·src·=·temporaryFileName()·|·"/"
  
32 o1·=·/tmp/M2-3689060-0/0/32 o1·=·/tmp/M2-2666599-0/0/
33 i2·:·dst·=·temporaryFileName()·|·"/"33 i2·:·dst·=·temporaryFileName()·|·"/"
  
34 o2·=·/tmp/M2-3689060-0/1/34 o2·=·/tmp/M2-2666599-0/1/
35 i3·:·makeDirectory·(src|"a/")35 i3·:·makeDirectory·(src|"a/")
  
36 o3·=·/tmp/M2-3689060-0/0/a/36 o3·=·/tmp/M2-2666599-0/0/a/
37 i4·:·makeDirectory·(src|"b/")37 i4·:·makeDirectory·(src|"b/")
  
38 o4·=·/tmp/M2-3689060-0/0/b/38 o4·=·/tmp/M2-2666599-0/0/b/
39 i5·:·makeDirectory·(src|"b/c/")39 i5·:·makeDirectory·(src|"b/c/")
  
40 o5·=·/tmp/M2-3689060-0/0/b/c/40 o5·=·/tmp/M2-2666599-0/0/b/c/
41 i6·:·src|"a/f"·<<·"hi·there"·<<·close41 i6·:·src|"a/f"·<<·"hi·there"·<<·close
  
42 o6·=·/tmp/M2-3689060-0/0/a/f42 o6·=·/tmp/M2-2666599-0/0/a/f
  
43 o6·:·File43 o6·:·File
44 i7·:·src|"a/g"·<<·"hi·there"·<<·close44 i7·:·src|"a/g"·<<·"hi·there"·<<·close
  
45 o7·=·/tmp/M2-3689060-0/0/a/g45 o7·=·/tmp/M2-2666599-0/0/a/g
  
46 o7·:·File46 o7·:·File
47 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close47 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close
  
48 o8·=·/tmp/M2-3689060-0/0/b/c/g48 o8·=·/tmp/M2-2666599-0/0/b/c/g
  
49 o8·:·File49 o8·:·File
50 i9·:·stack·findFiles·src50 i9·:·stack·findFiles·src
  
51 o9·=·/tmp/M2-3689060-0/0/51 o9·=·/tmp/M2-2666599-0/0/
52 ·····/tmp/M2-3689060-0/0/b/52 ·····/tmp/M2-2666599-0/0/b/
53 ·····/tmp/M2-3689060-0/0/b/c/53 ·····/tmp/M2-2666599-0/0/b/c/
54 ·····/tmp/M2-3689060-0/0/b/c/g54 ·····/tmp/M2-2666599-0/0/b/c/g
55 ·····/tmp/M2-3689060-0/0/a/55 ·····/tmp/M2-2666599-0/0/a/
56 ·····/tmp/M2-3689060-0/0/a/f 
57 ·····/tmp/M2-3689060-0/0/a/g56 ·····/tmp/M2-2666599-0/0/a/g
 57 ·····/tmp/M2-2666599-0/0/a/f
58 i10·:·copyDirectory(src,dst,Verbose=>true)58 i10·:·copyDirectory(src,dst,Verbose=>true)
59 ·--·copying:·/tmp/M2-3689060-0/0/b/c/g·->·/tmp/M2-3689060-0/1/b/c/g59 ·--·copying:·/tmp/M2-2666599-0/0/b/c/g·->·/tmp/M2-2666599-0/1/b/c/g
60 ·--·copying:·/tmp/M2-3689060-0/0/a/f·->·/tmp/M2-3689060-0/1/a/f 
61 ·--·copying:·/tmp/M2-3689060-0/0/a/g·->·/tmp/M2-3689060-0/1/a/g60 ·--·copying:·/tmp/M2-2666599-0/0/a/g·->·/tmp/M2-2666599-0/1/a/g
 61 ·--·copying:·/tmp/M2-2666599-0/0/a/f·->·/tmp/M2-2666599-0/1/a/f
62 i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)62 i11·:·copyDirectory(src,dst,Verbose=>true,UpdateOnly·=>·true)
63 ·--·skipping:·/tmp/M2-3689060-0/0/b/c/g·not·newer·than·/tmp/M2-3689060-0/1/b/c/63 ·--·skipping:·/tmp/M2-2666599-0/0/b/c/g·not·newer·than·/tmp/M2-2666599-0/1/b/c/
64 g64 g
65 ·--·skipping:·/tmp/M2-3689060-0/0/a/f·not·newer·than·/tmp/M2-3689060-0/1/a/f 
66 ·--·skipping:·/tmp/M2-3689060-0/0/a/g·not·newer·than·/tmp/M2-3689060-0/1/a/g65 ·--·skipping:·/tmp/M2-2666599-0/0/a/g·not·newer·than·/tmp/M2-2666599-0/1/a/g
 66 ·--·skipping:·/tmp/M2-2666599-0/0/a/f·not·newer·than·/tmp/M2-2666599-0/1/a/f
67 i12·:·stack·findFiles·dst67 i12·:·stack·findFiles·dst
  
68 o12·=·/tmp/M2-3689060-0/1/68 o12·=·/tmp/M2-2666599-0/1/
69 ······/tmp/M2-3689060-0/1/b/69 ······/tmp/M2-2666599-0/1/b/
70 ······/tmp/M2-3689060-0/1/b/c/70 ······/tmp/M2-2666599-0/1/b/c/
71 ······/tmp/M2-3689060-0/1/b/c/g71 ······/tmp/M2-2666599-0/1/b/c/g
72 ······/tmp/M2-3689060-0/1/a/72 ······/tmp/M2-2666599-0/1/a/
73 ······/tmp/M2-3689060-0/1/a/f 
74 ······/tmp/M2-3689060-0/1/a/g73 ······/tmp/M2-2666599-0/1/a/g
 74 ······/tmp/M2-2666599-0/1/a/f
75 i13·:·get·(dst|"b/c/g")75 i13·:·get·(dst|"b/c/g")
  
76 o13·=·ho·there76 o13·=·ho·there
77 Now·we·remove·the·files·and·directories·we·created.77 Now·we·remove·the·files·and·directories·we·created.
78 i14·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d78 i14·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d
  
79 o14·=·rm79 o14·=·rm
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81 ······</div>81 ······</div>
82 ······<div>82 ······<div>
83 ········<h2>Description</h2>83 ········<h2>Description</h2>
84 ········<table·class="examples">84 ········<table·class="examples">
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()86 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()
  
87 o1·=·/tmp/M2-3683335-0/0</code></pre>87 o1·=·/tmp/M2-2663535-0/0</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()90 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()
  
91 o2·=·/tmp/M2-3683335-0/1</code></pre>91 o2·=·/tmp/M2-2663535-0/1</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i3·:·src·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close94 <td>··············<pre><code·class="language-macaulay2">i3·:·src·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
95 o3·=·/tmp/M2-3683335-0/095 o3·=·/tmp/M2-2663535-0/0
  
96 o3·:·File</code></pre>96 o3·:·File</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i4·:·copyFile(src,dst,Verbose=>true)99 <td>··············<pre><code·class="language-macaulay2">i4·:·copyFile(src,dst,Verbose=>true)
100 ·--·copying:·/tmp/M2-3683335-0/0·->·/tmp/M2-3683335-0/1</code></pre>100 ·--·copying:·/tmp/M2-2663535-0/0·->·/tmp/M2-2663535-0/1</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i5·:·get·dst103 <td>··············<pre><code·class="language-macaulay2">i5·:·get·dst
  
104 o5·=·hi·there</code></pre>104 o5·=·hi·there</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)107 <td>··············<pre><code·class="language-macaulay2">i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
108 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/1</code></pre>108 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i7·:·src·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close111 <td>··············<pre><code·class="language-macaulay2">i7·:·src·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close
  
112 o7·=·/tmp/M2-3683335-0/0112 o7·=·/tmp/M2-2663535-0/0
  
113 o7·:·File</code></pre>113 o7·:·File</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)116 <td>··············<pre><code·class="language-macaulay2">i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
117 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/1</code></pre>117 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i9·:·get·dst120 <td>··············<pre><code·class="language-macaulay2">i9·:·get·dst
  
121 o9·=·hi·there</code></pre>121 o9·=·hi·there</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
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19 ··········o·Verbose·=>·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·default·value·false,·whether·to·report19 ··········o·Verbose·=>·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·default·value·false,·whether·to·report
20 ············individual·file·operations20 ············individual·file·operations
21 ····*·Consequences:21 ····*·Consequences:
22 ··········o·the·file·may·be·copied22 ··········o·the·file·may·be·copied
23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
24 i1·:·src·=·temporaryFileName()24 i1·:·src·=·temporaryFileName()
  
25 o1·=·/tmp/M2-3683335-0/025 o1·=·/tmp/M2-2663535-0/0
26 i2·:·dst·=·temporaryFileName()26 i2·:·dst·=·temporaryFileName()
  
27 o2·=·/tmp/M2-3683335-0/127 o2·=·/tmp/M2-2663535-0/1
28 i3·:·src·<<·"hi·there"·<<·close28 i3·:·src·<<·"hi·there"·<<·close
  
29 o3·=·/tmp/M2-3683335-0/029 o3·=·/tmp/M2-2663535-0/0
  
30 o3·:·File30 o3·:·File
31 i4·:·copyFile(src,dst,Verbose=>true)31 i4·:·copyFile(src,dst,Verbose=>true)
32 ·--·copying:·/tmp/M2-3683335-0/0·->·/tmp/M2-3683335-0/132 ·--·copying:·/tmp/M2-2663535-0/0·->·/tmp/M2-2663535-0/1
33 i5·:·get·dst33 i5·:·get·dst
  
34 o5·=·hi·there34 o5·=·hi·there
35 i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)35 i6·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
36 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/136 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1
37 i7·:·src·<<·"ho·there"·<<·close37 i7·:·src·<<·"ho·there"·<<·close
  
38 o7·=·/tmp/M2-3683335-0/038 o7·=·/tmp/M2-2663535-0/0
  
39 o7·:·File39 o7·:·File
40 i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)40 i8·:·copyFile(src,dst,Verbose=>true,UpdateOnly·=>·true)
41 ·--·skipping:·/tmp/M2-3683335-0/0·not·newer·than·/tmp/M2-3683335-0/141 ·--·skipping:·/tmp/M2-2663535-0/0·not·newer·than·/tmp/M2-2663535-0/1
42 i9·:·get·dst42 i9·:·get·dst
  
43 o9·=·hi·there43 o9·=·hi·there
44 i10·:·removeFile·src44 i10·:·removeFile·src
45 i11·:·removeFile·dst45 i11·:·removeFile·dst
46 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*46 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
47 ····*·_\x8c_\x8o_\x8p_\x8y_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y47 ····*·_\x8c_\x8o_\x8p_\x8y_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y
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61 ······</div>61 ······</div>
62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<table·class="examples">64 ········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·t1·=·cpuTime()66 <td>··············<pre><code·class="language-macaulay2">i1·:·t1·=·cpuTime()
  
67 o1·=·171.45965111867 o1·=·258.3199765700001
  
68 o1·:·RR·(of·precision·53)</code></pre>68 o1·:·RR·(of·precision·53)</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ··········<tr>70 ··········<tr>
71 <td>··············<pre><code·class="language-macaulay2">i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;</code></pre>71 <td>··············<pre><code·class="language-macaulay2">i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;</code></pre>
72 </td>··········</tr>72 </td>··········</tr>
73 ··········<tr>73 ··········<tr>
74 <td>··············<pre><code·class="language-macaulay2">i3·:·t2·=·cpuTime()74 <td>··············<pre><code·class="language-macaulay2">i3·:·t2·=·cpuTime()
  
75 o3·=·172.27195301275 o3·=·259.871625954
  
76 o3·:·RR·(of·precision·53)</code></pre>76 o3·:·RR·(of·precision·53)</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i4·:·t2-t179 <td>··············<pre><code·class="language-macaulay2">i4·:·t2-t1
  
80 o4·=·.812301894000000980 o4·=·1.551649383999973
  
81 o4·:·RR·(of·precision·53)</code></pre>81 o4·:·RR·(of·precision·53)</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ········</table>83 ········</table>
84 ······</div>84 ······</div>
85 ······<div>85 ······<div>
86 ········<h2>See·also</h2>86 ········<h2>See·also</h2>
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9 ············cpuTime()9 ············cpuTime()
10 ····*·Outputs:10 ····*·Outputs:
11 ··········o·a·_\x8r_\x8e_\x8a_\x8l_\x8·_\x8n_\x8u_\x8m_\x8b_\x8e_\x8r,·the·number·of·seconds·of·cpu·time·used·since·the11 ··········o·a·_\x8r_\x8e_\x8a_\x8l_\x8·_\x8n_\x8u_\x8m_\x8b_\x8e_\x8r,·the·number·of·seconds·of·cpu·time·used·since·the
12 ············program·was·started12 ············program·was·started
13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
14 i1·:·t1·=·cpuTime()14 i1·:·t1·=·cpuTime()
  
15 o1·=·171.45965111815 o1·=·258.3199765700001
  
16 o1·:·RR·(of·precision·53)16 o1·:·RR·(of·precision·53)
17 i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;17 i2·:·for·i·from·0·to·1000000·do·223131321321*324234324324;
18 i3·:·t2·=·cpuTime()18 i3·:·t2·=·cpuTime()
  
19 o3·=·172.27195301219 o3·=·259.871625954
  
20 o3·:·RR·(of·precision·53)20 o3·:·RR·(of·precision·53)
21 i4·:·t2-t121 i4·:·t2-t1
  
22 o4·=·.812301894000000922 o4·=·1.551649383999973
  
23 o4·:·RR·(of·precision·53)23 o4·:·RR·(of·precision·53)
24 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
25 ····*·_\x8t_\x8i_\x8m_\x8e·--·time·a·computation25 ····*·_\x8t_\x8i_\x8m_\x8e·--·time·a·computation
26 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation26 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation
27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·--·get·the·current·time27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·--·get·the·current·time
28 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*28 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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61 ······</div>61 ······</div>
62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<table·class="examples">64 ········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·currentTime()66 <td>··············<pre><code·class="language-macaulay2">i1·:·currentTime()
  
67 o1·=·1748961727</code></pre>67 o1·=·1783379688</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ········</table>69 ········</table>
70 ········<p>We·can·compute,·roughly,·how·many·years·ago·the·epoch·began·as·follows.</p>70 ········<p>We·can·compute,·roughly,·how·many·years·ago·the·epoch·began·as·follows.</p>
71 ········<table·class="examples">71 ········<table·class="examples">
72 ··········<tr>72 ··········<tr>
73 <td>··············<pre><code·class="language-macaulay2">i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)73 <td>··············<pre><code·class="language-macaulay2">i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)
  
74 o2·=·55.4223908253243374 o2·=·56.51305259139096
  
75 o2·:·RR·(of·precision·53)</code></pre>75 o2·:·RR·(of·precision·53)</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ········</table>77 ········</table>
78 ········<p>We·can·also·compute·how·many·months·account·for·the·fractional·part·of·that·number.</p>78 ········<p>We·can·also·compute·how·many·months·account·for·the·fractional·part·of·that·number.</p>
79 ········<table·class="examples">79 ········<table·class="examples">
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·12·*·(oo·-·floor·oo)81 <td>··············<pre><code·class="language-macaulay2">i3·:·12·*·(oo·-·floor·oo)
  
82 o3·=·5.06868990389193682 o3·=·6.156631096691513
  
83 o3·:·RR·(of·precision·53)</code></pre>83 o3·:·RR·(of·precision·53)</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ········</table>85 ········</table>
86 ········<p>Compare·that·to·the·current·date,·available·from·a·standard·Unix·command.</p>86 ········<p>Compare·that·to·the·current·date,·available·from·a·standard·Unix·command.</p>
87 ········<table·class="examples">87 ········<table·class="examples">
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i4·:·run·&quot;date&quot;89 <td>··············<pre><code·class="language-macaulay2">i4·:·run·&quot;date&quot;
90 Tue·Jun··3·02:42:07·-12·202590 Tue·Jul··7·13:14:48·+14·2026
  
91 o4·=·0</code></pre>91 o4·=·0</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ········</table>93 ········</table>
94 ······</div>94 ······</div>
95 ······<div·class="waystouse">95 ······<div·class="waystouse">
96 ········<h2>For·the·programmer</h2>96 ········<h2>For·the·programmer</h2>
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9 ············currentTime()9 ············currentTime()
10 ····*·Outputs:10 ····*·Outputs:
11 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·current·time,·in·seconds·since·00:00:00·1970-01-0111 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·current·time,·in·seconds·since·00:00:00·1970-01-01
12 ············UTC,·the·beginning·of·the·epoch12 ············UTC,·the·beginning·of·the·epoch
13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
14 i1·:·currentTime()14 i1·:·currentTime()
  
15 o1·=·174896172715 o1·=·1783379688
16 We·can·compute,·roughly,·how·many·years·ago·the·epoch·began·as·follows.16 We·can·compute,·roughly,·how·many·years·ago·the·epoch·began·as·follows.
17 i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)17 i2·:·currentTime()·/(·(365·+·97./400)·*·24·*·60·*·60·)
  
18 o2·=·55.4223908253243318 o2·=·56.51305259139096
  
19 o2·:·RR·(of·precision·53)19 o2·:·RR·(of·precision·53)
20 We·can·also·compute·how·many·months·account·for·the·fractional·part·of·that20 We·can·also·compute·how·many·months·account·for·the·fractional·part·of·that
21 number.21 number.
22 i3·:·12·*·(oo·-·floor·oo)22 i3·:·12·*·(oo·-·floor·oo)
  
23 o3·=·5.06868990389193623 o3·=·6.156631096691513
  
24 o3·:·RR·(of·precision·53)24 o3·:·RR·(of·precision·53)
25 Compare·that·to·the·current·date,·available·from·a·standard·Unix·command.25 Compare·that·to·the·current·date,·available·from·a·standard·Unix·command.
26 i4·:·run·"date"26 i4·:·run·"date"
27 Tue·Jun··3·02:42:07·-12·202527 Tue·Jul··7·13:14:48·+14·2026
  
28 o4·=·028 o4·=·0
29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
30 The·object·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.30 The·object·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 <span·class="tt">elapsedTiming·e</span>·evaluates·<span·class="tt">e</span>·and·returns·a·list·of·type·<a·title="the·class·of·all·timing·results"·href="___Time.html">Time</a>·of·the·form·<span·class="tt">{t,v}</span>,·where·<span·class="tt">t</span>·is·the·number·of·seconds·of·time·elapsed,·and·<span·class="tt">v</span>·is·the·value·of·the·expression.········<p></p>47 <span·class="tt">elapsedTiming·e</span>·evaluates·<span·class="tt">e</span>·and·returns·a·list·of·type·<a·title="the·class·of·all·timing·results"·href="___Time.html">Time</a>·of·the·form·<span·class="tt">{t,v}</span>,·where·<span·class="tt">t</span>·is·the·number·of·seconds·of·time·elapsed,·and·<span·class="tt">v</span>·is·the·value·of·the·expression.········<p></p>
48 The·default·method·for·printing·such·timing·results·is·to·display·the·timing·separately·in·a·comment·below·the·computed·value.········<table·class="examples">48 The·default·method·for·printing·such·timing·results·is·to·display·the·timing·separately·in·a·comment·below·the·computed·value.········<table·class="examples">
49 ··········<tr>49 ··········<tr>
50 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTiming·sleep·150 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTiming·sleep·1
  
51 o1·=·051 o1·=·0
52 ·····--·1.00014·seconds52 ·····--·1.00019·seconds
  
53 o1·:·Time</code></pre>53 o1·:·Time</code></pre>
54 </td>··········</tr>54 </td>··········</tr>
55 ··········<tr>55 ··········<tr>
56 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·oo56 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·oo
  
57 o2·=·Time{1.00014,·0}</code></pre>57 o2·=·Time{1.00019,·0}</code></pre>
58 </td>··········</tr>58 </td>··········</tr>
59 ········</table>59 ········</table>
60 ······</div>60 ······</div>
61 ······<div>61 ······<div>
62 ········<h2>See·also</h2>62 ········<h2>See·also</h2>
63 ········<ul>63 ········<ul>
64 ··········<li>64 ··········<li>
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10 where·t·is·the·number·of·seconds·of·time·elapsed,·and·v·is·the·value·of·the10 where·t·is·the·number·of·seconds·of·time·elapsed,·and·v·is·the·value·of·the
11 expression.11 expression.
12 The·default·method·for·printing·such·timing·results·is·to·display·the·timing12 The·default·method·for·printing·such·timing·results·is·to·display·the·timing
13 separately·in·a·comment·below·the·computed·value.13 separately·in·a·comment·below·the·computed·value.
14 i1·:·elapsedTiming·sleep·114 i1·:·elapsedTiming·sleep·1
  
15 o1·=·015 o1·=·0
16 ·····--·1.00014·seconds16 ·····--·1.00019·seconds
  
17 o1·:·Time17 o1·:·Time
18 i2·:·peek·oo18 i2·:·peek·oo
  
19 o2·=·Time{1.00014,·0}19 o2·=·Time{1.00019,·0}
20 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*20 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
21 ····*·_\x8T_\x8i_\x8m_\x8e·--·the·class·of·all·timing·results21 ····*·_\x8T_\x8i_\x8m_\x8e·--·the·class·of·all·timing·results
22 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed22 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed
23 ····*·_\x8c_\x8p_\x8u_\x8T_\x8i_\x8m_\x8e·--·seconds·of·cpu·time·used·since·Macaulay2·began23 ····*·_\x8c_\x8p_\x8u_\x8T_\x8i_\x8m_\x8e·--·seconds·of·cpu·time·used·since·Macaulay2·began
24 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation24 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation
25 ····*·_\x8t_\x8i_\x8m_\x8e·--·time·a·computation25 ····*·_\x8t_\x8i_\x8m_\x8e·--·time·a·computation
26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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53 ···············3···················3·····2···············353 ···············3···················3·····2···············3
54 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)54 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)
  
55 o2·:·Ideal·of·R</code></pre>55 o2·:·Ideal·of·R</code></pre>
56 </td>··········</tr>56 </td>··········</tr>
57 ··········<tr>57 ··········<tr>
58 <td>··············<pre><code·class="language-macaulay2">i3·:·time·leadTerm·gens·gb·I58 <td>··············<pre><code·class="language-macaulay2">i3·:·time·leadTerm·gens·gb·I
59 ·--·used·0.0945913s·(cpu);·0.0945912s·(thread);·0s·(gc)59 ·--·used·0.141724s·(cpu);·0.141723s·(thread);·0s·(gc)
  
60 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy460 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy4
61 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
62 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z262 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z2
63 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
64 ·····267076255345488270sy3z4·5256861933965245618410txyz664 ·····267076255345488270sy3z4·5256861933965245618410txyz6
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
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140 ···············3···················3·····2···············3140 ···············3···················3·····2···············3
141 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)141 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)
  
142 o7·:·Ideal·of·R</code></pre>142 o7·:·Ideal·of·R</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i8·:·time·G·=·eliminate(I,{s,t})145 <td>··············<pre><code·class="language-macaulay2">i8·:·time·G·=·eliminate(I,{s,t})
146 ·--·used·0.0997789s·(cpu);·0.0997815s·(thread);·0s·(gc)146 ·--·used·0.147031s·(cpu);·0.14662s·(thread);·0s·(gc)
  
147 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···147 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···
148 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+148 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
149 ·····------------------------------------------------------------------------149 ·····------------------------------------------------------------------------
150 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·150 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·
151 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z151 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
152 ·····------------------------------------------------------------------------152 ·····------------------------------------------------------------------------
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215 ··········<tr>215 ··········<tr>
216 <td>··············<pre><code·class="language-macaulay2">i11·:·I1·=·substitute(I,R1);216 <td>··············<pre><code·class="language-macaulay2">i11·:·I1·=·substitute(I,R1);
  
217 o11·:·Ideal·of·R1</code></pre>217 o11·:·Ideal·of·R1</code></pre>
218 </td>··········</tr>218 </td>··········</tr>
219 ··········<tr>219 ··········<tr>
220 <td>··············<pre><code·class="language-macaulay2">i12·:·time·G·=·eliminate(I1,{s,t})220 <td>··············<pre><code·class="language-macaulay2">i12·:·time·G·=·eliminate(I1,{s,t})
221 ·--·used·0.0325054s·(cpu);·0.0325089s·(thread);·0s·(gc)221 ·--·used·0.0468117s·(cpu);·0.0468164s·(thread);·0s·(gc)
  
222 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7··222 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7··
223 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-223 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-
224 ······-----------------------------------------------------------------------224 ······-----------------------------------------------------------------------
225 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2··225 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2··
226 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-226 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-
227 ······-----------------------------------------------------------------------227 ······-----------------------------------------------------------------------
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297 ···················3·············3·····2·········3297 ···················3·············3·····2·········3
298 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})298 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})
  
299 o16·:·RingMap·A·&lt;--·B</code></pre>299 o16·:·RingMap·A·&lt;--·B</code></pre>
300 </td>··········</tr>300 </td>··········</tr>
301 ··········<tr>301 ··········<tr>
302 <td>··············<pre><code·class="language-macaulay2">i17·:·time·G·=·kernel·F302 <td>··············<pre><code·class="language-macaulay2">i17·:·time·G·=·kernel·F
303 ·--·used·0.249418s·(cpu);·0.117309s·(thread);·0s·(gc)303 ·--·used·0.365832s·(cpu);·0.159494s·(thread);·0s·(gc)
  
304 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···304 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8···
305 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+305 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
306 ······-----------------------------------------------------------------------306 ······-----------------------------------------------------------------------
307 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·307 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7·
308 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z308 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
309 ······-----------------------------------------------------------------------309 ······-----------------------------------------------------------------------
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372 o19·=·R372 o19·=·R
  
373 o19·:·PolynomialRing</code></pre>373 o19·:·PolynomialRing</code></pre>
374 </td>··········</tr>374 </td>··········</tr>
375 ··········<tr>375 ··········<tr>
376 <td>··············<pre><code·class="language-macaulay2">i20·:·time·f1·=·resultant(I_0,I_2,s)376 <td>··············<pre><code·class="language-macaulay2">i20·:·time·f1·=·resultant(I_0,I_2,s)
377 ·--·used·0.00127094s·(cpu);·0.00126649s·(thread);·0s·(gc)377 ·--·used·0.00182866s·(cpu);·0.00182897s·(thread);·0s·(gc)
  
378 ·········9····9······7····3378 ·········9····9······7····3
379 o20·=·x*t··-·t··-·z*t··-·z379 o20·=·x*t··-·t··-·z*t··-·z
  
380 o20·:·R</code></pre>380 o20·:·R</code></pre>
381 </td>··········</tr>381 </td>··········</tr>
382 ··········<tr>382 ··········<tr>
383 <td>··············<pre><code·class="language-macaulay2">i21·:·time·f2·=·resultant(I_1,f1,t)383 <td>··············<pre><code·class="language-macaulay2">i21·:·time·f2·=·resultant(I_1,f1,t)
384 ·--·used·0.0285206s·(cpu);·0.0285309s·(thread);·0s·(gc)384 ·--·used·0.0445255s·(cpu);·0.0445387s·(thread);·0s·(gc)
  
385 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2··385 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2··
386 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+386 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+
387 ······-----------------------------------------------------------------------387 ······-----------------------------------------------------------------------
388 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8···388 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8···
389 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+389 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+
390 ······-----------------------------------------------------------------------390 ······-----------------------------------------------------------------------
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13 i2·:·I·=·ideal(x-s^3-s*t-1,·y-t^3-3*t^2-t,·z-s*t^3)13 i2·:·I·=·ideal(x-s^3-s*t-1,·y-t^3-3*t^2-t,·z-s*t^3)
  
14 ···············3···················3·····2···············314 ···············3···················3·····2···············3
15 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)15 o2·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··-·t·+·y,·-·s*t··+·z)
  
16 o2·:·Ideal·of·R16 o2·:·Ideal·of·R
17 i3·:·time·leadTerm·gens·gb·I17 i3·:·time·leadTerm·gens·gb·I
18 ·--·used·0.0945913s·(cpu);·0.0945912s·(thread);·0s·(gc)18 ·--·used·0.141724s·(cpu);·0.141723s·(thread);·0s·(gc)
  
19 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy419 o3·=·|·x3y9·5148txy3·108729sxy2z2·sy4z·46644741sxy3z·143sy5·6sxy4
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z221 ·····563515116021sx2y3·4374txy2z3·612704350498473090tx2yz3·217458ty4z2
22 ·····------------------------------------------------------------------------22 ·····------------------------------------------------------------------------
23 ·····267076255345488270sy3z4·5256861933965245618410txyz623 ·····267076255345488270sy3z4·5256861933965245618410txyz6
24 ·····------------------------------------------------------------------------24 ·····------------------------------------------------------------------------
Offset 89, 15 lines modifiedOffset 89, 15 lines modified
89 i7·:·I·=·ideal(x-s^3-s*t-1,·y-t^3-3*t^2-t,·z-s*t^3)89 i7·:·I·=·ideal(x-s^3-s*t-1,·y-t^3-3*t^2-t,·z-s*t^3)
  
90 ···············3···················3·····2···············390 ···············3···················3·····2···············3
91 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)91 o7·=·ideal·(-·s··-·s*t·+·x·-·1,·-·t··-·3t··+·y·-·t,·-·s*t··+·z)
  
92 o7·:·Ideal·of·R92 o7·:·Ideal·of·R
93 i8·:·time·G·=·eliminate(I,{s,t})93 i8·:·time·G·=·eliminate(I,{s,t})
94 ·--·used·0.0997789s·(cpu);·0.0997815s·(thread);·0s·(gc)94 ·--·used·0.147031s·(cpu);·0.14662s·(thread);·0s·(gc)
  
95 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········895 ············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8
96 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+96 o8·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
97 ·····------------------------------------------------------------------------97 ·····------------------------------------------------------------------------
98 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······798 ·········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7
99 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z99 ·····7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
Offset 156, 15 lines modifiedOffset 156, 15 lines modified
156 Sometimes·giving·the·variables·different·degrees·will·speed·up·the156 Sometimes·giving·the·variables·different·degrees·will·speed·up·the
157 computations.·Here,·we·set·the·degrees·of·x,·y,·and·z·to·be·the·total·degrees.157 computations.·Here,·we·set·the·degrees·of·x,·y,·and·z·to·be·the·total·degrees.
158 i10·:·R1·=·QQ[x,y,z,s,t,·Degrees=>{3,3,4,1,1}];158 i10·:·R1·=·QQ[x,y,z,s,t,·Degrees=>{3,3,4,1,1}];
159 i11·:·I1·=·substitute(I,R1);159 i11·:·I1·=·substitute(I,R1);
  
160 o11·:·Ideal·of·R1160 o11·:·Ideal·of·R1
161 i12·:·time·G·=·eliminate(I1,{s,t})161 i12·:·time·G·=·eliminate(I1,{s,t})
162 ·--·used·0.0325054s·(cpu);·0.0325089s·(thread);·0s·(gc)162 ·--·used·0.0468117s·(cpu);·0.0468164s·(thread);·0s·(gc)
  
163 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7163 ·············3·9·····2·6·3·······3·6····9·····2·8·········5·4······2·7
164 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-164 o12·=·ideal(x·y··-·3x·y·z··+·3x*y·z··-·z··-·6x·y·z·-·15x*y·z··+·21y·z··-
165 ······-----------------------------------------------------------------------165 ······-----------------------------------------------------------------------
166 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2166 ········2·9·······2·5·3·······6·3····7·3·········2·6·····3·6·······7·2
167 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-167 ······3x·y··-·324x·y·z··+·6x*y·z··-·y·z··-·405x*y·z··-·3y·z··+·7x*y·z··-
168 ······-----------------------------------------------------------------------168 ······-----------------------------------------------------------------------
Offset 227, 15 lines modifiedOffset 227, 15 lines modified
227 i16·:·F·=·map(A,B,{s^3+s*t+1,·t^3+3*t^2+t,·s*t^3})227 i16·:·F·=·map(A,B,{s^3+s*t+1,·t^3+3*t^2+t,·s*t^3})
  
228 ···················3·············3·····2·········3228 ···················3·············3·····2·········3
229 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})229 o16·=·map·(A,·B,·{s··+·s*t·+·1,·t··+·3t··+·t,·s*t·})
  
230 o16·:·RingMap·A·<--·B230 o16·:·RingMap·A·<--·B
231 i17·:·time·G·=·kernel·F231 i17·:·time·G·=·kernel·F
232 ·--·used·0.249418s·(cpu);·0.117309s·(thread);·0s·(gc)232 ·--·used·0.365832s·(cpu);·0.159494s·(thread);·0s·(gc)
  
233 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8233 ·············3·9·····2·9·····2·8······2·6·3·······9····2·7·········8
234 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+234 o17·=·ideal(x·y··-·3x·y··-·6x·y·z·-·3x·y·z··+·3x*y··-·x·y·z·+·12x*y·z·+
235 ······-----------------------------------------------------------------------235 ······-----------------------------------------------------------------------
236 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7236 ··········7·2·······2·5·3·······6·3····7·3········5·4·······3·6····9·······7
237 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z237 ······7x*y·z··-·324x·y·z··+·6x*y·z··-·y·z··-·15x*y·z··+·3x*y·z··-·y··+·2x*y·z
238 ······-----------------------------------------------------------------------238 ······-----------------------------------------------------------------------
Offset 296, 22 lines modifiedOffset 296, 22 lines modified
296 involve·the·variables·s·and·t.296 involve·the·variables·s·and·t.
297 i19·:·use·ring·I297 i19·:·use·ring·I
  
298 o19·=·R298 o19·=·R
  
299 o19·:·PolynomialRing299 o19·:·PolynomialRing
300 i20·:·time·f1·=·resultant(I_0,I_2,s)300 i20·:·time·f1·=·resultant(I_0,I_2,s)
301 ·--·used·0.00127094s·(cpu);·0.00126649s·(thread);·0s·(gc)301 ·--·used·0.00182866s·(cpu);·0.00182897s·(thread);·0s·(gc)
  
302 ·········9····9······7····3302 ·········9····9······7····3
303 o20·=·x*t··-·t··-·z*t··-·z303 o20·=·x*t··-·t··-·z*t··-·z
  
304 o20·:·R304 o20·:·R
305 i21·:·time·f2·=·resultant(I_1,f1,t)305 i21·:·time·f2·=·resultant(I_1,f1,t)
306 ·--·used·0.0285206s·(cpu);·0.0285309s·(thread);·0s·(gc)306 ·--·used·0.0445255s·(cpu);·0.0445387s·(thread);·0s·(gc)
  
307 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2307 ·········3·9·····2·9·····2·8······2·6·3·······9····2·7·········8········7·2
308 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+308 o21·=·-·x·y··+·3x·y··+·6x·y·z·+·3x·y·z··-·3x*y··+·x·y·z·-·12x*y·z·-·7x*y·z··+
309 ······-----------------------------------------------------------------------309 ······-----------------------------------------------------------------------
310 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8310 ··········2·5·3·······6·3····7·3········5·4·······3·6····9·······7······8
311 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+311 ······324x·y·z··-·6x*y·z··+·y·z··+·15x*y·z··-·3x*y·z··+·y··-·2x*y·z·+·6y·z·+
312 ······-----------------------------------------------------------------------312 ······-----------------------------------------------------------------------
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
    
Offset 144, 15 lines modifiedOffset 144, 15 lines modified
144 ····································Version·=>·0.0144 ····································Version·=>·0.0
145 ·············package·prefix·=>·/usr/145 ·············package·prefix·=>·/usr/
146 ·············PackageIsLoaded·=>·true146 ·············PackageIsLoaded·=>·true
147 ·············pkgname·=>·Foo147 ·············pkgname·=>·Foo
148 ·············private·dictionary·=>·Foo#&quot;private·dictionary&quot;148 ·············private·dictionary·=>·Foo#&quot;private·dictionary&quot;
149 ·············processed·documentation·=>·MutableHashTable{}149 ·············processed·documentation·=>·MutableHashTable{}
150 ·············raw·documentation·=>·MutableHashTable{}150 ·············raw·documentation·=>·MutableHashTable{}
151 ·············source·directory·=>·/tmp/M2-3680509-0/88-rundir/151 ·············source·directory·=>·/tmp/M2-2656573-0/88-rundir/
152 ·············source·file·=>·stdio152 ·············source·file·=>·stdio
153 ·············test·inputs·=>·MutableList{}</code></pre>153 ·············test·inputs·=>·MutableList{}</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i7·:·dictionaryPath156 <td>··············<pre><code·class="language-macaulay2">i7·:·dictionaryPath
  
157 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,157 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,
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Offset 78, 15 lines modifiedOffset 78, 15 lines modified
78 ····································Version·=>·0.078 ····································Version·=>·0.0
79 ·············package·prefix·=>·/usr/79 ·············package·prefix·=>·/usr/
80 ·············PackageIsLoaded·=>·true80 ·············PackageIsLoaded·=>·true
81 ·············pkgname·=>·Foo81 ·············pkgname·=>·Foo
82 ·············private·dictionary·=>·Foo#"private·dictionary"82 ·············private·dictionary·=>·Foo#"private·dictionary"
83 ·············processed·documentation·=>·MutableHashTable{}83 ·············processed·documentation·=>·MutableHashTable{}
84 ·············raw·documentation·=>·MutableHashTable{}84 ·············raw·documentation·=>·MutableHashTable{}
85 ·············source·directory·=>·/tmp/M2-3680509-0/88-rundir/85 ·············source·directory·=>·/tmp/M2-2656573-0/88-rundir/
86 ·············source·file·=>·stdio86 ·············source·file·=>·stdio
87 ·············test·inputs·=>·MutableList{}87 ·············test·inputs·=>·MutableList{}
88 i7·:·dictionaryPath88 i7·:·dictionaryPath
  
89 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,89 o7·=·{Foo.Dictionary,·PackageCitations.Dictionary,·Varieties.Dictionary,
90 ·····------------------------------------------------------------------------90 ·····------------------------------------------------------------------------
91 ·····Truncations.Dictionary,·Polyhedra.Dictionary,·Saturation.Dictionary,91 ·····Truncations.Dictionary,·Polyhedra.Dictionary,·Saturation.Dictionary,
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
    
Offset 67, 25 lines modifiedOffset 67, 25 lines modified
67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 ········<table·class="examples">70 ········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()72 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
73 o1·=·/tmp/M2-3681031-0/0</code></pre>73 o1·=·/tmp/M2-2657156-0/0</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·fileExists·fn76 <td>··············<pre><code·class="language-macaulay2">i2·:·fileExists·fn
  
77 o2·=·false</code></pre>77 o2·=·false</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close80 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
81 o3·=·/tmp/M2-3681031-0/081 o3·=·/tmp/M2-2657156-0/0
  
82 o3·:·File</code></pre>82 o3·:·File</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i4·:·fileExists·fn85 <td>··············<pre><code·class="language-macaulay2">i4·:·fileExists·fn
  
86 o4·=·true</code></pre>86 o4·=·true</code></pre>
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Offset 11, 21 lines modifiedOffset 11, 21 lines modified
11 ····*·Inputs:11 ····*·Inputs:
12 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g12 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g
13 ····*·Outputs:13 ····*·Outputs:
14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·a·file·with·the·filename·or·path·fn·exists14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·a·file·with·the·filename·or·path·fn·exists
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·fn·=·temporaryFileName()16 i1·:·fn·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3681031-0/017 o1·=·/tmp/M2-2657156-0/0
18 i2·:·fileExists·fn18 i2·:·fileExists·fn
  
19 o2·=·false19 o2·=·false
20 i3·:·fn·<<·"hi·there"·<<·close20 i3·:·fn·<<·"hi·there"·<<·close
  
21 o3·=·/tmp/M2-3681031-0/021 o3·=·/tmp/M2-2657156-0/0
  
22 o3·:·File22 o3·:·File
23 i4·:·fileExists·fn23 i4·:·fileExists·fn
  
24 o4·=·true24 o4·=·true
25 i5·:·removeFile·fn25 i5·:·removeFile·fn
26 If·fn·refers·to·a·symbolic·link,·then·whether·the·file·exists·is·determined·by26 If·fn·refers·to·a·symbolic·link,·then·whether·the·file·exists·is·determined·by
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
    
Offset 68, 34 lines modifiedOffset 68, 34 lines modified
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 ········<p>The·length·of·an·open·output·file·is·determined·from·the·internal·count·of·the·number·of·bytes·written·so·far.</p>70 ········<p>The·length·of·an·open·output·file·is·determined·from·the·internal·count·of·the·number·of·bytes·written·so·far.</p>
71 ········<table·class="examples">71 ········<table·class="examples">
72 ··········<tr>72 ··········<tr>
73 <td>··············<pre><code·class="language-macaulay2">i1·:·f·=·temporaryFileName()·&lt;&lt;·&quot;hi·there&quot;73 <td>··············<pre><code·class="language-macaulay2">i1·:·f·=·temporaryFileName()·&lt;&lt;·&quot;hi·there&quot;
  
74 o1·=·/tmp/M2-3696937-0/074 o1·=·/tmp/M2-2679084-0/0
  
75 o1·:·File</code></pre>75 o1·:·File</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·fileLength·f78 <td>··············<pre><code·class="language-macaulay2">i2·:·fileLength·f
  
79 o2·=·8</code></pre>79 o2·=·8</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·close·f82 <td>··············<pre><code·class="language-macaulay2">i3·:·close·f
  
83 o3·=·/tmp/M2-3696937-0/083 o3·=·/tmp/M2-2679084-0/0
  
84 o3·:·File</code></pre>84 o3·:·File</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·filename·=·toString·f87 <td>··············<pre><code·class="language-macaulay2">i4·:·filename·=·toString·f
  
88 o4·=·/tmp/M2-3696937-0/0</code></pre>88 o4·=·/tmp/M2-2679084-0/0</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·fileLength·filename91 <td>··············<pre><code·class="language-macaulay2">i5·:·fileLength·filename
  
92 o5·=·8</code></pre>92 o5·=·8</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
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12 ····*·Outputs:12 ····*·Outputs:
13 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·length·of·the·file·f·or·the·file·whose·name·is·f13 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·length·of·the·file·f·or·the·file·whose·name·is·f
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 The·length·of·an·open·output·file·is·determined·from·the·internal·count·of·the15 The·length·of·an·open·output·file·is·determined·from·the·internal·count·of·the
16 number·of·bytes·written·so·far.16 number·of·bytes·written·so·far.
17 i1·:·f·=·temporaryFileName()·<<·"hi·there"17 i1·:·f·=·temporaryFileName()·<<·"hi·there"
  
18 o1·=·/tmp/M2-3696937-0/018 o1·=·/tmp/M2-2679084-0/0
  
19 o1·:·File19 o1·:·File
20 i2·:·fileLength·f20 i2·:·fileLength·f
  
21 o2·=·821 o2·=·8
22 i3·:·close·f22 i3·:·close·f
  
23 o3·=·/tmp/M2-3696937-0/023 o3·=·/tmp/M2-2679084-0/0
  
24 o3·:·File24 o3·:·File
25 i4·:·filename·=·toString·f25 i4·:·filename·=·toString·f
  
26 o4·=·/tmp/M2-3696937-0/026 o4·=·/tmp/M2-2679084-0/0
27 i5·:·fileLength·filename27 i5·:·fileLength·filename
  
28 o5·=·828 o5·=·8
29 i6·:·get·filename29 i6·:·get·filename
  
30 o6·=·hi·there30 o6·=·hi·there
31 i7·:·length·oo31 i7·:·length·oo
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69 ······</div>69 ······</div>
70 ······<div>70 ······<div>
71 ········<h2>Description</h2>71 ········<h2>Description</h2>
72 ········<table·class="examples">72 ········<table·class="examples">
73 ··········<tr>73 ··········<tr>
74 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()74 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
75 o1·=·/tmp/M2-3694014-0/0</code></pre>75 o1·=·/tmp/M2-2675848-0/0</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·f·=·fn·&lt;&lt;·&quot;hi·there&quot;78 <td>··············<pre><code·class="language-macaulay2">i2·:·f·=·fn·&lt;&lt;·&quot;hi·there&quot;
  
79 o2·=·/tmp/M2-3694014-0/079 o2·=·/tmp/M2-2675848-0/0
  
80 o2·:·File</code></pre>80 o2·:·File</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileMode·f83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileMode·f
  
84 o3·=·420</code></pre>84 o3·=·420</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·close·f87 <td>··············<pre><code·class="language-macaulay2">i4·:·close·f
  
88 o4·=·/tmp/M2-3694014-0/088 o4·=·/tmp/M2-2675848-0/0
  
89 o4·:·File</code></pre>89 o4·:·File</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i5·:·removeFile·fn</code></pre>92 <td>··············<pre><code·class="language-macaulay2">i5·:·removeFile·fn</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ········</table>94 ········</table>
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11 ····*·Inputs:11 ····*·Inputs:
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14 ··········o·the·mode·of·the·open·file·f14 ··········o·the·mode·of·the·open·file·f
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·fn·=·temporaryFileName()16 i1·:·fn·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3694014-0/017 o1·=·/tmp/M2-2675848-0/0
18 i2·:·f·=·fn·<<·"hi·there"18 i2·:·f·=·fn·<<·"hi·there"
  
19 o2·=·/tmp/M2-3694014-0/019 o2·=·/tmp/M2-2675848-0/0
  
20 o2·:·File20 o2·:·File
21 i3·:·fileMode·f21 i3·:·fileMode·f
  
22 o3·=·42022 o3·=·420
23 i4·:·close·f23 i4·:·close·f
  
24 o4·=·/tmp/M2-3694014-0/024 o4·=·/tmp/M2-2675848-0/0
  
25 o4·:·File25 o4·:·File
26 i5·:·removeFile·fn26 i5·:·removeFile·fn
27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
28 ····*·_\x8f_\x8i_\x8l_\x8e_\x8M_\x8o_\x8d_\x8e_\x8(_\x8F_\x8i_\x8l_\x8e_\x8)·--·get·file·mode28 ····*·_\x8f_\x8i_\x8l_\x8e_\x8M_\x8o_\x8d_\x8e_\x8(_\x8F_\x8i_\x8l_\x8e_\x8)·--·get·file·mode
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70 ······<div>70 ······<div>
71 ········<h2>Description</h2>71 ········<h2>Description</h2>
72 ········<table·class="examples">72 ········<table·class="examples">
73 ··········<tr>73 ··········<tr>
74 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()74 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
75 o1·=·/tmp/M2-3683492-0/0</code></pre>75 o1·=·/tmp/M2-2663710-0/0</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close78 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
79 o2·=·/tmp/M2-3683492-0/079 o2·=·/tmp/M2-2663710-0/0
  
80 o2·:·File</code></pre>80 o2·:·File</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileMode·fn83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileMode·fn
  
84 o3·=·420</code></pre>84 o3·=·420</code></pre>
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11 ····*·Inputs:11 ····*·Inputs:
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14 ··········o·the·mode·of·the·file·located·at·the·filename·or·path·fn14 ··········o·the·mode·of·the·file·located·at·the·filename·or·path·fn
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·fn·=·temporaryFileName()16 i1·:·fn·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3683492-0/017 o1·=·/tmp/M2-2663710-0/0
18 i2·:·fn·<<·"hi·there"·<<·close18 i2·:·fn·<<·"hi·there"·<<·close
  
19 o2·=·/tmp/M2-3683492-0/019 o2·=·/tmp/M2-2663710-0/0
  
20 o2·:·File20 o2·:·File
21 i3·:·fileMode·fn21 i3·:·fileMode·fn
  
22 o3·=·42022 o3·=·420
23 i4·:·removeFile·fn23 i4·:·removeFile·fn
24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
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73 ······</div>73 ······</div>
74 ······<div>74 ······<div>
75 ········<h2>Description</h2>75 ········<h2>Description</h2>
76 ········<table·class="examples">76 ········<table·class="examples">
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()78 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
79 o1·=·/tmp/M2-3682085-0/0</code></pre>79 o1·=·/tmp/M2-2662405-0/0</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i2·:·f·=·fn·&lt;&lt;·&quot;hi·there&quot;82 <td>··············<pre><code·class="language-macaulay2">i2·:·f·=·fn·&lt;&lt;·&quot;hi·there&quot;
  
83 o2·=·/tmp/M2-3682085-0/083 o2·=·/tmp/M2-2662405-0/0
  
84 o2·:·File</code></pre>84 o2·:·File</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i3·:·m·=·7·+·7*8·+·7*6487 <td>··············<pre><code·class="language-macaulay2">i3·:·m·=·7·+·7*8·+·7*64
  
88 o3·=·511</code></pre>88 o3·=·511</code></pre>
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98 <td>··············<pre><code·class="language-macaulay2">i5·:·fileMode·f98 <td>··············<pre><code·class="language-macaulay2">i5·:·fileMode·f
  
99 o5·=·511</code></pre>99 o5·=·511</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i6·:·close·f102 <td>··············<pre><code·class="language-macaulay2">i6·:·close·f
  
103 o6·=·/tmp/M2-3682085-0/0103 o6·=·/tmp/M2-2662405-0/0
  
104 o6·:·File</code></pre>104 o6·:·File</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i7·:·fileMode·fn107 <td>··············<pre><code·class="language-macaulay2">i7·:·fileMode·fn
  
108 o7·=·511</code></pre>108 o7·=·511</code></pre>
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14 ····*·Consequences:14 ····*·Consequences:
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16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
17 i1·:·fn·=·temporaryFileName()17 i1·:·fn·=·temporaryFileName()
  
18 o1·=·/tmp/M2-3682085-0/018 o1·=·/tmp/M2-2662405-0/0
19 i2·:·f·=·fn·<<·"hi·there"19 i2·:·f·=·fn·<<·"hi·there"
  
20 o2·=·/tmp/M2-3682085-0/020 o2·=·/tmp/M2-2662405-0/0
  
21 o2·:·File21 o2·:·File
22 i3·:·m·=·7·+·7*8·+·7*6422 i3·:·m·=·7·+·7*8·+·7*64
  
23 o3·=·51123 o3·=·511
24 i4·:·fileMode(m,f)24 i4·:·fileMode(m,f)
25 i5·:·fileMode·f25 i5·:·fileMode·f
  
26 o5·=·51126 o5·=·511
27 i6·:·close·f27 i6·:·close·f
  
28 o6·=·/tmp/M2-3682085-0/028 o6·=·/tmp/M2-2662405-0/0
  
29 o6·:·File29 o6·:·File
30 i7·:·fileMode·fn30 i7·:·fileMode·fn
  
31 o7·=·51131 o7·=·511
32 i8·:·removeFile·fn32 i8·:·removeFile·fn
33 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*33 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
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74 ······<div>74 ······<div>
75 ········<h2>Description</h2>75 ········<h2>Description</h2>
76 ········<table·class="examples">76 ········<table·class="examples">
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()78 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
79 o1·=·/tmp/M2-3696540-0/0</code></pre>79 o1·=·/tmp/M2-2677458-0/0</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close82 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
83 o2·=·/tmp/M2-3696540-0/083 o2·=·/tmp/M2-2677458-0/0
  
84 o2·:·File</code></pre>84 o2·:·File</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i3·:·m·=·fileMode·fn87 <td>··············<pre><code·class="language-macaulay2">i3·:·m·=·fileMode·fn
  
88 o3·=·420</code></pre>88 o3·=·420</code></pre>
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13 ··········o·fn13 ··········o·fn
14 ····*·Consequences:14 ····*·Consequences:
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16 ············mo16 ············mo
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 i1·:·fn·=·temporaryFileName()18 i1·:·fn·=·temporaryFileName()
  
19 o1·=·/tmp/M2-3696540-0/019 o1·=·/tmp/M2-2677458-0/0
20 i2·:·fn·<<·"hi·there"·<<·close20 i2·:·fn·<<·"hi·there"·<<·close
  
21 o2·=·/tmp/M2-3696540-0/021 o2·=·/tmp/M2-2677458-0/0
  
22 o2·:·File22 o2·:·File
23 i3·:·m·=·fileMode·fn23 i3·:·m·=·fileMode·fn
  
24 o3·=·42024 o3·=·420
25 i4·:·fileMode(m|7,fn)25 i4·:·fileMode(m|7,fn)
26 i5·:·fileMode·fn26 i5·:·fileMode·fn
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78 ······<div>78 ······<div>
79 ········<h2>Description</h2>79 ········<h2>Description</h2>
80 The·value·is·the·number·of·seconds·since·00:00:00·1970-01-01·UTC,·the·beginning·of·the·epoch,·so·the·number·of·seconds·ago·a·file·or·directory·was·modified·may·be·found·by·using·the·following·code.········<table·class="examples">80 The·value·is·the·number·of·seconds·since·00:00:00·1970-01-01·UTC,·the·beginning·of·the·epoch,·so·the·number·of·seconds·ago·a·file·or·directory·was·modified·may·be·found·by·using·the·following·code.········<table·class="examples">
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i1·:·currentTime()·-·fileTime·&quot;.&quot;82 <td>··············<pre><code·class="language-macaulay2">i1·:·currentTime()·-·fileTime·&quot;.&quot;
  
83 o1·=·28</code></pre>83 o1·=·41</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ········</table>85 ········</table>
86 ······</div>86 ······</div>
87 ······<div>87 ······<div>
88 ········<h2>See·also</h2>88 ········<h2>See·also</h2>
89 ········<ul>89 ········<ul>
90 ··········<li>90 ··········<li>
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19 ············returns·null·if·no·error·occurs19 ············returns·null·if·no·error·occurs
20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*20 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
21 The·value·is·the·number·of·seconds·since·00:00:00·1970-01-01·UTC,·the·beginning21 The·value·is·the·number·of·seconds·since·00:00:00·1970-01-01·UTC,·the·beginning
22 of·the·epoch,·so·the·number·of·seconds·ago·a·file·or·directory·was·modified·may22 of·the·epoch,·so·the·number·of·seconds·ago·a·file·or·directory·was·modified·may
23 be·found·by·using·the·following·code.23 be·found·by·using·the·following·code.
24 i1·:·currentTime()·-·fileTime·"."24 i1·:·currentTime()·-·fileTime·"."
  
25 o1·=·2825 o1·=·41
26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·--·get·the·current·time27 ····*·_\x8c_\x8u_\x8r_\x8r_\x8e_\x8n_\x8t_\x8T_\x8i_\x8m_\x8e·--·get·the·current·time
28 ····*·_\x8f_\x8i_\x8l_\x8e_\x8·_\x8m_\x8a_\x8n_\x8i_\x8p_\x8u_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n·--·Unix·file·manipulation·functions28 ····*·_\x8f_\x8i_\x8l_\x8e_\x8·_\x8m_\x8a_\x8n_\x8i_\x8p_\x8u_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n·--·Unix·file·manipulation·functions
29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
30 The·object·_\x8f_\x8i_\x8l_\x8e_\x8T_\x8i_\x8m_\x8e·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.30 The·object·_\x8f_\x8i_\x8l_\x8e_\x8T_\x8i_\x8m_\x8e·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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106 ·············3······3106 ·············3······3
107 o6·:·Matrix·R··&lt;--·R</code></pre>107 o6·:·Matrix·R··&lt;--·R</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i7·:·syz·f110 <td>··············<pre><code·class="language-macaulay2">i7·:·syz·f
  
111 ···--·registering·gb·0·at·0x7fee57b43000111 ···--·registering·gb·0·at·0x7f493ae25000
  
112 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb·elements·=·3112 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb·elements·=·3
113 ···--·number·of·monomials················=·9113 ···--·number·of·monomials················=·9
114 ···--·#reduction·steps·=·6114 ···--·#reduction·steps·=·6
115 ···--·#spairs·done·=·6115 ···--·#spairs·done·=·6
116 ···--·ncalls·=·0116 ···--·ncalls·=·0
117 ···--·nloop·=·0117 ···--·nloop·=·0
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37 ·····{3}·|·x2-3··0·····-z4+2·|37 ·····{3}·|·x2-3··0·····-z4+2·|
38 ·····{4}·|·0·····x2-3··y3-1··|38 ·····{4}·|·0·····x2-3··y3-1··|
  
39 ·············3······339 ·············3······3
40 o6·:·Matrix·R··<--·R40 o6·:·Matrix·R··<--·R
41 i7·:·syz·f41 i7·:·syz·f
  
42 ···--·registering·gb·0·at·0x7fee57b4300042 ···--·registering·gb·0·at·0x7f493ae25000
  
43 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb43 ···--·[gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber·of·(nonminimal)·gb
44 elements·=·344 elements·=·3
45 ···--·number·of·monomials················=·945 ···--·number·of·monomials················=·9
46 ···--·#reduction·steps·=·646 ···--·#reduction·steps·=·6
47 ···--·#spairs·done·=·647 ···--·#spairs·done·=·6
48 ···--·ncalls·=·048 ···--·ncalls·=·0
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90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i3·:·removeFile·&quot;test-file&quot;</code></pre>92 <td>··············<pre><code·class="language-macaulay2">i3·:·removeFile·&quot;test-file&quot;</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i4·:·get·&quot;!date&quot;95 <td>··············<pre><code·class="language-macaulay2">i4·:·get·&quot;!date&quot;
  
96 o4·=·Tue·Jun··3·02:41:41·-12·2025</code></pre>96 o4·=·Tue·Jul··7·13:14:10·+14·2026</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ········</table>98 ········</table>
99 ······</div>99 ······</div>
100 ······<div>100 ······<div>
101 ········<h2>See·also</h2>101 ········<h2>See·also</h2>
102 ········<ul>102 ········<ul>
103 ··········<li>103 ··········<li>
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25 o1·:·File25 o1·:·File
26 i2·:·get·"test-file"26 i2·:·get·"test-file"
  
27 o2·=·hi·there27 o2·=·hi·there
28 i3·:·removeFile·"test-file"28 i3·:·removeFile·"test-file"
29 i4·:·get·"!date"29 i4·:·get·"!date"
  
30 o4·=·Tue·Jun··3·02:41:41·-12·202530 o4·=·Tue·Jul··7·13:14:10·+14·2026
31 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*31 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
32 ····*·_\x8r_\x8e_\x8a_\x8d·--·read·from·a·file32 ····*·_\x8r_\x8e_\x8a_\x8d·--·read·from·a·file
33 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e·--·remove·a·file33 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e·--·remove·a·file
34 ····*·_\x8c_\x8l_\x8o_\x8s_\x8e·--·close·a·file34 ····*·_\x8c_\x8l_\x8o_\x8s_\x8e·--·close·a·file
35 ····*·_\x8F_\x8i_\x8l_\x8e_\x8·_\x8<_\x8<_\x8·_\x8T_\x8h_\x8i_\x8n_\x8g·--·print·to·a·file35 ····*·_\x8F_\x8i_\x8l_\x8e_\x8·_\x8<_\x8<_\x8·_\x8T_\x8h_\x8i_\x8n_\x8g·--·print·to·a·file
36 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8ge\x8et\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
37 ····*·get(File)37 ····*·get(File)
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61 ······</div>61 ······</div>
62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<table·class="examples">64 ········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·groupID()66 <td>··············<pre><code·class="language-macaulay2">i1·:·groupID()
  
67 o1·=·3552659</code></pre>67 o1·=·2549134</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ········</table>69 ········</table>
70 ······</div>70 ······</div>
71 ······<div>71 ······<div>
72 ········<h2>See·also</h2>72 ········<h2>See·also</h2>
73 ········<ul>73 ········<ul>
74 ··········<li>74 ··········<li>
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9 ············groupID()9 ············groupID()
10 ····*·Outputs:10 ····*·Outputs:
11 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·process·group·identifier·of·the·current·Macaulay211 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·process·group·identifier·of·the·current·Macaulay2
12 ············process12 ············process
13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
14 i1·:·groupID()14 i1·:·groupID()
  
15 o1·=·355265915 o1·=·2549134
16 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
17 ····*·_\x8p_\x8r_\x8o_\x8c_\x8e_\x8s_\x8s_\x8I_\x8D·--·the·process·identifier17 ····*·_\x8p_\x8r_\x8o_\x8c_\x8e_\x8s_\x8s_\x8I_\x8D·--·the·process·identifier
18 ····*·_\x8s_\x8e_\x8t_\x8G_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·set·the·process·group·identifier18 ····*·_\x8s_\x8e_\x8t_\x8G_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·set·the·process·group·identifier
19 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
20 The·object·_\x8g_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.20 The·object·_\x8g_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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72 <td>··············<pre><code·class="language-macaulay2">i1·:·isDirectory·&quot;.&quot;72 <td>··············<pre><code·class="language-macaulay2">i1·:·isDirectory·&quot;.&quot;
  
73 o1·=·true</code></pre>73 o1·=·true</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·=·temporaryFileName()76 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·=·temporaryFileName()
  
77 o2·=·/tmp/M2-3680745-0/0</code></pre>77 o2·=·/tmp/M2-2656842-0/0</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close80 <td>··············<pre><code·class="language-macaulay2">i3·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
81 o3·=·/tmp/M2-3680745-0/081 o3·=·/tmp/M2-2656842-0/0
  
82 o3·:·File</code></pre>82 o3·:·File</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i4·:·isDirectory·fn85 <td>··············<pre><code·class="language-macaulay2">i4·:·isDirectory·fn
  
86 o4·=·false</code></pre>86 o4·=·false</code></pre>
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14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·directory14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·directory
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·isDirectory·"."16 i1·:·isDirectory·"."
  
17 o1·=·true17 o1·=·true
18 i2·:·fn·=·temporaryFileName()18 i2·:·fn·=·temporaryFileName()
  
19 o2·=·/tmp/M2-3680745-0/019 o2·=·/tmp/M2-2656842-0/0
20 i3·:·fn·<<·"hi·there"·<<·close20 i3·:·fn·<<·"hi·there"·<<·close
  
21 o3·=·/tmp/M2-3680745-0/021 o3·=·/tmp/M2-2656842-0/0
  
22 o3·:·File22 o3·:·File
23 i4·:·isDirectory·fn23 i4·:·isDirectory·fn
  
24 o4·=·false24 o4·=·false
25 i5·:·removeFile·fn25 i5·:·removeFile·fn
26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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175 ··········<tr>175 ··········<tr>
176 <td>··············<pre><code·class="language-macaulay2">i18·:·isPrime(m*m*m1*m1*m2^6)176 <td>··············<pre><code·class="language-macaulay2">i18·:·isPrime(m*m*m1*m1*m2^6)
  
177 o18·=·false</code></pre>177 o18·=·false</code></pre>
178 </td>··········</tr>178 </td>··········</tr>
179 ··········<tr>179 ··········<tr>
180 <td>··············<pre><code·class="language-macaulay2">i19·:·elapsedTime·facs·=·factor(m*m1)180 <td>··············<pre><code·class="language-macaulay2">i19·:·elapsedTime·facs·=·factor(m*m1)
181 ·--·6.7843s·elapsed181 ·--·4.57625s·elapsed
  
182 o19·=·1000000000000000000000000000057*1000000000000000000010000000083182 o19·=·1000000000000000000000000000057*1000000000000000000010000000083
  
183 o19·:·Expression·of·class·Product</code></pre>183 o19·:·Expression·of·class·Product</code></pre>
184 </td>··········</tr>184 </td>··········</tr>
185 ··········<tr>185 ··········<tr>
186 <td>··············<pre><code·class="language-macaulay2">i20·:·facs·=·facs//toList/toList186 <td>··············<pre><code·class="language-macaulay2">i20·:·facs·=·facs//toList/toList
Offset 201, 21 lines modifiedOffset 201, 21 lines modified
201 <td>··············<pre><code·class="language-macaulay2">i22·:·m3·=·nextPrime·(m^3)201 <td>··············<pre><code·class="language-macaulay2">i22·:·m3·=·nextPrime·(m^3)
  
202 o22·=·10000000000000000000000000001710000000000000000000000000097470000000000202 o22·=·10000000000000000000000000001710000000000000000000000000097470000000000
203 ······00000000000000185613</code></pre>203 ······00000000000000185613</code></pre>
204 </td>··········</tr>204 </td>··········</tr>
205 ··········<tr>205 ··········<tr>
206 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·isPrime·m3206 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·isPrime·m3
207 ·--·.0452662s·elapsed207 ·--·.0643698s·elapsed
  
208 o23·=·true</code></pre>208 o23·=·true</code></pre>
209 </td>··········</tr>209 </td>··········</tr>
210 ··········<tr>210 ··········<tr>
211 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·isPseudoprime·m3211 <td>··············<pre><code·class="language-macaulay2">i24·:·elapsedTime·isPseudoprime·m3
212 ·--·.000131339s·elapsed212 ·--·.000153268s·elapsed
  
213 o24·=·true</code></pre>213 o24·=·true</code></pre>
214 </td>··········</tr>214 </td>··········</tr>
215 ········</table>215 ········</table>
216 ······</div>216 ······</div>
217 ······<div>217 ······<div>
218 ········<h2>See·also</h2>218 ········<h2>See·also</h2>
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80 i17·:·isPrime·(m*m1)80 i17·:·isPrime·(m*m1)
  
81 o17·=·false81 o17·=·false
82 i18·:·isPrime(m*m*m1*m1*m2^6)82 i18·:·isPrime(m*m*m1*m1*m2^6)
  
83 o18·=·false83 o18·=·false
84 i19·:·elapsedTime·facs·=·factor(m*m1)84 i19·:·elapsedTime·facs·=·factor(m*m1)
85 ·--·6.7843s·elapsed85 ·--·4.57625s·elapsed
  
86 o19·=·1000000000000000000000000000057*100000000000000000001000000008386 o19·=·1000000000000000000000000000057*1000000000000000000010000000083
  
87 o19·:·Expression·of·class·Product87 o19·:·Expression·of·class·Product
88 i20·:·facs·=·facs//toList/toList88 i20·:·facs·=·facs//toList/toList
  
89 o20·=·{{1000000000000000000000000000057,·1},89 o20·=·{{1000000000000000000000000000057,·1},
Offset 98, 19 lines modifiedOffset 98, 19 lines modified
98 o20·:·List98 o20·:·List
99 i21·:·assert(set·facs·===·set·{{m,1},·{m1,1}})99 i21·:·assert(set·facs·===·set·{{m,1},·{m1,1}})
100 i22·:·m3·=·nextPrime·(m^3)100 i22·:·m3·=·nextPrime·(m^3)
  
101 o22·=·10000000000000000000000000001710000000000000000000000000097470000000000101 o22·=·10000000000000000000000000001710000000000000000000000000097470000000000
102 ······00000000000000185613102 ······00000000000000185613
103 i23·:·elapsedTime·isPrime·m3103 i23·:·elapsedTime·isPrime·m3
104 ·--·.0452662s·elapsed104 ·--·.0643698s·elapsed
  
105 o23·=·true105 o23·=·true
106 i24·:·elapsedTime·isPseudoprime·m3106 i24·:·elapsedTime·isPseudoprime·m3
107 ·--·.000131339s·elapsed107 ·--·.000153268s·elapsed
  
108 o24·=·true108 o24·=·true
109 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*109 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
110 ····*·_\x8i_\x8s_\x8P_\x8r_\x8i_\x8m_\x8e_\x8(_\x8Z_\x8Z_\x8)·--·whether·a·integer·or·polynomial·is·prime110 ····*·_\x8i_\x8s_\x8P_\x8r_\x8i_\x8m_\x8e_\x8(_\x8Z_\x8Z_\x8)·--·whether·a·integer·or·polynomial·is·prime
111 ····*·_\x8f_\x8a_\x8c_\x8t_\x8o_\x8r_\x8(_\x8Z_\x8Z_\x8)·--·factor·a·ring·element111 ····*·_\x8f_\x8a_\x8c_\x8t_\x8o_\x8r_\x8(_\x8Z_\x8Z_\x8)·--·factor·a·ring·element
112 ····*·_\x8n_\x8e_\x8x_\x8t_\x8P_\x8r_\x8i_\x8m_\x8e_\x8(_\x8N_\x8u_\x8m_\x8b_\x8e_\x8r_\x8)·--·compute·the·smallest·prime·greater·than·or·equal·to112 ····*·_\x8n_\x8e_\x8x_\x8t_\x8P_\x8r_\x8i_\x8m_\x8e_\x8(_\x8N_\x8u_\x8m_\x8b_\x8e_\x8r_\x8)·--·compute·the·smallest·prime·greater·than·or·equal·to
113 ······a·given·number113 ······a·given·number
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67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 In·UNIX,·a·regular·file·is·one·that·is·not·special·in·some·way.··Special·files·include·symbolic·links·and·directories.··A·regular·file·is·a·sequence·of·bytes·stored·permanently·in·a·file·system.········<table·class="examples">70 In·UNIX,·a·regular·file·is·one·that·is·not·special·in·some·way.··Special·files·include·symbolic·links·and·directories.··A·regular·file·is·a·sequence·of·bytes·stored·permanently·in·a·file·system.········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()72 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
73 o1·=·/tmp/M2-3697277-0/0</code></pre>73 o1·=·/tmp/M2-2679154-0/0</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close76 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
77 o2·=·/tmp/M2-3697277-0/077 o2·=·/tmp/M2-2679154-0/0
  
78 o2·:·File</code></pre>78 o2·:·File</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·isRegularFile·fn81 <td>··············<pre><code·class="language-macaulay2">i3·:·isRegularFile·fn
  
82 o3·=·true</code></pre>82 o3·=·true</code></pre>
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14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·regular·file14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·regular·file
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 In·UNIX,·a·regular·file·is·one·that·is·not·special·in·some·way.·Special·files16 In·UNIX,·a·regular·file·is·one·that·is·not·special·in·some·way.·Special·files
17 include·symbolic·links·and·directories.·A·regular·file·is·a·sequence·of·bytes17 include·symbolic·links·and·directories.·A·regular·file·is·a·sequence·of·bytes
18 stored·permanently·in·a·file·system.18 stored·permanently·in·a·file·system.
19 i1·:·fn·=·temporaryFileName()19 i1·:·fn·=·temporaryFileName()
  
20 o1·=·/tmp/M2-3697277-0/020 o1·=·/tmp/M2-2679154-0/0
21 i2·:·fn·<<·"hi·there"·<<·close21 i2·:·fn·<<·"hi·there"·<<·close
  
22 o2·=·/tmp/M2-3697277-0/022 o2·=·/tmp/M2-2679154-0/0
  
23 o2·:·File23 o2·:·File
24 i3·:·isRegularFile·fn24 i3·:·isRegularFile·fn
  
25 o3·=·true25 o3·=·true
26 i4·:·removeFile·fn26 i4·:·removeFile·fn
27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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77 ······</div>77 ······</div>
78 ······<div>78 ······<div>
79 ········<h2>Description</h2>79 ········<h2>Description</h2>
80 ········<table·class="examples">80 ········<table·class="examples">
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()82 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()
  
83 o1·=·/tmp/M2-3681260-0/0</code></pre>83 o1·=·/tmp/M2-2659965-0/0</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·(dir|&quot;/a/b/c&quot;)86 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·(dir|&quot;/a/b/c&quot;)
  
87 o2·=·/tmp/M2-3681260-0/0/a/b/c</code></pre>87 o2·=·/tmp/M2-2659965-0/0/a/b/c</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i3·:·removeDirectory·(dir|&quot;/a/b/c&quot;)</code></pre>90 <td>··············<pre><code·class="language-macaulay2">i3·:·removeDirectory·(dir|&quot;/a/b/c&quot;)</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·removeDirectory·(dir|&quot;/a/b&quot;)</code></pre>93 <td>··············<pre><code·class="language-macaulay2">i4·:·removeDirectory·(dir|&quot;/a/b&quot;)</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
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14 ····*·Outputs:14 ····*·Outputs:
15 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·the·name·of·the·newly·made·directory15 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·the·name·of·the·newly·made·directory
16 ····*·Consequences:16 ····*·Consequences:
17 ··········o·the·directory·is·made,·with·as·many·new·path·components·as·needed17 ··········o·the·directory·is·made,·with·as·many·new·path·components·as·needed
18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
19 i1·:·dir·=·temporaryFileName()19 i1·:·dir·=·temporaryFileName()
  
20 o1·=·/tmp/M2-3681260-0/020 o1·=·/tmp/M2-2659965-0/0
21 i2·:·makeDirectory·(dir|"/a/b/c")21 i2·:·makeDirectory·(dir|"/a/b/c")
  
22 o2·=·/tmp/M2-3681260-0/0/a/b/c22 o2·=·/tmp/M2-2659965-0/0/a/b/c
23 i3·:·removeDirectory·(dir|"/a/b/c")23 i3·:·removeDirectory·(dir|"/a/b/c")
24 i4·:·removeDirectory·(dir|"/a/b")24 i4·:·removeDirectory·(dir|"/a/b")
25 i5·:·removeDirectory·(dir|"/a")25 i5·:·removeDirectory·(dir|"/a")
26 A·filename·starting·with·~/·will·have·the·tilde·replaced·by·the·home·directory.26 A·filename·starting·with·~/·will·have·the·tilde·replaced·by·the·home·directory.
27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
28 ····*·_\x8m_\x8k_\x8d_\x8i_\x8r28 ····*·_\x8m_\x8k_\x8d_\x8i_\x8r
29 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
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51 o1·=·fib51 o1·=·fib
  
52 o1·:·FunctionClosure</code></pre>52 o1·:·FunctionClosure</code></pre>
53 </td>··········</tr>53 </td>··········</tr>
54 ··········<tr>54 ··········<tr>
55 <td>··············<pre><code·class="language-macaulay2">i2·:·time·fib·2855 <td>··············<pre><code·class="language-macaulay2">i2·:·time·fib·28
56 ·--·used·0.625112s·(cpu);·0.489916s·(thread);·0s·(gc)56 ·--·used·0.850079s·(cpu);·0.673465s·(thread);·0s·(gc)
  
57 o2·=·514229</code></pre>57 o2·=·514229</code></pre>
58 </td>··········</tr>58 </td>··········</tr>
59 ··········<tr>59 ··········<tr>
60 <td>··············<pre><code·class="language-macaulay2">i3·:·fib·=·memoize·fib60 <td>··············<pre><code·class="language-macaulay2">i3·:·fib·=·memoize·fib
  
61 o3·=·fib61 o3·=·fib
  
62 o3·:·FunctionClosure</code></pre>62 o3·:·FunctionClosure</code></pre>
63 </td>··········</tr>63 </td>··········</tr>
64 ··········<tr>64 ··········<tr>
65 <td>··············<pre><code·class="language-macaulay2">i4·:·time·fib·2865 <td>··············<pre><code·class="language-macaulay2">i4·:·time·fib·28
66 ·--·used·4.67e-05s·(cpu);·4.637e-05s·(thread);·0s·(gc)66 ·--·used·5.8079e-05s·(cpu);·5.7037e-05s·(thread);·0s·(gc)
  
67 o4·=·514229</code></pre>67 o4·=·514229</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ··········<tr>69 ··········<tr>
70 <td>··············<pre><code·class="language-macaulay2">i5·:·time·fib·2870 <td>··············<pre><code·class="language-macaulay2">i5·:·time·fib·28
71 ·--·used·1.402e-06s·(cpu);·1.352e-06s·(thread);·0s·(gc)71 ·--·used·3.397e-06s·(cpu);·2.885e-06s·(thread);·0s·(gc)
  
72 o5·=·514229</code></pre>72 o5·=·514229</code></pre>
73 </td>··········</tr>73 </td>··········</tr>
74 ········</table>74 ········</table>
75 ········<p>An·optional·second·argument·to·memoize·provides·a·list·of·initial·values,·each·of·the·form·<span·class="tt">x·=>·v</span>,·where·<span·class="tt">v</span>·is·the·value·to·be·provided·for·the·argument·<span·class="tt">x</span>.</p>75 ········<p>An·optional·second·argument·to·memoize·provides·a·list·of·initial·values,·each·of·the·form·<span·class="tt">x·=>·v</span>,·where·<span·class="tt">v</span>·is·the·value·to·be·provided·for·the·argument·<span·class="tt">x</span>.</p>
76 ········<p>Alternatively,·values·can·be·provided·after·defining·the·memoized·function·using·the·syntax·<span·class="tt">f·x·=·v</span>.·A·slightly·more·efficient·implementation·of·the·above·would·be</p>76 ········<p>Alternatively,·values·can·be·provided·after·defining·the·memoized·function·using·the·syntax·<span·class="tt">f·x·=·v</span>.·A·slightly·more·efficient·implementation·of·the·above·would·be</p>
77 ········<table·class="examples">77 ········<table·class="examples">
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10 arguments·are·presented.10 arguments·are·presented.
11 i1·:·fib·=·n·->·if·n·<=·1·then·1·else·fib(n-1)·+·fib(n-2)11 i1·:·fib·=·n·->·if·n·<=·1·then·1·else·fib(n-1)·+·fib(n-2)
  
12 o1·=·fib12 o1·=·fib
  
13 o1·:·FunctionClosure13 o1·:·FunctionClosure
14 i2·:·time·fib·2814 i2·:·time·fib·28
15 ·--·used·0.625112s·(cpu);·0.489916s·(thread);·0s·(gc)15 ·--·used·0.850079s·(cpu);·0.673465s·(thread);·0s·(gc)
  
16 o2·=·51422916 o2·=·514229
17 i3·:·fib·=·memoize·fib17 i3·:·fib·=·memoize·fib
  
18 o3·=·fib18 o3·=·fib
  
19 o3·:·FunctionClosure19 o3·:·FunctionClosure
20 i4·:·time·fib·2820 i4·:·time·fib·28
21 ·--·used·4.67e-05s·(cpu);·4.637e-05s·(thread);·0s·(gc)21 ·--·used·5.8079e-05s·(cpu);·5.7037e-05s·(thread);·0s·(gc)
  
22 o4·=·51422922 o4·=·514229
23 i5·:·time·fib·2823 i5·:·time·fib·28
24 ·--·used·1.402e-06s·(cpu);·1.352e-06s·(thread);·0s·(gc)24 ·--·used·3.397e-06s·(cpu);·2.885e-06s·(thread);·0s·(gc)
  
25 o5·=·51422925 o5·=·514229
26 An·optional·second·argument·to·memoize·provides·a·list·of·initial·values,·each26 An·optional·second·argument·to·memoize·provides·a·list·of·initial·values,·each
27 of·the·form·x·=>·v,·where·v·is·the·value·to·be·provided·for·the·argument·x.27 of·the·form·x·=>·v,·where·v·is·the·value·to·be·provided·for·the·argument·x.
28 Alternatively,·values·can·be·provided·after·defining·the·memoized·function28 Alternatively,·values·can·be·provided·after·defining·the·memoized·function
29 using·the·syntax·f·x·=·v.·A·slightly·more·efficient·implementation·of·the·above29 using·the·syntax·f·x·=·v.·A·slightly·more·efficient·implementation·of·the·above
30 would·be30 would·be
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86 ·····{13·=>·(poincare,·BettiTally)································}86 ·····{13·=>·(poincare,·BettiTally)································}
87 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}87 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}
88 ·····{15·=>·(degree,·BettiTally)··································}88 ·····{15·=>·(degree,·BettiTally)··································}
89 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}89 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}
90 ·····{17·=>·(^,·Ring,·BettiTally)·································}90 ·····{17·=>·(^,·Ring,·BettiTally)·································}
91 ·····{18·=>·(regularity,·BettiTally)······························}91 ·····{18·=>·(regularity,·BettiTally)······························}
92 ·····{19·=>·(mathML,·BettiTally)··································}92 ·····{19·=>·(mathML,·BettiTally)··································}
93 ·····{20·=>·(codim,·BettiTally)···································}93 ·····{20·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············}
94 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}94 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}
95 ·····{22·=>·(dual,·BettiTally)····································} 
96 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············} 
97 ·····{24·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}95 ·····{22·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}
98 ·····{25·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}96 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}
 97 ·····{24·=>·(dual,·BettiTally)····································}
 98 ·····{25·=>·(codim,·BettiTally)···································}
  
99 o1·:·NumberedVerticalList</code></pre>99 o1·:·NumberedVerticalList</code></pre>
100 </td>············</tr>100 </td>············</tr>
101 ············<tr>101 ············<tr>
102 <td>················<pre><code·class="language-macaulay2">i2·:·methods·resolution102 <td>················<pre><code·class="language-macaulay2">i2·:·methods·resolution
  
103 o2·=·{0·=>·(resolution,·Ideal)·}103 o2·=·{0·=>·(resolution,·Ideal)·}
Offset 186, 20 lines modifiedOffset 186, 20 lines modified
186 ··············</ul>186 ··············</ul>
187 ············</li>187 ············</li>
188 ··········</ul>188 ··········</ul>
189 ··········<table·class="examples">189 ··········<table·class="examples">
190 ············<tr>190 ············<tr>
191 <td>················<pre><code·class="language-macaulay2">i5·:·methods(·Matrix,·Matrix·)191 <td>················<pre><code·class="language-macaulay2">i5·:·methods(·Matrix,·Matrix·)
  
192 o5·=·{0·=>·(diff',·Matrix,·Matrix)·······························}192 o5·=·{0·=>·(-,·Matrix,·Matrix)···································}
193 ·····{1·=>·(contract',·Matrix,·Matrix)···························} 
194 ·····{2·=>·(+,·Matrix,·Matrix)···································}193 ·····{1·=>·(+,·Matrix,·Matrix)···································}
 194 ·····{2·=>·(diff',·Matrix,·Matrix)·······························}
195 ·····{3·=>·(diff,·Matrix,·Matrix)································}195 ·····{3·=>·(diff,·Matrix,·Matrix)································}
196 ·····{4·=>·(contract,·Matrix,·Matrix)····························}196 ·····{4·=>·(contract',·Matrix,·Matrix)···························}
197 ·····{5·=>·(-,·Matrix,·Matrix)···································}197 ·····{5·=>·(contract,·Matrix,·Matrix)····························}
198 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}198 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}
199 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}199 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}
200 ·····{8·=>·(==,·Matrix,·Matrix)··································}200 ·····{8·=>·(==,·Matrix,·Matrix)··································}
201 ·····{9·=>·(*,·Matrix,·Matrix)···································}201 ·····{9·=>·(*,·Matrix,·Matrix)···································}
202 ·····{10·=>·(|,·Matrix,·Matrix)··································}202 ·····{10·=>·(|,·Matrix,·Matrix)··································}
203 ·····{11·=>·(||,·Matrix,·Matrix)·································}203 ·····{11·=>·(||,·Matrix,·Matrix)·································}
204 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}204 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}
Offset 210, 16 lines modifiedOffset 210, 16 lines modified
210 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}210 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}
211 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}211 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}
212 ·····{19·=>·(//,·Matrix,·Matrix)·································}212 ·····{19·=>·(//,·Matrix,·Matrix)·································}
213 ·····{20·=>·(\\,·Matrix,·Matrix)·································}213 ·····{20·=>·(\\,·Matrix,·Matrix)·································}
214 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}214 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}
215 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}215 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}
216 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}216 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}
217 ·····{24·=>·(%,·Matrix,·Matrix)··································} 
218 ·····{25·=>·(remainder,·Matrix,·Matrix)··························}217 ·····{24·=>·(remainder,·Matrix,·Matrix)··························}
 218 ·····{25·=>·(%,·Matrix,·Matrix)··································}
219 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}219 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}
220 ·····{27·=>·(solve,·Matrix,·Matrix)······························}220 ·····{27·=>·(solve,·Matrix,·Matrix)······························}
221 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}221 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}
222 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}222 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}
223 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}223 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}
224 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}224 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}
225 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}225 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}
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Offset 30, 20 lines modifiedOffset 30, 20 lines modified
30 ·····{13·=>·(poincare,·BettiTally)································}30 ·····{13·=>·(poincare,·BettiTally)································}
31 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}31 ·····{14·=>·(hilbertPolynomial,·ZZ,·BettiTally)···················}
32 ·····{15·=>·(degree,·BettiTally)··································}32 ·····{15·=>·(degree,·BettiTally)··································}
33 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}33 ·····{16·=>·(hilbertSeries,·ZZ,·BettiTally)·······················}
34 ·····{17·=>·(^,·Ring,·BettiTally)·································}34 ·····{17·=>·(^,·Ring,·BettiTally)·································}
35 ·····{18·=>·(regularity,·BettiTally)······························}35 ·····{18·=>·(regularity,·BettiTally)······························}
36 ·····{19·=>·(mathML,·BettiTally)··································}36 ·····{19·=>·(mathML,·BettiTally)··································}
37 ·····{20·=>·(codim,·BettiTally)···································}37 ·····{20·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············}
38 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}38 ·····{21·=>·(truncate,·BettiTally,·ZZ,·ZZ)························}
39 ·····{22·=>·(dual,·BettiTally)····································} 
40 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·ZZ)············} 
41 ·····{24·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}39 ·····{22·=>·(truncate,·BettiTally,·ZZ,·InfiniteNumber)············}
42 ·····{25·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}40 ·····{23·=>·(truncate,·BettiTally,·InfiniteNumber,·InfiniteNumber)}
 41 ·····{24·=>·(dual,·BettiTally)····································}
 42 ·····{25·=>·(codim,·BettiTally)···································}
  
43 o1·:·NumberedVerticalList43 o1·:·NumberedVerticalList
44 i2·:·methods·resolution44 i2·:·methods·resolution
  
45 o2·=·{0·=>·(resolution,·Ideal)·}45 o2·=·{0·=>·(resolution,·Ideal)·}
46 ·····{1·=>·(resolution,·Module)}46 ·····{1·=>·(resolution,·Module)}
47 ·····{2·=>·(resolution,·Matrix)}47 ·····{2·=>·(resolution,·Matrix)}
Offset 85, 20 lines modifiedOffset 85, 20 lines modified
85 ····*·Inputs:85 ····*·Inputs:
86 ··········o·X,·a·_\x8t_\x8y_\x8p_\x8e86 ··········o·X,·a·_\x8t_\x8y_\x8p_\x8e
87 ··········o·Y,·a·_\x8t_\x8y_\x8p_\x8e87 ··········o·Y,·a·_\x8t_\x8y_\x8p_\x8e
88 ····*·Outputs:88 ····*·Outputs:
89 ··········o·a·_\x8v_\x8e_\x8r_\x8t_\x8i_\x8c_\x8a_\x8l_\x8·_\x8l_\x8i_\x8s_\x8t·of·those·methods·associated·with89 ··········o·a·_\x8v_\x8e_\x8r_\x8t_\x8i_\x8c_\x8a_\x8l_\x8·_\x8l_\x8i_\x8s_\x8t·of·those·methods·associated·with
90 i5·:·methods(·Matrix,·Matrix·)90 i5·:·methods(·Matrix,·Matrix·)
  
91 o5·=·{0·=>·(diff',·Matrix,·Matrix)·······························}91 o5·=·{0·=>·(-,·Matrix,·Matrix)···································}
92 ·····{1·=>·(contract',·Matrix,·Matrix)···························} 
93 ·····{2·=>·(+,·Matrix,·Matrix)···································}92 ·····{1·=>·(+,·Matrix,·Matrix)···································}
 93 ·····{2·=>·(diff',·Matrix,·Matrix)·······························}
94 ·····{3·=>·(diff,·Matrix,·Matrix)································}94 ·····{3·=>·(diff,·Matrix,·Matrix)································}
95 ·····{4·=>·(contract,·Matrix,·Matrix)····························}95 ·····{4·=>·(contract',·Matrix,·Matrix)···························}
96 ·····{5·=>·(-,·Matrix,·Matrix)···································}96 ·····{5·=>·(contract,·Matrix,·Matrix)····························}
97 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}97 ·····{6·=>·(markedGB,·Matrix,·Matrix)····························}
98 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}98 ·····{7·=>·(Hom,·Matrix,·Matrix)·································}
99 ·····{8·=>·(==,·Matrix,·Matrix)··································}99 ·····{8·=>·(==,·Matrix,·Matrix)··································}
100 ·····{9·=>·(*,·Matrix,·Matrix)···································}100 ·····{9·=>·(*,·Matrix,·Matrix)···································}
101 ·····{10·=>·(|,·Matrix,·Matrix)··································}101 ·····{10·=>·(|,·Matrix,·Matrix)··································}
102 ·····{11·=>·(||,·Matrix,·Matrix)·································}102 ·····{11·=>·(||,·Matrix,·Matrix)·································}
103 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}103 ·····{12·=>·(subquotient,·Matrix,·Matrix)························}
Offset 109, 16 lines modifiedOffset 109, 16 lines modified
109 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}109 ·····{17·=>·(quotientRemainder',·Matrix,·Matrix)·················}
110 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}110 ·····{18·=>·(quotientRemainder,·Matrix,·Matrix)··················}
111 ·····{19·=>·(//,·Matrix,·Matrix)·································}111 ·····{19·=>·(//,·Matrix,·Matrix)·································}
112 ·····{20·=>·(\\,·Matrix,·Matrix)·································}112 ·····{20·=>·(\\,·Matrix,·Matrix)·································}
113 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}113 ·····{21·=>·(quotient',·Matrix,·Matrix)··························}
114 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}114 ·····{22·=>·(quotient,·Matrix,·Matrix)···························}
115 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}115 ·····{23·=>·(remainder',·Matrix,·Matrix)·························}
116 ·····{24·=>·(%,·Matrix,·Matrix)··································} 
117 ·····{25·=>·(remainder,·Matrix,·Matrix)··························}116 ·····{24·=>·(remainder,·Matrix,·Matrix)··························}
 117 ·····{25·=>·(%,·Matrix,·Matrix)··································}
118 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}118 ·····{26·=>·(pushout,·Matrix,·Matrix)····························}
119 ·····{27·=>·(solve,·Matrix,·Matrix)······························}119 ·····{27·=>·(solve,·Matrix,·Matrix)······························}
120 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}120 ·····{28·=>·(intersection,·Matrix,·Matrix,·Matrix,·Matrix)·······}
121 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}121 ·····{29·=>·(pullback,·Matrix,·Matrix)···························}
122 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}122 ·····{30·=>·(tensor,·Matrix,·Matrix)·····························}
123 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}123 ·····{31·=>·(intersection,·Matrix,·Matrix)·······················}
124 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}124 ·····{32·=>·(substitute,·Matrix,·Matrix)·························}
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
    
Offset 97, 15 lines modifiedOffset 97, 15 lines modified
  
97 o2·=·S97 o2·=·S
  
98 o2·:·PolynomialRing</code></pre>98 o2·:·PolynomialRing</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·minimalBetti·I101 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·C·=·minimalBetti·I
102 ·--·1.70586s·elapsed102 ·--·2.62963s·elapsed
  
103 ············0··1···2···3···4····5···6···7···8··9·10103 ············0··1···2···3···4····5···6···7···8··9·10
104 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··1104 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··1
105 ·········0:·1··.···.···.···.····.···.···.···.··.··.105 ·········0:·1··.···.···.···.····.···.···.···.··.··.
106 ·········1:·.·35·140·189··84····.···.···.···.··.··.106 ·········1:·.·35·140·189··84····.···.···.···.··.··.
107 ·········2:·.··.···.·196·735·1080·735·196···.··.··.107 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
108 ·········3:·.··.···.···.···.····.··84·189·140·35··.108 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 121, 15 lines modifiedOffset 121, 15 lines modified
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i4·:·I·=·ideal·I_*;122 <td>··············<pre><code·class="language-macaulay2">i4·:·I·=·ideal·I_*;
  
123 o4·:·Ideal·of·S</code></pre>123 o4·:·Ideal·of·S</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)126 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)
127 ·--·.77006s·elapsed127 ·--·.805845s·elapsed
  
128 ············0··1···2···3···4····5···6···7128 ············0··1···2···3···4····5···6···7
129 o5·=·total:·1·35·140·385·819·1080·735·196129 o5·=·total:·1·35·140·385·819·1080·735·196
130 ·········0:·1··.···.···.···.····.···.···.130 ·········0:·1··.···.···.···.····.···.···.
131 ·········1:·.·35·140·189··84····.···.···.131 ·········1:·.·35·140·189··84····.···.···.
132 ·········2:·.··.···.·196·735·1080·735·196132 ·········2:·.··.···.·196·735·1080·735·196
  
Offset 138, 15 lines modifiedOffset 138, 15 lines modified
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·ideal·I_*;139 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·ideal·I_*;
  
140 o6·:·Ideal·of·S</code></pre>140 o6·:·Ideal·of·S</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)143 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)
144 ·--·.0289581s·elapsed144 ·--·.0301415s·elapsed
  
145 ············0··1···2···3··4145 ············0··1···2···3··4
146 o7·=·total:·1·35·140·189·84146 o7·=·total:·1·35·140·189·84
147 ·········0:·1··.···.···.··.147 ·········0:·1··.···.···.··.
148 ·········1:·.·35·140·189·84148 ·········1:·.·35·140·189·84
  
149 o7·:·BettiTally</code></pre>149 o7·:·BettiTally</code></pre>
Offset 154, 15 lines modifiedOffset 154, 15 lines modified
154 ··········<tr>154 ··········<tr>
155 <td>··············<pre><code·class="language-macaulay2">i8·:·I·=·ideal·I_*;155 <td>··············<pre><code·class="language-macaulay2">i8·:·I·=·ideal·I_*;
  
156 o8·:·Ideal·of·S</code></pre>156 o8·:·Ideal·of·S</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)159 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)
160 ·--·1.20943s·elapsed160 ·--·1.52404s·elapsed
  
161 ············0··1···2···3···4····5161 ············0··1···2···3···4····5
162 o9·=·total:·1·35·140·385·819·1080162 o9·=·total:·1·35·140·385·819·1080
163 ·········0:·1··.···.···.···.····.163 ·········0:·1··.···.···.···.····.
164 ·········1:·.·35·140·189··84····.164 ·········1:·.·35·140·189··84····.
165 ·········2:·.··.···.·196·735·1080165 ·········2:·.··.···.·196·735·1080
  
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44 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,644 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,6
45 i2·:·S·=·ring·I45 i2·:·S·=·ring·I
  
46 o2·=·S46 o2·=·S
  
47 o2·:·PolynomialRing47 o2·:·PolynomialRing
48 i3·:·elapsedTime·C·=·minimalBetti·I48 i3·:·elapsedTime·C·=·minimalBetti·I
49 ·--·1.70586s·elapsed49 ·--·2.62963s·elapsed
  
50 ············0··1···2···3···4····5···6···7···8··9·1050 ············0··1···2···3···4····5···6···7···8··9·10
51 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··151 o3·=·total:·1·35·140·385·819·1080·819·385·140·35··1
52 ·········0:·1··.···.···.···.····.···.···.···.··.··.52 ·········0:·1··.···.···.···.····.···.···.···.··.··.
53 ·········1:·.·35·140·189··84····.···.···.···.··.··.53 ·········1:·.·35·140·189··84····.···.···.···.··.··.
54 ·········2:·.··.···.·196·735·1080·735·196···.··.··.54 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
55 ·········3:·.··.···.···.···.····.··84·189·140·35··.55 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 61, 40 lines modifiedOffset 61, 40 lines modified
61 o3·:·BettiTally61 o3·:·BettiTally
62 One·can·compute·smaller·parts·of·the·Betti·table,·by·using·_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8L_\x8i_\x8m_\x8i_\x8t·and/or62 One·can·compute·smaller·parts·of·the·Betti·table,·by·using·_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8L_\x8i_\x8m_\x8i_\x8t·and/or
63 _\x8L_\x8e_\x8n_\x8g_\x8t_\x8h_\x8L_\x8i_\x8m_\x8i_\x8t.63 _\x8L_\x8e_\x8n_\x8g_\x8t_\x8h_\x8L_\x8i_\x8m_\x8i_\x8t.
64 i4·:·I·=·ideal·I_*;64 i4·:·I·=·ideal·I_*;
  
65 o4·:·Ideal·of·S65 o4·:·Ideal·of·S
66 i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)66 i5·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>2)
67 ·--·.77006s·elapsed67 ·--·.805845s·elapsed
  
68 ············0··1···2···3···4····5···6···768 ············0··1···2···3···4····5···6···7
69 o5·=·total:·1·35·140·385·819·1080·735·19669 o5·=·total:·1·35·140·385·819·1080·735·196
70 ·········0:·1··.···.···.···.····.···.···.70 ·········0:·1··.···.···.···.····.···.···.
71 ·········1:·.·35·140·189··84····.···.···.71 ·········1:·.·35·140·189··84····.···.···.
72 ·········2:·.··.···.·196·735·1080·735·19672 ·········2:·.··.···.·196·735·1080·735·196
  
73 o5·:·BettiTally73 o5·:·BettiTally
74 i6·:·I·=·ideal·I_*;74 i6·:·I·=·ideal·I_*;
  
75 o6·:·Ideal·of·S75 o6·:·Ideal·of·S
76 i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)76 i7·:·elapsedTime·C·=·minimalBetti(I,·DegreeLimit=>1,·LengthLimit=>5)
77 ·--·.0289581s·elapsed77 ·--·.0301415s·elapsed
  
78 ············0··1···2···3··478 ············0··1···2···3··4
79 o7·=·total:·1·35·140·189·8479 o7·=·total:·1·35·140·189·84
80 ·········0:·1··.···.···.··.80 ·········0:·1··.···.···.··.
81 ·········1:·.·35·140·189·8481 ·········1:·.·35·140·189·84
  
82 o7·:·BettiTally82 o7·:·BettiTally
83 i8·:·I·=·ideal·I_*;83 i8·:·I·=·ideal·I_*;
  
84 o8·:·Ideal·of·S84 o8·:·Ideal·of·S
85 i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)85 i9·:·elapsedTime·C·=·minimalBetti(I,·LengthLimit=>5)
86 ·--·1.20943s·elapsed86 ·--·1.52404s·elapsed
  
87 ············0··1···2···3···4····587 ············0··1···2···3···4····5
88 o9·=·total:·1·35·140·385·819·108088 o9·=·total:·1·35·140·385·819·1080
89 ·········0:·1··.···.···.···.····.89 ·········0:·1··.···.···.···.····.
90 ·········1:·.·35·140·189··84····.90 ·········1:·.·35·140·189··84····.
91 ·········2:·.··.···.·196·735·108091 ·········2:·.··.···.·196·735·1080
  
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70 ······<div>70 ······<div>
71 ········<h2>Description</h2>71 ········<h2>Description</h2>
72 ········<p>Only·one·directory·will·be·made,·so·the·components·of·the·path·p·other·than·the·last·must·already·exist.</p>72 ········<p>Only·one·directory·will·be·made,·so·the·components·of·the·path·p·other·than·the·last·must·already·exist.</p>
73 ········<table·class="examples">73 ········<table·class="examples">
74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i1·:·p·=·temporaryFileName()·|·&quot;/&quot;75 <td>··············<pre><code·class="language-macaulay2">i1·:·p·=·temporaryFileName()·|·&quot;/&quot;
  
76 o1·=·/tmp/M2-3681409-0/0/</code></pre>76 o1·=·/tmp/M2-2660565-0/0/</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·mkdir·p</code></pre>79 <td>··············<pre><code·class="language-macaulay2">i2·:·mkdir·p</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·isDirectory·p82 <td>··············<pre><code·class="language-macaulay2">i3·:·isDirectory·p
  
83 o3·=·true</code></pre>83 o3·=·true</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i4·:·(fn·=·p·|·&quot;foo&quot;)·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close86 <td>··············<pre><code·class="language-macaulay2">i4·:·(fn·=·p·|·&quot;foo&quot;)·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
87 o4·=·/tmp/M2-3681409-0/0/foo87 o4·=·/tmp/M2-2660565-0/0/foo
  
88 o4·:·File</code></pre>88 o4·:·File</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·get·fn91 <td>··············<pre><code·class="language-macaulay2">i5·:·get·fn
  
92 o5·=·hi·there</code></pre>92 o5·=·hi·there</code></pre>
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12 ····*·Consequences:12 ····*·Consequences:
13 ··········o·a·directory·will·be·created·at·the·path·p13 ··········o·a·directory·will·be·created·at·the·path·p
14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*14 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
15 Only·one·directory·will·be·made,·so·the·components·of·the·path·p·other·than·the15 Only·one·directory·will·be·made,·so·the·components·of·the·path·p·other·than·the
16 last·must·already·exist.16 last·must·already·exist.
17 i1·:·p·=·temporaryFileName()·|·"/"17 i1·:·p·=·temporaryFileName()·|·"/"
  
18 o1·=·/tmp/M2-3681409-0/0/18 o1·=·/tmp/M2-2660565-0/0/
19 i2·:·mkdir·p19 i2·:·mkdir·p
20 i3·:·isDirectory·p20 i3·:·isDirectory·p
  
21 o3·=·true21 o3·=·true
22 i4·:·(fn·=·p·|·"foo")·<<·"hi·there"·<<·close22 i4·:·(fn·=·p·|·"foo")·<<·"hi·there"·<<·close
  
23 o4·=·/tmp/M2-3681409-0/0/foo23 o4·=·/tmp/M2-2660565-0/0/foo
  
24 o4·:·File24 o4·:·File
25 i5·:·get·fn25 i5·:·get·fn
  
26 o5·=·hi·there26 o5·=·hi·there
27 i6·:·removeFile·fn27 i6·:·removeFile·fn
28 i7·:·removeDirectory·p28 i7·:·removeDirectory·p
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85 ······</div>85 ······</div>
86 ······<div>86 ······<div>
87 ········<h2>Description</h2>87 ········<h2>Description</h2>
88 ········<table·class="examples">88 ········<table·class="examples">
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()
  
91 o1·=·/tmp/M2-3681093-0/0</code></pre>91 o1·=·/tmp/M2-2657350-0/0</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()
  
95 o2·=·/tmp/M2-3681093-0/1</code></pre>95 o2·=·/tmp/M2-2657350-0/1</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i3·:·src·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close98 <td>··············<pre><code·class="language-macaulay2">i3·:·src·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
99 o3·=·/tmp/M2-3681093-0/099 o3·=·/tmp/M2-2657350-0/0
  
100 o3·:·File</code></pre>100 o3·:·File</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i4·:·moveFile(src,dst,Verbose=>true)103 <td>··············<pre><code·class="language-macaulay2">i4·:·moveFile(src,dst,Verbose=>true)
104 --moving:·/tmp/M2-3681093-0/0·->·/tmp/M2-3681093-0/1</code></pre>104 --moving:·/tmp/M2-2657350-0/0·->·/tmp/M2-2657350-0/1</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i5·:·get·dst107 <td>··············<pre><code·class="language-macaulay2">i5·:·get·dst
  
108 o5·=·hi·there</code></pre>108 o5·=·hi·there</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i6·:·bak·=·moveFile(dst,Verbose=>true)111 <td>··············<pre><code·class="language-macaulay2">i6·:·bak·=·moveFile(dst,Verbose=>true)
112 --backup·file·created:·/tmp/M2-3681093-0/1.bak112 --backup·file·created:·/tmp/M2-2657350-0/1.bak
  
113 o6·=·/tmp/M2-3681093-0/1.bak</code></pre>113 o6·=·/tmp/M2-2657350-0/1.bak</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i7·:·removeFile·bak</code></pre>116 <td>··············<pre><code·class="language-macaulay2">i7·:·removeFile·bak</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ········</table>118 ········</table>
119 ······</div>119 ······</div>
120 ······<div>120 ······<div>
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21 ··········o·the·name·of·the·backup·file·if·one·was·created,·or·_\x8n_\x8u_\x8l_\x8l21 ··········o·the·name·of·the·backup·file·if·one·was·created,·or·_\x8n_\x8u_\x8l_\x8l
22 ····*·Consequences:22 ····*·Consequences:
23 ··········o·the·file·will·be·moved·by·creating·a·new·link·to·the·file·and23 ··········o·the·file·will·be·moved·by·creating·a·new·link·to·the·file·and
24 ············removing·the·old·one24 ············removing·the·old·one
25 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
26 i1·:·src·=·temporaryFileName()26 i1·:·src·=·temporaryFileName()
  
27 o1·=·/tmp/M2-3681093-0/027 o1·=·/tmp/M2-2657350-0/0
28 i2·:·dst·=·temporaryFileName()28 i2·:·dst·=·temporaryFileName()
  
29 o2·=·/tmp/M2-3681093-0/129 o2·=·/tmp/M2-2657350-0/1
30 i3·:·src·<<·"hi·there"·<<·close30 i3·:·src·<<·"hi·there"·<<·close
  
31 o3·=·/tmp/M2-3681093-0/031 o3·=·/tmp/M2-2657350-0/0
  
32 o3·:·File32 o3·:·File
33 i4·:·moveFile(src,dst,Verbose=>true)33 i4·:·moveFile(src,dst,Verbose=>true)
34 --moving:·/tmp/M2-3681093-0/0·->·/tmp/M2-3681093-0/134 --moving:·/tmp/M2-2657350-0/0·->·/tmp/M2-2657350-0/1
35 i5·:·get·dst35 i5·:·get·dst
  
36 o5·=·hi·there36 o5·=·hi·there
37 i6·:·bak·=·moveFile(dst,Verbose=>true)37 i6·:·bak·=·moveFile(dst,Verbose=>true)
38 --backup·file·created:·/tmp/M2-3681093-0/1.bak38 --backup·file·created:·/tmp/M2-2657350-0/1.bak
  
39 o6·=·/tmp/M2-3681093-0/1.bak39 o6·=·/tmp/M2-2657350-0/1.bak
40 i7·:·removeFile·bak40 i7·:·removeFile·bak
41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
42 ····*·_\x8c_\x8o_\x8p_\x8y_\x8F_\x8i_\x8l_\x8e42 ····*·_\x8c_\x8o_\x8p_\x8y_\x8F_\x8i_\x8l_\x8e
43 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*43 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
44 ····*·moveFile(String)44 ····*·moveFile(String)
45 ····*·_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e_\x8(_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8,_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8)45 ····*·_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e_\x8(_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8,_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8)
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43 <hr>····<div>43 <hr>····<div>
44 ······<h1>nanosleep·--·sleep·for·a·given·number·of·nanoseconds</h1>44 ······<h1>nanosleep·--·sleep·for·a·given·number·of·nanoseconds</h1>
45 ······<div>45 ······<div>
46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 <span·class="tt">nanosleep·n</span>·--·sleeps·for·<span·class="tt">n</span>·nanoseconds.········<table·class="examples">47 <span·class="tt">nanosleep·n</span>·--·sleeps·for·<span·class="tt">n</span>·nanoseconds.········<table·class="examples">
48 ··········<tr>48 ··········<tr>
49 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTime·nanosleep·50000000049 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTime·nanosleep·500000000
50 ·--·.500137s·elapsed50 ·--·.50013s·elapsed
  
51 o1·=·0</code></pre>51 o1·=·0</code></pre>
52 </td>··········</tr>52 </td>··········</tr>
53 ········</table>53 ········</table>
54 ······</div>54 ······</div>
55 ······<div>55 ······<div>
56 ········<h2>See·also</h2>56 ········<h2>See·also</h2>
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4 [q···················]4 [q···················]
5 _\x8n_\x8e_\x8x_\x8t·|·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·|·_\x8f_\x8o_\x8r_\x8w_\x8a_\x8r_\x8d·|·_\x8b_\x8a_\x8c_\x8k_\x8w_\x8a_\x8r_\x8d·|·_\x8u_\x8p·|·_\x8i_\x8n_\x8d_\x8e_\x8x·|·_\x8t_\x8o_\x8c5 _\x8n_\x8e_\x8x_\x8t·|·_\x8p_\x8r_\x8e_\x8v_\x8i_\x8o_\x8u_\x8s·|·_\x8f_\x8o_\x8r_\x8w_\x8a_\x8r_\x8d·|·_\x8b_\x8a_\x8c_\x8k_\x8w_\x8a_\x8r_\x8d·|·_\x8u_\x8p·|·_\x8i_\x8n_\x8d_\x8e_\x8x·|·_\x8t_\x8o_\x8c
6 ===============================================================================6 ===============================================================================
7 *\x8**\x8**\x8**\x8**\x8**\x8*·n\x8na\x8an\x8no\x8os\x8sl\x8le\x8ee\x8ep\x8p·-\x8--\x8-·s\x8sl\x8le\x8ee\x8ep\x8p·f\x8fo\x8or\x8r·a\x8a·g\x8gi\x8iv\x8ve\x8en\x8n·n\x8nu\x8um\x8mb\x8be\x8er\x8r·o\x8of\x8f·n\x8na\x8an\x8no\x8os\x8se\x8ec\x8co\x8on\x8nd\x8ds\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*7 *\x8**\x8**\x8**\x8**\x8**\x8*·n\x8na\x8an\x8no\x8os\x8sl\x8le\x8ee\x8ep\x8p·-\x8--\x8-·s\x8sl\x8le\x8ee\x8ep\x8p·f\x8fo\x8or\x8r·a\x8a·g\x8gi\x8iv\x8ve\x8en\x8n·n\x8nu\x8um\x8mb\x8be\x8er\x8r·o\x8of\x8f·n\x8na\x8an\x8no\x8os\x8se\x8ec\x8co\x8on\x8nd\x8ds\x8s·*\x8**\x8**\x8**\x8**\x8**\x8*
8 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*8 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
9 nanosleep·n·--·sleeps·for·n·nanoseconds.9 nanosleep·n·--·sleeps·for·n·nanoseconds.
10 i1·:·elapsedTime·nanosleep·50000000010 i1·:·elapsedTime·nanosleep·500000000
11 ·--·.500137s·elapsed11 ·--·.50013s·elapsed
  
12 o1·=·012 o1·=·0
13 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
14 ····*·_\x8s_\x8l_\x8e_\x8e_\x8p·--·sleep·for·a·while14 ····*·_\x8s_\x8l_\x8e_\x8e_\x8p·--·sleep·for·a·while
15 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
16 The·object·_\x8n_\x8a_\x8n_\x8o_\x8s_\x8l_\x8e_\x8e_\x8p·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.16 The·object·_\x8n_\x8a_\x8n_\x8o_\x8s_\x8l_\x8e_\x8e_\x8p·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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60 ········</div>60 ········</div>
61 ········<table·class="examples">61 ········<table·class="examples">
62 ··········<tr>62 ··········<tr>
63 <td>··············<pre><code·class="language-macaulay2">i2·:·L·=·random·toList·(1..10000);</code></pre>63 <td>··············<pre><code·class="language-macaulay2">i2·:·L·=·random·toList·(1..10000);</code></pre>
64 </td>··········</tr>64 </td>··········</tr>
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);66 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);
67 ·--·.680556s·elapsed</code></pre>67 ·--·1.29299s·elapsed</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ··········<tr>69 ··········<tr>
70 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);70 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);
71 ·--·.180746s·elapsed</code></pre>71 ·--·.23955s·elapsed</code></pre>
72 </td>··········</tr>72 </td>··········</tr>
73 ········</table>73 ········</table>
74 ········<div>74 ········<div>
75 ··········<p>You·will·have·to·try·it·on·your·examples·to·see·how·much·they·speed·up.</p>75 ··········<p>You·will·have·to·try·it·on·your·examples·to·see·how·much·they·speed·up.</p>
76 ··········<p>Warning:·Threads·computing·in·parallel·can·give·wrong·answers·if·their·code·is·not·&quot;thread·safe&quot;,·meaning·they·make·modifications·to·memory·without·ensuring·the·modifications·get·safely·communicated·to·other·threads.·(Thread·safety·can·slow·computations·some.)·Currently,·modifications·to·Macaulay2·variables·and·mutable·hash·tables·are·thread·safe,·but·not·changes·inside·mutable·lists.·Also,·access·to·external·libraries·such·as·singular,·etc.,·may·not·currently·be·thread·safe.</p>76 ··········<p>Warning:·Threads·computing·in·parallel·can·give·wrong·answers·if·their·code·is·not·&quot;thread·safe&quot;,·meaning·they·make·modifications·to·memory·without·ensuring·the·modifications·get·safely·communicated·to·other·threads.·(Thread·safety·can·slow·computations·some.)·Currently,·modifications·to·Macaulay2·variables·and·mutable·hash·tables·are·thread·safe,·but·not·changes·inside·mutable·lists.·Also,·access·to·external·libraries·such·as·singular,·etc.,·may·not·currently·be·thread·safe.</p>
77 ··········<p>The·rest·of·this·document·describes·how·to·control·parallel·tasks·more·directly.</p>77 ··········<p>The·rest·of·this·document·describes·how·to·control·parallel·tasks·more·directly.</p>
78 ··········<p>The·task·system·schedules·functions·and·inputs·to·run·on·a·preset·number·of·threads.·The·number·of·threads·to·be·used·is·given·by·the·variable·<a·title="the·current·maximum·number·of·simultaneously·running·tasks"·href="_allowable__Threads.html">allowableThreads</a>,·and·may·be·examined·and·changed·as·follows.·(<a·title="the·current·maximum·number·of·simultaneously·running·tasks"·href="_allowable__Threads.html">allowableThreads</a>·is·temporarily·increased·if·necessary·inside·<a·title="apply·a·function·to·each·element·in·parallel"·href="_parallel__Apply.html">parallelApply</a>.)</p>78 ··········<p>The·task·system·schedules·functions·and·inputs·to·run·on·a·preset·number·of·threads.·The·number·of·threads·to·be·used·is·given·by·the·variable·<a·title="the·current·maximum·number·of·simultaneously·running·tasks"·href="_allowable__Threads.html">allowableThreads</a>,·and·may·be·examined·and·changed·as·follows.·(<a·title="the·current·maximum·number·of·simultaneously·running·tasks"·href="_allowable__Threads.html">allowableThreads</a>·is·temporarily·increased·if·necessary·inside·<a·title="apply·a·function·to·each·element·in·parallel"·href="_parallel__Apply.html">parallelApply</a>.)</p>
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17 big·computation.·If·the·list·is·long,·it·will·be·split·into·chunks·for·each17 big·computation.·If·the·list·is·long,·it·will·be·split·into·chunks·for·each
18 core,·reducing·the·overhead.·But·the·speedup·is·still·limited·by·the·different18 core,·reducing·the·overhead.·But·the·speedup·is·still·limited·by·the·different
19 threads·competing·for·memory,·including·cpu·caches;·it·is·like·running19 threads·competing·for·memory,·including·cpu·caches;·it·is·like·running
20 Macaulay2·on·a·computer·that·is·running·other·big·programs·at·the·same·time.·We20 Macaulay2·on·a·computer·that·is·running·other·big·programs·at·the·same·time.·We
21 can·see·this·using·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e.21 can·see·this·using·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e.
22 i2·:·L·=·random·toList·(1..10000);22 i2·:·L·=·random·toList·(1..10000);
23 i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);23 i3·:·elapsedTime·········apply(1..100,·n·->·sort·L);
24 ·--·.680556s·elapsed24 ·--·1.29299s·elapsed
25 i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);25 i4·:·elapsedTime·parallelApply(1..100,·n·->·sort·L);
26 ·--·.180746s·elapsed26 ·--·.23955s·elapsed
27 You·will·have·to·try·it·on·your·examples·to·see·how·much·they·speed·up.27 You·will·have·to·try·it·on·your·examples·to·see·how·much·they·speed·up.
28 Warning:·Threads·computing·in·parallel·can·give·wrong·answers·if·their·code·is28 Warning:·Threads·computing·in·parallel·can·give·wrong·answers·if·their·code·is
29 not·"thread·safe",·meaning·they·make·modifications·to·memory·without·ensuring29 not·"thread·safe",·meaning·they·make·modifications·to·memory·without·ensuring
30 the·modifications·get·safely·communicated·to·other·threads.·(Thread·safety·can30 the·modifications·get·safely·communicated·to·other·threads.·(Thread·safety·can
31 slow·computations·some.)·Currently,·modifications·to·Macaulay2·variables·and31 slow·computations·some.)·Currently,·modifications·to·Macaulay2·variables·and
32 mutable·hash·tables·are·thread·safe,·but·not·changes·inside·mutable·lists.32 mutable·hash·tables·are·thread·safe,·but·not·changes·inside·mutable·lists.
33 Also,·access·to·external·libraries·such·as·singular,·etc.,·may·not·currently·be33 Also,·access·to·external·libraries·such·as·singular,·etc.,·may·not·currently·be
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Offset 123, 15 lines modifiedOffset 123, 15 lines modified
  
123 o3·=·S123 o3·=·S
  
124 o3·:·PolynomialRing</code></pre>124 o3·:·PolynomialRing</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ··········<tr>126 ··········<tr>
127 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·minimalBetti·I127 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·minimalBetti·I
128 ·--·2.09728s·elapsed128 ·--·3.10166s·elapsed
  
129 ············0··1···2···3···4····5···6···7···8··9·10129 ············0··1···2···3···4····5···6···7···8··9·10
130 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··1130 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··1
131 ·········0:·1··.···.···.···.····.···.···.···.··.··.131 ·········0:·1··.···.···.···.····.···.···.···.··.··.
132 ·········1:·.·35·140·189··84····.···.···.···.··.··.132 ·········1:·.·35·140·189··84····.···.···.···.··.··.
133 ·········2:·.··.···.·196·735·1080·735·196···.··.··.133 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
134 ·········3:·.··.···.···.···.····.··84·189·140·35··.134 ·········3:·.··.···.···.···.····.··84·189·140·35··.
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142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i5·:·I·=·ideal·I_*;143 <td>··············<pre><code·class="language-macaulay2">i5·:·I·=·ideal·I_*;
  
144 o5·:·Ideal·of·S</code></pre>144 o5·:·Ideal·of·S</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)147 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)
148 ·--·2.08346s·elapsed148 ·--·2.44486s·elapsed
  
149 ············0··1···2···3···4····5···6···7···8··9·10149 ············0··1···2···3···4····5···6···7···8··9·10
150 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··1150 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··1
151 ·········0:·1··.···.···.···.····.···.···.···.··.··.151 ·········0:·1··.···.···.···.····.···.···.···.··.··.
152 ·········1:·.·35·140·189··84····.···.···.···.··.··.152 ·········1:·.·35·140·189··84····.···.···.···.··.··.
153 ·········2:·.··.···.·196·735·1080·735·196···.··.··.153 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
154 ·········3:·.··.···.···.···.····.··84·189·140·35··.154 ·········3:·.··.···.···.···.····.··84·189·140·35··.
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166 ··········<tr>166 ··········<tr>
167 <td>··············<pre><code·class="language-macaulay2">i8·:·numTBBThreads·=·1167 <td>··············<pre><code·class="language-macaulay2">i8·:·numTBBThreads·=·1
  
168 o8·=·1</code></pre>168 o8·=·1</code></pre>
169 </td>··········</tr>169 </td>··········</tr>
170 ··········<tr>170 ··········<tr>
171 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·minimalBetti(I)171 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·minimalBetti(I)
172 ·--·1.93825s·elapsed172 ·--·2.1033s·elapsed
  
173 ············0··1···2···3···4····5···6···7···8··9·10173 ············0··1···2···3···4····5···6···7···8··9·10
174 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1174 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1
175 ·········0:·1··.···.···.···.····.···.···.···.··.··.175 ·········0:·1··.···.···.···.····.···.···.···.··.··.
176 ·········1:·.·35·140·189··84····.···.···.···.··.··.176 ·········1:·.·35·140·189··84····.···.···.···.··.··.
177 ·········2:·.··.···.·196·735·1080·735·196···.··.··.177 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
178 ·········3:·.··.···.···.···.····.··84·189·140·35··.178 ·········3:·.··.···.···.···.····.··84·189·140·35··.
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199 ··········<tr>199 ··········<tr>
200 <td>··············<pre><code·class="language-macaulay2">i12·:·I·=·ideal·I_*;200 <td>··············<pre><code·class="language-macaulay2">i12·:·I·=·ideal·I_*;
  
201 o12·:·Ideal·of·S</code></pre>201 o12·:·Ideal·of·S</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ··········<tr>203 ··········<tr>
204 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)204 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
205 ·--·1.94911s·elapsed205 ·--·4.44298s·elapsed
  
206 ·······1······35······241······841······1781······2464······2294······1432······576······135······14206 ·······1······35······241······841······1781······2464······2294······1432······576······135······14
207 o13·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S207 o13·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S
208 ··································································································208 ··································································································
209 ······0······1·······2········3········4·········5·········6·········7·········8········9········10209 ······0······1·······2········3········4·········5·········6·········7·········8········9········10
  
210 o13·:·Complex</code></pre>210 o13·:·Complex</code></pre>
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220 ··········<tr>220 ··········<tr>
221 <td>··············<pre><code·class="language-macaulay2">i15·:·I·=·ideal·I_*;221 <td>··············<pre><code·class="language-macaulay2">i15·:·I·=·ideal·I_*;
  
222 o15·:·Ideal·of·S</code></pre>222 o15·:·Ideal·of·S</code></pre>
223 </td>··········</tr>223 </td>··········</tr>
224 ··········<tr>224 ··········<tr>
225 <td>··············<pre><code·class="language-macaulay2">i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)225 <td>··············<pre><code·class="language-macaulay2">i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
226 ·--·1.96008s·elapsed226 ·--·2.75209s·elapsed
  
227 ·······1······35······241······841······1781······2464······2294······1432······576······135······14227 ·······1······35······241······841······1781······2464······2294······1432······576······135······14
228 o16·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S228 o16·=·S··&lt;--·S···&lt;--·S····&lt;--·S····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S·····&lt;--·S····&lt;--·S····&lt;--·S
229 ··································································································229 ··································································································
230 ······0······1·······2········3········4·········5·········6·········7·········8········9········10230 ······0······1·······2········3········4·········5·········6·········7·········8········9········10
  
231 o16·:·Complex</code></pre>231 o16·:·Complex</code></pre>
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253 ··········<tr>253 ··········<tr>
254 <td>··············<pre><code·class="language-macaulay2">i19·:·I·=·ideal·random(S^1,·S^{4:-5});254 <td>··············<pre><code·class="language-macaulay2">i19·:·I·=·ideal·random(S^1,·S^{4:-5});
  
255 o19·:·Ideal·of·S</code></pre>255 o19·:·Ideal·of·S</code></pre>
256 </td>··········</tr>256 </td>··········</tr>
257 ··········<tr>257 ··········<tr>
258 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);258 <td>··············<pre><code·class="language-macaulay2">i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);
259 ·--·3.15699s·elapsed259 ·--·4.27149s·elapsed
  
260 ··············1······108260 ··············1······108
261 o20·:·Matrix·S··&lt;--·S</code></pre>261 o20·:·Matrix·S··&lt;--·S</code></pre>
262 </td>··········</tr>262 </td>··········</tr>
263 ··········<tr>263 ··········<tr>
264 <td>··············<pre><code·class="language-macaulay2">i21·:·numTBBThreads·=·1264 <td>··············<pre><code·class="language-macaulay2">i21·:·numTBBThreads·=·1
  
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270 ··········<tr>270 ··········<tr>
271 <td>··············<pre><code·class="language-macaulay2">i22·:·I·=·ideal·I_*;271 <td>··············<pre><code·class="language-macaulay2">i22·:·I·=·ideal·I_*;
  
272 o22·:·Ideal·of·S</code></pre>272 o22·:·Ideal·of·S</code></pre>
273 </td>··········</tr>273 </td>··········</tr>
274 ··········<tr>274 ··········<tr>
275 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);275 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);
276 ·--·5.57081s·elapsed276 ·--·5.95563s·elapsed
  
277 ··············1······108277 ··············1······108
278 o23·:·Matrix·S··&lt;--·S</code></pre>278 o23·:·Matrix·S··&lt;--·S</code></pre>
279 </td>··········</tr>279 </td>··········</tr>
280 ··········<tr>280 ··········<tr>
281 <td>··············<pre><code·class="language-macaulay2">i24·:·numTBBThreads·=·10281 <td>··············<pre><code·class="language-macaulay2">i24·:·numTBBThreads·=·10
  
Offset 287, 15 lines modifiedOffset 287, 15 lines modified
287 ··········<tr>287 ··········<tr>
288 <td>··············<pre><code·class="language-macaulay2">i25·:·I·=·ideal·I_*;288 <td>··············<pre><code·class="language-macaulay2">i25·:·I·=·ideal·I_*;
  
289 o25·:·Ideal·of·S</code></pre>289 o25·:·Ideal·of·S</code></pre>
290 </td>··········</tr>290 </td>··········</tr>
291 ··········<tr>291 ··········<tr>
292 <td>··············<pre><code·class="language-macaulay2">i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);292 <td>··············<pre><code·class="language-macaulay2">i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·&quot;F4&quot;);
293 ·--·2.66126s·elapsed293 ·--·3.85508s·elapsed
  
294 ··············1······108294 ··············1······108
295 o26·:·Matrix·S··&lt;--·S</code></pre>295 o26·:·Matrix·S··&lt;--·S</code></pre>
296 </td>··········</tr>296 </td>··········</tr>
297 ········</table>297 ········</table>
298 ········<div>298 ········<div>
299 ··········<p>For·Groebner·basis·computation·in·associative·algebras,·ParallelizeByDegree·is·not·relevant.··In·this·case,·use·<span·class="tt">numTBBThreads</span>·to·control·the·amount·of·parallelism.</p>299 ··········<p>For·Groebner·basis·computation·in·associative·algebras,·ParallelizeByDegree·is·not·relevant.··In·this·case,·use·<span·class="tt">numTBBThreads</span>·to·control·the·amount·of·parallelism.</p>
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Offset 92, 30 lines modifiedOffset 92, 30 lines modified
92 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,692 0,5···1,5···2,5···3,5···4,5···0,6···1,6···2,6···3,6···4,6···5,6
93 i3·:·S·=·ring·I93 i3·:·S·=·ring·I
  
94 o3·=·S94 o3·=·S
  
95 o3·:·PolynomialRing95 o3·:·PolynomialRing
96 i4·:·elapsedTime·minimalBetti·I96 i4·:·elapsedTime·minimalBetti·I
97 ·--·2.09728s·elapsed97 ·--·3.10166s·elapsed
  
98 ············0··1···2···3···4····5···6···7···8··9·1098 ············0··1···2···3···4····5···6···7···8··9·10
99 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··199 o4·=·total:·1·35·140·385·819·1080·819·385·140·35··1
100 ·········0:·1··.···.···.···.····.···.···.···.··.··.100 ·········0:·1··.···.···.···.····.···.···.···.··.··.
101 ·········1:·.·35·140·189··84····.···.···.···.··.··.101 ·········1:·.·35·140·189··84····.···.···.···.··.··.
102 ·········2:·.··.···.·196·735·1080·735·196···.··.··.102 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
103 ·········3:·.··.···.···.···.····.··84·189·140·35··.103 ·········3:·.··.···.···.···.····.··84·189·140·35··.
104 ·········4:·.··.···.···.···.····.···.···.···.··.··1104 ·········4:·.··.···.···.···.····.···.···.···.··.··1
  
105 o4·:·BettiTally105 o4·:·BettiTally
106 i5·:·I·=·ideal·I_*;106 i5·:·I·=·ideal·I_*;
  
107 o5·:·Ideal·of·S107 o5·:·Ideal·of·S
108 i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)108 i6·:·elapsedTime·minimalBetti(I,·ParallelizeByDegree·=>·true)
109 ·--·2.08346s·elapsed109 ·--·2.44486s·elapsed
  
110 ············0··1···2···3···4····5···6···7···8··9·10110 ············0··1···2···3···4····5···6···7···8··9·10
111 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··1111 o6·=·total:·1·35·140·385·819·1080·819·385·140·35··1
112 ·········0:·1··.···.···.···.····.···.···.···.··.··.112 ·········0:·1··.···.···.···.····.···.···.···.··.··.
113 ·········1:·.·35·140·189··84····.···.···.···.··.··.113 ·········1:·.·35·140·189··84····.···.···.···.··.··.
114 ·········2:·.··.···.·196·735·1080·735·196···.··.··.114 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
115 ·········3:·.··.···.···.···.····.··84·189·140·35··.115 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 125, 15 lines modifiedOffset 125, 15 lines modified
125 i7·:·I·=·ideal·I_*;125 i7·:·I·=·ideal·I_*;
  
126 o7·:·Ideal·of·S126 o7·:·Ideal·of·S
127 i8·:·numTBBThreads·=·1127 i8·:·numTBBThreads·=·1
  
128 o8·=·1128 o8·=·1
129 i9·:·elapsedTime·minimalBetti(I)129 i9·:·elapsedTime·minimalBetti(I)
130 ·--·1.93825s·elapsed130 ·--·2.1033s·elapsed
  
131 ············0··1···2···3···4····5···6···7···8··9·10131 ············0··1···2···3···4····5···6···7···8··9·10
132 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1132 o9·=·total:·1·35·140·385·819·1080·819·385·140·35··1
133 ·········0:·1··.···.···.···.····.···.···.···.··.··.133 ·········0:·1··.···.···.···.····.···.···.···.··.··.
134 ·········1:·.·35·140·189··84····.···.···.···.··.··.134 ·········1:·.·35·140·189··84····.···.···.···.··.··.
135 ·········2:·.··.···.·196·735·1080·735·196···.··.··.135 ·········2:·.··.···.·196·735·1080·735·196···.··.··.
136 ·········3:·.··.···.···.···.····.··84·189·140·35··.136 ·········3:·.··.···.···.···.····.··84·189·140·35··.
Offset 148, 15 lines modifiedOffset 148, 15 lines modified
148 i11·:·numTBBThreads·=·0148 i11·:·numTBBThreads·=·0
  
149 o11·=·0149 o11·=·0
150 i12·:·I·=·ideal·I_*;150 i12·:·I·=·ideal·I_*;
  
151 o12·:·Ideal·of·S151 o12·:·Ideal·of·S
152 i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)152 i13·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
153 ·--·1.94911s·elapsed153 ·--·4.44298s·elapsed
  
154 ·······1······35······241······841······1781······2464······2294······1432154 ·······1······35······241······841······1781······2464······2294······1432
155 576······135······14155 576······135······14
156 o13·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<-156 o13·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<-
157 -·S····<--·S····<--·S157 -·S····<--·S····<--·S
  
  
Offset 167, 15 lines modifiedOffset 167, 15 lines modified
167 i14·:·numTBBThreads·=·1167 i14·:·numTBBThreads·=·1
  
168 o14·=·1168 o14·=·1
169 i15·:·I·=·ideal·I_*;169 i15·:·I·=·ideal·I_*;
  
170 o15·:·Ideal·of·S170 o15·:·Ideal·of·S
171 i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)171 i16·:·elapsedTime·freeResolution(I,·Strategy·=>·Nonminimal)
172 ·--·1.96008s·elapsed172 ·--·2.75209s·elapsed
  
173 ·······1······35······241······841······1781······2464······2294······1432173 ·······1······35······241······841······1781······2464······2294······1432
174 576······135······14174 576······135······14
175 o16·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<-175 o16·=·S··<--·S···<--·S····<--·S····<--·S·····<--·S·····<--·S·····<--·S·····<-
176 -·S····<--·S····<--·S176 -·S····<--·S····<--·S
  
  
Offset 194, 37 lines modifiedOffset 194, 37 lines modified
194 o18·=·S194 o18·=·S
  
195 o18·:·PolynomialRing195 o18·:·PolynomialRing
196 i19·:·I·=·ideal·random(S^1,·S^{4:-5});196 i19·:·I·=·ideal·random(S^1,·S^{4:-5});
  
197 o19·:·Ideal·of·S197 o19·:·Ideal·of·S
198 i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");198 i20·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
199 ·--·3.15699s·elapsed199 ·--·4.27149s·elapsed
  
200 ··············1······108200 ··············1······108
201 o20·:·Matrix·S··<--·S201 o20·:·Matrix·S··<--·S
202 i21·:·numTBBThreads·=·1202 i21·:·numTBBThreads·=·1
  
203 o21·=·1203 o21·=·1
204 i22·:·I·=·ideal·I_*;204 i22·:·I·=·ideal·I_*;
  
205 o22·:·Ideal·of·S205 o22·:·Ideal·of·S
206 i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");206 i23·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
207 ·--·5.57081s·elapsed207 ·--·5.95563s·elapsed
  
208 ··············1······108208 ··············1······108
209 o23·:·Matrix·S··<--·S209 o23·:·Matrix·S··<--·S
210 i24·:·numTBBThreads·=·10210 i24·:·numTBBThreads·=·10
  
211 o24·=·10211 o24·=·10
212 i25·:·I·=·ideal·I_*;212 i25·:·I·=·ideal·I_*;
  
213 o25·:·Ideal·of·S213 o25·:·Ideal·of·S
214 i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");214 i26·:·elapsedTime·groebnerBasis(I,·Strategy·=>·"F4");
215 ·--·2.66126s·elapsed215 ·--·3.85508s·elapsed
  
216 ··············1······108216 ··············1······108
217 o26·:·Matrix·S··<--·S217 o26·:·Matrix·S··<--·S
218 For·Groebner·basis·computation·in·associative·algebras,·ParallelizeByDegree·is218 For·Groebner·basis·computation·in·associative·algebras,·ParallelizeByDegree·is
219 not·relevant.·In·this·case,·use·numTBBThreads·to·control·the·amount·of219 not·relevant.·In·this·case,·use·numTBBThreads·to·control·the·amount·of
220 parallelism.220 parallelism.
221 i27·:·needsPackage·"AssociativeAlgebras"221 i27·:·needsPackage·"AssociativeAlgebras"
Offset 245, 15 lines modifiedOffset 245, 15 lines modified
245 ···············2·························2·························2245 ···············2·························2·························2
246 o30·=·ideal·(5a··+·2b*c·+·3c*b,·3a*c·+·5b··+·2c*a,·2a*b·+·3b*a·+·5c·)246 o30·=·ideal·(5a··+·2b*c·+·3c*b,·3a*c·+·5b··+·2c*a,·2a*b·+·3b*a·+·5c·)
  
247 ················ZZ247 ················ZZ
248 o30·:·Ideal·of·---<|a,·b,·c|>248 o30·:·Ideal·of·---<|a,·b,·c|>
249 ···············101249 ···············101
250 i31·:·elapsedTime·NCGB(I,·22);250 i31·:·elapsedTime·NCGB(I,·22);
251 ·--·.728771s·elapsed251 ·--·1.08623s·elapsed
  
252 ···············ZZ············1·······ZZ············148252 ···············ZZ············1·······ZZ············148
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Offset 314, 34 lines modifiedOffset 314, 34 lines modified
314 ··········<tr>314 ··········<tr>
315 <td>··············<pre><code·class="language-macaulay2">i27·:·gbTrace·=·3315 <td>··············<pre><code·class="language-macaulay2">i27·:·gbTrace·=·3
  
316 o27·=·3</code></pre>316 o27·=·3</code></pre>
317 </td>··········</tr>317 </td>··········</tr>
318 ··········<tr>318 ··········<tr>
319 <td>··············<pre><code·class="language-macaulay2">i28·:·time·poincare·I319 <td>··············<pre><code·class="language-macaulay2">i28·:·time·poincare·I
320 ·--·used·0.00272158s·(cpu);·8.723e-06s·(thread);·0s·(gc)320 ·--·used·0.00115029s·(cpu);·9.398e-06s·(thread);·0s·(gc)
  
321 ············3·····6····9321 ············3·····6····9
322 o28·=·1·-·3T··+·3T··-·T322 o28·=·1·-·3T··+·3T··-·T
  
323 o28·:·ZZ[T]</code></pre>323 o28·:·ZZ[T]</code></pre>
324 </td>··········</tr>324 </td>··········</tr>
325 ··········<tr>325 ··········<tr>
326 <td>··············<pre><code·class="language-macaulay2">i29·:·time·gens·gb·I;326 <td>··············<pre><code·class="language-macaulay2">i29·:·time·gens·gb·I;
  
327 ···--·registering·gb·19·at·0x7fd8ebd7ee00327 ···--·registering·gb·19·at·0x7f23f5491e00
  
328 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of·(nonminimal)·gb·elements·=·11328 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of·(nonminimal)·gb·elements·=·11
329 ···--·number·of·monomials················=·4186329 ···--·number·of·monomials················=·4186
330 ···--·#reduction·steps·=·38330 ···--·#reduction·steps·=·38
331 ···--·#spairs·done·=·11331 ···--·#spairs·done·=·11
332 ···--·ncalls·=·10332 ···--·ncalls·=·10
333 ···--·nloop·=·29333 ···--·nloop·=·29
334 ···--·nsaved·=·0334 ···--·nsaved·=·0
335 ···--··--·used·0.00926653s·(cpu);·0.00991467s·(thread);·0s·(gc)335 ···--··--·used·0.010838s·(cpu);·0.0117141s·(thread);·0s·(gc)
  
336 ··············1······11336 ··············1······11
337 o29·:·Matrix·R··&lt;--·R</code></pre>337 o29·:·Matrix·R··&lt;--·R</code></pre>
338 </td>··········</tr>338 </td>··········</tr>
339 ········</table>339 ········</table>
340 ········<div>340 ········<div>
341 ··········<p>In·this·case,·the·savings·is·minimal,·but·often·it·can·be·dramatic.·Another·important·situation·is·to·compute·a·Gröbner·basis·using·a·different·monomial·order.</p>341 ··········<p>In·this·case,·the·savings·is·minimal,·but·often·it·can·be·dramatic.·Another·important·situation·is·to·compute·a·Gröbner·basis·using·a·different·monomial·order.</p>
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349 ········<table·class="examples">349 ········<table·class="examples">
350 ··········<tr>350 ··········<tr>
351 <td>··············<pre><code·class="language-macaulay2">i30·:·R·=·QQ[a..d];</code></pre>351 <td>··············<pre><code·class="language-macaulay2">i30·:·R·=·QQ[a..d];</code></pre>
352 </td>··········</tr>352 </td>··········</tr>
353 ··········<tr>353 ··········<tr>
354 <td>··············<pre><code·class="language-macaulay2">i31·:·I·=·ideal·random(R^1,·R^{3:-3});354 <td>··············<pre><code·class="language-macaulay2">i31·:·I·=·ideal·random(R^1,·R^{3:-3});
  
355 ···--·registering·gb·20·at·0x7fd8ebd7ec40355 ···--·registering·gb·20·at·0x7f23f5491c40
  
356 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0356 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0
357 ···--·number·of·monomials················=·0357 ···--·number·of·monomials················=·0
358 ···--·#reduction·steps·=·0358 ···--·#reduction·steps·=·0
359 ···--·#spairs·done·=·0359 ···--·#spairs·done·=·0
360 ···--·ncalls·=·0360 ···--·ncalls·=·0
361 ···--·nloop·=·0361 ···--·nloop·=·0
362 ···--·nsaved·=·0362 ···--·nsaved·=·0
363 ···--·363 ···--·
364 o31·:·Ideal·of·R</code></pre>364 o31·:·Ideal·of·R</code></pre>
365 </td>··········</tr>365 </td>··········</tr>
366 ··········<tr>366 ··········<tr>
367 <td>··············<pre><code·class="language-macaulay2">i32·:·time·p·=·poincare·I367 <td>··············<pre><code·class="language-macaulay2">i32·:·time·p·=·poincare·I
  
368 ···--·registering·gb·21·at·0x7fd8ebd7e8c0368 ···--·registering·gb·21·at·0x7f23f54918c0
  
369 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of·(nonminimal)·gb·elements·=·11369 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of·(nonminimal)·gb·elements·=·11
370 ···--·number·of·monomials················=·267370 ···--·number·of·monomials················=·267
371 ···--·#reduction·steps·=·236371 ···--·#reduction·steps·=·236
372 ···--·#spairs·done·=·30372 ···--·#spairs·done·=·30
373 ···--·ncalls·=·10373 ···--·ncalls·=·10
374 ···--·nloop·=·20374 ···--·nloop·=·20
375 ···--·nsaved·=·0375 ···--·nsaved·=·0
376 ···--··--·used·0.00320777s·(cpu);·0.00422687s·(thread);·0s·(gc)376 ···--··--·used·0.0040092s·(cpu);·0.00515655s·(thread);·0s·(gc)
  
377 ············3·····6····9377 ············3·····6····9
378 o32·=·1·-·3T··+·3T··-·T378 o32·=·1·-·3T··+·3T··-·T
  
379 o32·:·ZZ[T]</code></pre>379 o32·:·ZZ[T]</code></pre>
380 </td>··········</tr>380 </td>··········</tr>
381 ··········<tr>381 ··········<tr>
Offset 436, 27 lines modifiedOffset 436, 27 lines modified
436 <td>··············<pre><code·class="language-macaulay2">i36·:·gbTrace·=·3436 <td>··············<pre><code·class="language-macaulay2">i36·:·gbTrace·=·3
  
437 o36·=·3</code></pre>437 o36·=·3</code></pre>
438 </td>··········</tr>438 </td>··········</tr>
439 ··········<tr>439 ··········<tr>
440 <td>··············<pre><code·class="language-macaulay2">i37·:·time·gens·gb·J;440 <td>··············<pre><code·class="language-macaulay2">i37·:·time·gens·gb·J;
  
441 ···--·registering·gb·22·at·0x7fd8ebd7e700441 ···--·registering·gb·22·at·0x7f23f5491700
  
442 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m442 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}(3,9)m
443 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m443 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
444 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m444 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m{24}(1,3)m
445 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements·=·39445 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements·=·39
446 ···--·number·of·monomials················=·1051446 ···--·number·of·monomials················=·1051
447 ···--·#reduction·steps·=·284447 ···--·#reduction·steps·=·284
448 ···--·#spairs·done·=·53448 ···--·#spairs·done·=·53
449 ···--·ncalls·=·46449 ···--·ncalls·=·46
450 ···--·nloop·=·54450 ···--·nloop·=·54
451 ···--·nsaved·=·0451 ···--·nsaved·=·0
452 ···--··--·used·0.0359964s·(cpu);·0.0365404s·(thread);·0s·(gc)452 ···--··--·used·0.044002s·(cpu);·0.0450079s·(thread);·0s·(gc)
  
453 ··············1······39453 ··············1······39
454 o37·:·Matrix·S··&lt;--·S</code></pre>454 o37·:·Matrix·S··&lt;--·S</code></pre>
455 </td>··········</tr>455 </td>··········</tr>
456 ··········<tr>456 ··········<tr>
457 <td>··············<pre><code·class="language-macaulay2">i38·:·selectInSubring(1,·gens·gb·J)457 <td>··············<pre><code·class="language-macaulay2">i38·:·selectInSubring(1,·gens·gb·J)
  
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179 o26·=·1·-·3T··+·3T··-·T179 o26·=·1·-·3T··+·3T··-·T
  
180 o26·:·ZZ[T]180 o26·:·ZZ[T]
181 i27·:·gbTrace·=·3181 i27·:·gbTrace·=·3
  
182 o27·=·3182 o27·=·3
183 i28·:·time·poincare·I183 i28·:·time·poincare·I
184 ·--·used·0.00272158s·(cpu);·8.723e-06s·(thread);·0s·(gc)184 ·--·used·0.00115029s·(cpu);·9.398e-06s·(thread);·0s·(gc)
  
185 ············3·····6····9185 ············3·····6····9
186 o28·=·1·-·3T··+·3T··-·T186 o28·=·1·-·3T··+·3T··-·T
  
187 o28·:·ZZ[T]187 o28·:·ZZ[T]
188 i29·:·time·gens·gb·I;188 i29·:·time·gens·gb·I;
  
189 ···--·registering·gb·19·at·0x7fd8ebd7ee00189 ···--·registering·gb·19·at·0x7f23f5491e00
  
190 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of190 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(2,6)mm{7}(1,4)m{8}(0,2)number·of
191 (nonminimal)·gb·elements·=·11191 (nonminimal)·gb·elements·=·11
192 ···--·number·of·monomials················=·4186192 ···--·number·of·monomials················=·4186
193 ···--·#reduction·steps·=·38193 ···--·#reduction·steps·=·38
194 ···--·#spairs·done·=·11194 ···--·#spairs·done·=·11
195 ···--·ncalls·=·10195 ···--·ncalls·=·10
196 ···--·nloop·=·29196 ···--·nloop·=·29
197 ···--·nsaved·=·0197 ···--·nsaved·=·0
198 ···--··--·used·0.00926653s·(cpu);·0.00991467s·(thread);·0s·(gc)198 ···--··--·used·0.010838s·(cpu);·0.0117141s·(thread);·0s·(gc)
  
199 ··············1······11199 ··············1······11
200 o29·:·Matrix·R··<--·R200 o29·:·Matrix·R··<--·R
201 In·this·case,·the·savings·is·minimal,·but·often·it·can·be·dramatic.·Another201 In·this·case,·the·savings·is·minimal,·but·often·it·can·be·dramatic.·Another
202 important·situation·is·to·compute·a·Gröbner·basis·using·a·different·monomial202 important·situation·is·to·compute·a·Gröbner·basis·using·a·different·monomial
203 order.203 order.
204 i30·:·R·=·QQ[a..d];204 i30·:·R·=·QQ[a..d];
205 i31·:·I·=·ideal·random(R^1,·R^{3:-3});205 i31·:·I·=·ideal·random(R^1,·R^{3:-3});
  
206 ···--·registering·gb·20·at·0x7fd8ebd7ec40206 ···--·registering·gb·20·at·0x7f23f5491c40
  
207 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0207 ···--·[gb]number·of·(nonminimal)·gb·elements·=·0
208 ···--·number·of·monomials················=·0208 ···--·number·of·monomials················=·0
209 ···--·#reduction·steps·=·0209 ···--·#reduction·steps·=·0
210 ···--·#spairs·done·=·0210 ···--·#spairs·done·=·0
211 ···--·ncalls·=·0211 ···--·ncalls·=·0
212 ···--·nloop·=·0212 ···--·nloop·=·0
213 ···--·nsaved·=·0213 ···--·nsaved·=·0
214 ···--214 ···--
215 o31·:·Ideal·of·R215 o31·:·Ideal·of·R
216 i32·:·time·p·=·poincare·I216 i32·:·time·p·=·poincare·I
  
217 ···--·registering·gb·21·at·0x7fd8ebd7e8c0217 ···--·registering·gb·21·at·0x7f23f54918c0
  
218 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of218 ···--·[gb]{3}(3)mmm{4}(2)mm{5}(3)mmm{6}(6)mmoooo{7}(4)mooo{8}(2)oonumber·of
219 (nonminimal)·gb·elements·=·11219 (nonminimal)·gb·elements·=·11
220 ···--·number·of·monomials················=·267220 ···--·number·of·monomials················=·267
221 ···--·#reduction·steps·=·236221 ···--·#reduction·steps·=·236
222 ···--·#spairs·done·=·30222 ···--·#spairs·done·=·30
223 ···--·ncalls·=·10223 ···--·ncalls·=·10
224 ···--·nloop·=·20224 ···--·nloop·=·20
225 ···--·nsaved·=·0225 ···--·nsaved·=·0
226 ···--··--·used·0.00320777s·(cpu);·0.00422687s·(thread);·0s·(gc)226 ···--··--·used·0.0040092s·(cpu);·0.00515655s·(thread);·0s·(gc)
  
227 ············3·····6····9227 ············3·····6····9
228 o32·=·1·-·3T··+·3T··-·T228 o32·=·1·-·3T··+·3T··-·T
  
229 o32·:·ZZ[T]229 o32·:·ZZ[T]
230 i33·:·S·=·QQ[a..d,·MonomialOrder·=>·Eliminate·2]230 i33·:·S·=·QQ[a..d,·MonomialOrder·=>·Eliminate·2]
  
Offset 283, 30 lines modifiedOffset 283, 30 lines modified
  
283 o35·:·ZZ[T]283 o35·:·ZZ[T]
284 i36·:·gbTrace·=·3284 i36·:·gbTrace·=·3
  
285 o36·=·3285 o36·=·3
286 i37·:·time·gens·gb·J;286 i37·:·time·gens·gb·J;
  
287 ···--·registering·gb·22·at·0x7fd8ebd7e700287 ···--·registering·gb·22·at·0x7f23f5491700
  
288 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}288 ···--·[gb]{3}(3,3)mmm{4}(2,2)mm{5}(3,3)mmm{6}(3,7)mmm{7}(3,8)mmm{8}(3,9)mmm{9}
289 (3,9)m289 (3,9)m
290 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m290 ···--·mm{10}(2,8)mm{11}(1,5)m{12}(1,3)m{13}(1,3)m{14}(1,3)m{15}(1,3)m{16}(1,3)m
291 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m291 ···--·{17}(1,3)m{18}(1,3)m{19}(1,3)m{20}(1,3)m{21}(1,3)m{22}(1,3)m{23}(1,3)m
292 {24}(1,3)m292 {24}(1,3)m
293 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements293 ···--·{25}(1,3)m{26}(1,3)m{27}(1,3)m{28}(0,2)number·of·(nonminimal)·gb·elements
294 =·39294 =·39
295 ···--·number·of·monomials················=·1051295 ···--·number·of·monomials················=·1051
296 ···--·#reduction·steps·=·284296 ···--·#reduction·steps·=·284
297 ···--·#spairs·done·=·53297 ···--·#spairs·done·=·53
298 ···--·ncalls·=·46298 ···--·ncalls·=·46
299 ···--·nloop·=·54299 ···--·nloop·=·54
300 ···--·nsaved·=·0300 ···--·nsaved·=·0
301 ···--··--·used·0.0359964s·(cpu);·0.0365404s·(thread);·0s·(gc)301 ···--··--·used·0.044002s·(cpu);·0.0450079s·(thread);·0s·(gc)
  
302 ··············1······39302 ··············1······39
303 o37·:·Matrix·S··<--·S303 o37·:·Matrix·S··<--·S
304 i38·:·selectInSubring(1,·gens·gb·J)304 i38·:·selectInSubring(1,·gens·gb·J)
  
305 o38·=·|·243873059890414515367459726418219472801881021280016638460434780718278305 o38·=·|·243873059890414515367459726418219472801881021280016638460434780718278
306 ······-----------------------------------------------------------------------306 ······-----------------------------------------------------------------------
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Offset 61, 15 lines modifiedOffset 61, 15 lines modified
61 ······</div>61 ······</div>
62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<table·class="examples">64 ········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·processID()66 <td>··············<pre><code·class="language-macaulay2">i1·:·processID()
  
67 o1·=·3680509</code></pre>67 o1·=·2656573</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ········</table>69 ········</table>
70 ······</div>70 ······</div>
71 ······<div>71 ······<div>
72 ········<h2>See·also</h2>72 ········<h2>See·also</h2>
73 ········<ul>73 ········<ul>
74 ··········<li>74 ··········<li>
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9 ····*···Usage:9 ····*···Usage:
10 ············processID()10 ············processID()
11 ····*·Outputs:11 ····*·Outputs:
12 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·process·identifier·of·the·current·Macaulay2·process12 ··········o·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,·the·process·identifier·of·the·current·Macaulay2·process
13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
14 i1·:·processID()14 i1·:·processID()
  
15 o1·=·368050915 o1·=·2656573
16 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
17 ····*·_\x8g_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·the·process·group·identifier17 ····*·_\x8g_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·the·process·group·identifier
18 ····*·_\x8s_\x8e_\x8t_\x8G_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·set·the·process·group·identifier18 ····*·_\x8s_\x8e_\x8t_\x8G_\x8r_\x8o_\x8u_\x8p_\x8I_\x8D·--·set·the·process·group·identifier
19 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
20 The·object·_\x8p_\x8r_\x8o_\x8c_\x8e_\x8s_\x8s_\x8I_\x8D·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.20 The·object·_\x8p_\x8r_\x8o_\x8c_\x8e_\x8s_\x8s_\x8I_\x8D·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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59 ······123····110····1359 ······123····110····13
60 o2·=·x····+·x····+·x···+·160 o2·=·x····+·x····+·x···+·1
  
61 o2·:·R</code></pre>61 o2·:·R</code></pre>
62 </td>··········</tr>62 </td>··········</tr>
63 ··········<tr>63 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·factor·f64 <td>··············<pre><code·class="language-macaulay2">i3·:·time·factor·f
65 ·--·used·0.00258722s·(cpu);·0.00258178s·(thread);·0s·(gc)65 ·--·used·0.00491573s·(cpu);·0.00490021s·(thread);·0s·(gc)
  
66 ··············2········2·······2·······2·······2·······4·····3······2············4·····3·····2············4·····3·····2············10·····8·····6·····4······2·······10·····8······6·····4······2·······10······8····6····4·····2·······10······8·····6·····4······2·······10······8·····6·····4·····2·······10······8·····6······4·····2·······10······8·····6·····4······2·······10······8·····6······4·····2·······10·····8····6····4······2·······10·····8······6·····4······266 ··············2········2·······2·······2·······2·······4·····3······2············4·····3·····2············4·····3·····2············10·····8·····6·····4······2·······10·····8······6·····4······2·······10······8····6····4·····2·······10······8·····6·····4······2·······10······8·····6·····4·····2·······10······8·····6······4·····2·······10······8·····6·····4······2·······10······8·····6······4·····2·······10·····8····6····4······2·······10·····8······6·····4······2
67 o3·=·(x·+·1)(x··-·15)(x··+·8)(x··+·4)(x··+·2)(x··+·1)(x··-·4x··+·11x··-·4x·+·1)(x··-·6x··-·2x··-·6x·+·1)(x··+·9x··-·4x··+·9x·+·1)(x···-·5x··-·8x··-·4x··-·13x··+·1)(x···-·9x··+·15x··-·2x··+·10x··+·1)(x···-·10x··-·x··-·x··+·9x··+·1)(x···-·11x··-·8x··-·4x··+·11x··+·1)(x···-·13x··-·4x··-·8x··-·5x··+·1)(x···+·13x··-·2x··+·15x··+·5x··+·1)(x···+·11x··-·4x··-·8x··-·11x··+·1)(x···+·10x··-·2x··+·15x··-·9x··+·1)(x···+·9x··-·x··-·x··-·10x··+·1)(x···+·5x··+·15x··-·2x··+·13x··+·1)67 o3·=·(x·+·1)(x··-·15)(x··+·8)(x··+·4)(x··+·2)(x··+·1)(x··-·4x··+·11x··-·4x·+·1)(x··-·6x··-·2x··-·6x·+·1)(x··+·9x··-·4x··+·9x·+·1)(x···-·5x··-·8x··-·4x··-·13x··+·1)(x···-·9x··+·15x··-·2x··+·10x··+·1)(x···-·10x··-·x··-·x··+·9x··+·1)(x···-·11x··-·8x··-·4x··+·11x··+·1)(x···-·13x··-·4x··-·8x··-·5x··+·1)(x···+·13x··-·2x··+·15x··+·5x··+·1)(x···+·11x··-·4x··-·8x··-·11x··+·1)(x···+·10x··-·2x··+·15x··-·9x··+·1)(x···+·9x··-·x··-·x··-·10x··+·1)(x···+·5x··+·15x··-·2x··+·13x··+·1)
  
68 o3·:·Expression·of·class·Product</code></pre>68 o3·:·Expression·of·class·Product</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ··········<tr>70 ··········<tr>
Offset 92, 16 lines modifiedOffset 92, 16 lines modified
92 o6·:·FunctionClosure</code></pre>92 o6·:·FunctionClosure</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i7·:·for·i·to·10·do·(g();h();h())</code></pre>95 <td>··············<pre><code·class="language-macaulay2">i7·:·for·i·to·10·do·(g();h();h())</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i8·:·profileSummary98 <td>··············<pre><code·class="language-macaulay2">i8·:·profileSummary
99 g:·11·times,·used·.0250758·seconds99 g:·11·times,·used·.0390346·seconds
100 h:·22·times,·used·.0509056·seconds</code></pre>100 h:·22·times,·used·.0774187·seconds</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ········</table>102 ········</table>
103 ······</div>103 ······</div>
104 ······<div·class="waystouse">104 ······<div·class="waystouse">
105 ········<h2>Ways·to·use·<span·class="tt">profile</span>:</h2>105 ········<h2>Ways·to·use·<span·class="tt">profile</span>:</h2>
106 ········<ul>106 ········<ul>
107 ··········<li>107 ··········<li>
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17 i2·:·f·=·(x^110+1)*(x^13+1)17 i2·:·f·=·(x^110+1)*(x^13+1)
  
18 ······123····110····1318 ······123····110····13
19 o2·=·x····+·x····+·x···+·119 o2·=·x····+·x····+·x···+·1
  
20 o2·:·R20 o2·:·R
21 i3·:·time·factor·f21 i3·:·time·factor·f
22 ·--·used·0.00258722s·(cpu);·0.00258178s·(thread);·0s·(gc)22 ·--·used·0.00491573s·(cpu);·0.00490021s·(thread);·0s·(gc)
  
23 ··············2········2·······2·······2·······2·······4·····3······223 ··············2········2·······2·······2·······2·······4·····3······2
24 4·····3·····2············4·····3·····2············10·····8·····6·····4······224 4·····3·····2············4·····3·····2············10·····8·····6·····4······2
25 10·····8······6·····4······2·······10······8····6····4·····2·······10······825 10·····8······6·····4······2·······10······8····6····4·····2·······10······8
26 6·····4······2·······10······8·····6·····4·····2·······10······8·····6······426 6·····4······2·······10······8·····6·····4·····2·······10······8·····6······4
27 2·······10······8·····6·····4······2·······10······8·····6······4·····227 2·······10······8·····6·····4······2·······10······8·····6······4·····2
28 10·····8····6····4······2·······10·····8······6·····4······228 10·····8····6····4······2·······10·····8······6·····4······2
Offset 50, 14 lines modifiedOffset 50, 14 lines modified
50 i6·:·h·=·profile("h",·()·->·factor·f)50 i6·:·h·=·profile("h",·()·->·factor·f)
  
51 o6·=·h51 o6·=·h
  
52 o6·:·FunctionClosure52 o6·:·FunctionClosure
53 i7·:·for·i·to·10·do·(g();h();h())53 i7·:·for·i·to·10·do·(g();h();h())
54 i8·:·profileSummary54 i8·:·profileSummary
55 g:·11·times,·used·.0250758·seconds55 g:·11·times,·used·.0390346·seconds
56 h:·22·times,·used·.0509056·seconds56 h:·22·times,·used·.0774187·seconds
57 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8pr\x8ro\x8of\x8fi\x8il\x8le\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*57 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8pr\x8ro\x8of\x8fi\x8il\x8le\x8e:\x8:·*\x8**\x8**\x8**\x8**\x8*
58 ····*·profile(Function)58 ····*·profile(Function)
59 ····*·profile(String,Function)59 ····*·profile(String,Function)
60 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*60 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
61 The·object·_\x8p_\x8r_\x8o_\x8f_\x8i_\x8l_\x8e·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.61 The·object·_\x8p_\x8r_\x8o_\x8f_\x8i_\x8l_\x8e·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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Offset 90, 15 lines modifiedOffset 90, 15 lines modified
  
90 o5·=·(2,·10)90 o5·=·(2,·10)
  
91 o5·:·Sequence</code></pre>91 o5·:·Sequence</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i6·:·time·randomKRationalPoint(I)94 <td>··············<pre><code·class="language-macaulay2">i6·:·time·randomKRationalPoint(I)
95 ·--·used·0.120879s·(cpu);·0.0600286s·(thread);·0s·(gc)95 ·--·used·0.138709s·(cpu);·0.0707743s·(thread);·0s·(gc)
  
96 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)96 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)
97 ·············2······3···1·····3···0·····397 ·············2······3···1·····3···0·····3
  
98 o6·:·Ideal·of·R</code></pre>98 o6·:·Ideal·of·R</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
Offset 114, 15 lines modifiedOffset 114, 15 lines modified
  
114 o9·=·(3,·10)114 o9·=·(3,·10)
  
115 o9·:·Sequence</code></pre>115 o9·:·Sequence</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i10·:·time·randomKRationalPoint(I)118 <td>··············<pre><code·class="language-macaulay2">i10·:·time·randomKRationalPoint(I)
119 ·--·used·0.285537s·(cpu);·0.166293s·(thread);·0s·(gc)119 ·--·used·0.331339s·(cpu);·0.194846s·(thread);·0s·(gc)
  
120 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)120 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)
121 ··············4······5···3······5···2·····5···1······5···0······5121 ··············4······5···3······5···2·····5···1······5···0······5
  
122 o10·:·Ideal·of·R</code></pre>122 o10·:·Ideal·of·R</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ········</table>124 ········</table>
Offset 148, 15 lines modifiedOffset 148, 15 lines modified
148 <td>··············<pre><code·class="language-macaulay2">i14·:·I=ideal·random(n,R);148 <td>··············<pre><code·class="language-macaulay2">i14·:·I=ideal·random(n,R);
  
149 o14·:·Ideal·of·R</code></pre>149 o14·:·Ideal·of·R</code></pre>
150 </td>··········</tr>150 </td>··········</tr>
151 ··········<tr>151 ··········<tr>
152 <td>··············<pre><code·class="language-macaulay2">i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c->degree·c);152 <td>··············<pre><code·class="language-macaulay2">i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c->degree·c);
153 ·····················#select(degs,d->d==1))),f->f>0))153 ·····················#select(degs,d->d==1))),f->f>0))
154 ·--·used·2.47859s·(cpu);·1.39746s·(thread);·0s·(gc)154 ·--·used·4.09656s·(cpu);·1.66143s·(thread);·0s·(gc)
  
155 o15·=·58</code></pre>155 o15·=·58</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
157 ········</table>157 ········</table>
158 ······</div>158 ······</div>
159 ······<div·class="waystouse">159 ······<div·class="waystouse">
160 ········<h2>Ways·to·use·<span·class="tt">randomKRationalPoint</span>:</h2>160 ········<h2>Ways·to·use·<span·class="tt">randomKRationalPoint</span>:</h2>
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30 o4·:·Ideal·of·R30 o4·:·Ideal·of·R
31 i5·:·codim·I,·degree·I31 i5·:·codim·I,·degree·I
  
32 o5·=·(2,·10)32 o5·=·(2,·10)
  
33 o5·:·Sequence33 o5·:·Sequence
34 i6·:·time·randomKRationalPoint(I)34 i6·:·time·randomKRationalPoint(I)
35 ·--·used·0.120879s·(cpu);·0.0600286s·(thread);·0s·(gc)35 ·--·used·0.138709s·(cpu);·0.0707743s·(thread);·0s·(gc)
  
36 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)36 o6·=·ideal·(x··-·53x·,·x··+·8x·,·x··-·4x·)
37 ·············2······3···1·····3···0·····337 ·············2······3···1·····3···0·····3
  
38 o6·:·Ideal·of·R38 o6·:·Ideal·of·R
39 i7·:·R=kk[x_0..x_5];39 i7·:·R=kk[x_0..x_5];
40 i8·:·I=minors(3,random(R^5,R^{3:-1}));40 i8·:·I=minors(3,random(R^5,R^{3:-1}));
Offset 46, 15 lines modifiedOffset 46, 15 lines modified
46 o8·:·Ideal·of·R46 o8·:·Ideal·of·R
47 i9·:·codim·I,·degree·I47 i9·:·codim·I,·degree·I
  
48 o9·=·(3,·10)48 o9·=·(3,·10)
  
49 o9·:·Sequence49 o9·:·Sequence
50 i10·:·time·randomKRationalPoint(I)50 i10·:·time·randomKRationalPoint(I)
51 ·--·used·0.285537s·(cpu);·0.166293s·(thread);·0s·(gc)51 ·--·used·0.331339s·(cpu);·0.194846s·(thread);·0s·(gc)
  
52 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)52 o10·=·ideal·(x··-·27x·,·x··-·16x·,·x··-·9x·,·x··+·44x·,·x··-·52x·)
53 ··············4······5···3······5···2·····5···1······5···0······553 ··············4······5···3······5···2·····5···1······5···0······5
  
54 o10·:·Ideal·of·R54 o10·:·Ideal·of·R
55 The·claim·that·$63·\%$·of·the·intersections·contain·a·K-rational·point·can·be55 The·claim·that·$63·\%$·of·the·intersections·contain·a·K-rational·point·can·be
56 experimentally·tested:56 experimentally·tested:
Offset 70, 14 lines modifiedOffset 70, 14 lines modified
70 o13·:·RR·(of·precision·53)70 o13·:·RR·(of·precision·53)
71 i14·:·I=ideal·random(n,R);71 i14·:·I=ideal·random(n,R);
  
72 o14·:·Ideal·of·R72 o14·:·Ideal·of·R
73 i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c-73 i15·:·time·(#select(apply(100,i->(degs=apply(decompose(I+ideal·random(1,R)),c-
74 >degree·c);74 >degree·c);
75 ·····················#select(degs,d->d==1))),f->f>0))75 ·····················#select(degs,d->d==1))),f->f>0))
76 ·--·used·2.47859s·(cpu);·1.39746s·(thread);·0s·(gc)76 ·--·used·4.09656s·(cpu);·1.66143s·(thread);·0s·(gc)
  
77 o15·=·5877 o15·=·58
78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mK\x8KR\x8Ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lP\x8Po\x8oi\x8in\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mK\x8KR\x8Ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lP\x8Po\x8oi\x8in\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
79 ····*·randomKRationalPoint(Ideal)79 ····*·randomKRationalPoint(Ideal)
80 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*80 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
81 The·object·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8K_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8P_\x8o_\x8i_\x8n_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.81 The·object·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8K_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8P_\x8o_\x8i_\x8n_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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Offset 67, 32 lines modifiedOffset 67, 32 lines modified
67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 ········<table·class="examples">70 ········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()72 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()
  
73 o1·=·/tmp/M2-3695513-0/0</code></pre>73 o1·=·/tmp/M2-2676437-0/0</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir76 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir
  
77 o2·=·/tmp/M2-3695513-0/0</code></pre>77 o2·=·/tmp/M2-2676437-0/0</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·(fn·=·dir·|·&quot;/&quot;·|·&quot;foo&quot;)·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close80 <td>··············<pre><code·class="language-macaulay2">i3·:·(fn·=·dir·|·&quot;/&quot;·|·&quot;foo&quot;)·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
81 o3·=·/tmp/M2-3695513-0/0/foo81 o3·=·/tmp/M2-2676437-0/0/foo
  
82 o3·:·File</code></pre>82 o3·:·File</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i4·:·readDirectory·dir85 <td>··············<pre><code·class="language-macaulay2">i4·:·readDirectory·dir
  
86 o4·=·{..,·.,·foo}86 o4·=·{foo,·.,·..}
  
87 o4·:·List</code></pre>87 o4·:·List</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i5·:·removeFile·fn</code></pre>90 <td>··············<pre><code·class="language-macaulay2">i5·:·removeFile·fn</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
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11 ····*·Inputs:11 ····*·Inputs:
12 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·directory12 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·directory
13 ····*·Outputs:13 ····*·Outputs:
14 ··········o·a·_\x8l_\x8i_\x8s_\x8t,·the·list·of·filenames·stored·in·the·directory14 ··········o·a·_\x8l_\x8i_\x8s_\x8t,·the·list·of·filenames·stored·in·the·directory
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·dir·=·temporaryFileName()16 i1·:·dir·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3695513-0/017 o1·=·/tmp/M2-2676437-0/0
18 i2·:·makeDirectory·dir18 i2·:·makeDirectory·dir
  
19 o2·=·/tmp/M2-3695513-0/019 o2·=·/tmp/M2-2676437-0/0
20 i3·:·(fn·=·dir·|·"/"·|·"foo")·<<·"hi·there"·<<·close20 i3·:·(fn·=·dir·|·"/"·|·"foo")·<<·"hi·there"·<<·close
  
21 o3·=·/tmp/M2-3695513-0/0/foo21 o3·=·/tmp/M2-2676437-0/0/foo
  
22 o3·:·File22 o3·:·File
23 i4·:·readDirectory·dir23 i4·:·readDirectory·dir
  
24 o4·=·{..,·.,·foo}24 o4·=·{foo,·.,·..}
  
25 o4·:·List25 o4·:·List
26 i5·:·removeFile·fn26 i5·:·removeFile·fn
27 i6·:·removeDirectory·dir27 i6·:·removeDirectory·dir
28 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*28 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
29 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·remove·a·directory29 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·remove·a·directory
30 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e·--·remove·a·file30 ····*·_\x8r_\x8e_\x8m_\x8o_\x8v_\x8e_\x8F_\x8i_\x8l_\x8e·--·remove·a·file
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44 ······<h1>reading·files</h1>44 ······<h1>reading·files</h1>
45 ······<div>45 ······<div>
46 Sometimes·a·file·will·contain·a·single·expression·whose·value·you·wish·to·have·access·to.··For·example,·it·might·be·a·polynomial·produced·by·another·program.··The·function·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>·can·be·used·to·obtain·the·entire·contents·of·a·file·as·a·single·string.··We·illustrate·this·here·with·a·file·whose·name·is·<span·class="tt">expression</span>.········<p></p>46 Sometimes·a·file·will·contain·a·single·expression·whose·value·you·wish·to·have·access·to.··For·example,·it·might·be·a·polynomial·produced·by·another·program.··The·function·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>·can·be·used·to·obtain·the·entire·contents·of·a·file·as·a·single·string.··We·illustrate·this·here·with·a·file·whose·name·is·<span·class="tt">expression</span>.········<p></p>
47 First·we·create·the·file·by·writing·the·desired·text·to·it.········<table·class="examples">47 First·we·create·the·file·by·writing·the·desired·text·to·it.········<table·class="examples">
48 ··········<tr>48 ··········<tr>
49 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()49 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
50 o1·=·/tmp/M2-3685338-0/0</code></pre>50 o1·=·/tmp/M2-2665365-0/0</code></pre>
51 </td>··········</tr>51 </td>··········</tr>
52 ··········<tr>52 ··········<tr>
53 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3&quot;·&lt;&lt;·endl·&lt;&lt;·close53 <td>··············<pre><code·class="language-macaulay2">i2·:·fn·&lt;&lt;·&quot;z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3&quot;·&lt;&lt;·endl·&lt;&lt;·close
  
54 o2·=·/tmp/M2-3685338-0/054 o2·=·/tmp/M2-2665365-0/0
  
55 o2·:·File</code></pre>55 o2·:·File</code></pre>
56 </td>··········</tr>56 </td>··········</tr>
57 ········</table>57 ········</table>
58 Now·we·get·the·contents·of·the·file,·as·a·single·string.········<table·class="examples">58 Now·we·get·the·contents·of·the·file,·as·a·single·string.········<table·class="examples">
59 ··········<tr>59 ··········<tr>
60 <td>··············<pre><code·class="language-macaulay2">i3·:·get·fn60 <td>··············<pre><code·class="language-macaulay2">i3·:·get·fn
Offset 96, 15 lines modifiedOffset 96, 15 lines modified
96 ········</table>96 ········</table>
97 Often·a·file·will·contain·code·written·in·the·Macaulay2·language.·Let's·create·such·a·file.········<table·class="examples">97 Often·a·file·will·contain·code·written·in·the·Macaulay2·language.·Let's·create·such·a·file.········<table·class="examples">
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i7·:·fn·&lt;&lt;·&quot;sample·=·2^10099 <td>··············<pre><code·class="language-macaulay2">i7·:·fn·&lt;&lt;·&quot;sample·=·2^100
100 ·····print·sample100 ·····print·sample
101 ·····&quot;·&lt;&lt;·close101 ·····&quot;·&lt;&lt;·close
  
102 o7·=·/tmp/M2-3685338-0/0102 o7·=·/tmp/M2-2665365-0/0
  
103 o7·:·File</code></pre>103 o7·:·File</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ········</table>105 ········</table>
106 Now·verify·that·it·contains·the·desired·text·with·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>.········<table·class="examples">106 Now·verify·that·it·contains·the·desired·text·with·<a·title="get·the·contents·of·a·file"·href="_get.html">get</a>.········<table·class="examples">
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i8·:·get·fn108 <td>··············<pre><code·class="language-macaulay2">i8·:·get·fn
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Offset 7, 20 lines modifiedOffset 7, 20 lines modified
7 Sometimes·a·file·will·contain·a·single·expression·whose·value·you·wish·to·have7 Sometimes·a·file·will·contain·a·single·expression·whose·value·you·wish·to·have
8 access·to.·For·example,·it·might·be·a·polynomial·produced·by·another·program.8 access·to.·For·example,·it·might·be·a·polynomial·produced·by·another·program.
9 The·function·_\x8g_\x8e_\x8t·can·be·used·to·obtain·the·entire·contents·of·a·file·as·a9 The·function·_\x8g_\x8e_\x8t·can·be·used·to·obtain·the·entire·contents·of·a·file·as·a
10 single·string.·We·illustrate·this·here·with·a·file·whose·name·is·expression.10 single·string.·We·illustrate·this·here·with·a·file·whose·name·is·expression.
11 First·we·create·the·file·by·writing·the·desired·text·to·it.11 First·we·create·the·file·by·writing·the·desired·text·to·it.
12 i1·:·fn·=·temporaryFileName()12 i1·:·fn·=·temporaryFileName()
  
13 o1·=·/tmp/M2-3685338-0/013 o1·=·/tmp/M2-2665365-0/0
14 i2·:·fn·<<14 i2·:·fn·<<
15 "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"15 "z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2+8*y^3"
16 <<·endl·<<·close16 <<·endl·<<·close
  
17 o2·=·/tmp/M2-3685338-0/017 o2·=·/tmp/M2-2665365-0/0
  
18 o2·:·File18 o2·:·File
19 Now·we·get·the·contents·of·the·file,·as·a·single·string.19 Now·we·get·the·contents·of·the·file,·as·a·single·string.
20 i3·:·get·fn20 i3·:·get·fn
  
21 o3·=·z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^221 o3·=·z^6+3*x*z^4+6*y*z^4+3*x^2*z^2+12*x*y*z^2+12*y^2*z^2+x^3+6*x^2*y+12*x*y^2
22 ·····+8*y^322 ·····+8*y^3
Offset 50, 15 lines modifiedOffset 50, 15 lines modified
50 o6·:·Expression·of·class·Product50 o6·:·Expression·of·class·Product
51 Often·a·file·will·contain·code·written·in·the·Macaulay2·language.·Let's·create51 Often·a·file·will·contain·code·written·in·the·Macaulay2·language.·Let's·create
52 such·a·file.52 such·a·file.
53 i7·:·fn·<<·"sample·=·2^10053 i7·:·fn·<<·"sample·=·2^100
54 ·····print·sample54 ·····print·sample
55 ·····"·<<·close55 ·····"·<<·close
  
56 o7·=·/tmp/M2-3685338-0/056 o7·=·/tmp/M2-2665365-0/0
  
57 o7·:·File57 o7·:·File
58 Now·verify·that·it·contains·the·desired·text·with·_\x8g_\x8e_\x8t.58 Now·verify·that·it·contains·the·desired·text·with·_\x8g_\x8e_\x8t.
59 i8·:·get·fn59 i8·:·get·fn
  
60 o8·=·sample·=·2^10060 o8·=·sample·=·2^100
61 ·····print·sample61 ·····print·sample
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
    
Offset 67, 15 lines modifiedOffset 67, 15 lines modified
67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 ········<table·class="examples">70 ········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·p·=·temporaryFileName·()72 <td>··············<pre><code·class="language-macaulay2">i1·:·p·=·temporaryFileName·()
  
73 o1·=·/tmp/M2-3696255-0/0</code></pre>73 o1·=·/tmp/M2-2677144-0/0</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·symlinkFile·(&quot;foo&quot;,·p)</code></pre>76 <td>··············<pre><code·class="language-macaulay2">i2·:·symlinkFile·(&quot;foo&quot;,·p)</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i3·:·readlink·p79 <td>··············<pre><code·class="language-macaulay2">i3·:·readlink·p
  
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11 ····*·Inputs:11 ····*·Inputs:
12 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·file12 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·file
13 ····*·Outputs:13 ····*·Outputs:
14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·symbolic·link14 ··········o·a·_\x8B_\x8o_\x8o_\x8l_\x8e_\x8a_\x8n_\x8·_\x8v_\x8a_\x8l_\x8u_\x8e,·whether·fn·is·the·path·to·a·symbolic·link
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·p·=·temporaryFileName·()16 i1·:·p·=·temporaryFileName·()
  
17 o1·=·/tmp/M2-3696255-0/017 o1·=·/tmp/M2-2677144-0/0
18 i2·:·symlinkFile·("foo",·p)18 i2·:·symlinkFile·("foo",·p)
19 i3·:·readlink·p19 i3·:·readlink·p
  
20 o3·=·foo20 o3·=·foo
21 i4·:·removeFile·p21 i4·:·removeFile·p
22 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*22 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
23 The·object·_\x8r_\x8e_\x8a_\x8d_\x8l_\x8i_\x8n_\x8k·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.23 The·object·_\x8r_\x8e_\x8a_\x8d_\x8l_\x8i_\x8n_\x8k·is·a·_\x8c_\x8o_\x8m_\x8p_\x8i_\x8l_\x8e_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
    
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67 ······</div>67 ······</div>
68 ······<div>68 ······<div>
69 ········<h2>Description</h2>69 ········<h2>Description</h2>
70 ········<table·class="examples">70 ········<table·class="examples">
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i1·:·realpath·&quot;.&quot;72 <td>··············<pre><code·class="language-macaulay2">i1·:·realpath·&quot;.&quot;
  
73 o1·=·/tmp/M2-3680509-0/83-rundir/</code></pre>73 o1·=·/tmp/M2-2656573-0/83-rundir/</code></pre>
74 </td>··········</tr>74 </td>··········</tr>
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i2·:·p·=·temporaryFileName()76 <td>··············<pre><code·class="language-macaulay2">i2·:·p·=·temporaryFileName()
  
77 o2·=·/tmp/M2-3696286-0/0</code></pre>77 o2·=·/tmp/M2-2677183-0/0</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·q·=·temporaryFileName()80 <td>··············<pre><code·class="language-macaulay2">i3·:·q·=·temporaryFileName()
  
81 o3·=·/tmp/M2-3696286-0/1</code></pre>81 o3·=·/tmp/M2-2677183-0/1</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·symlinkFile(p,q)</code></pre>84 <td>··············<pre><code·class="language-macaulay2">i4·:·symlinkFile(p,q)</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i5·:·p·&lt;&lt;·close87 <td>··············<pre><code·class="language-macaulay2">i5·:·p·&lt;&lt;·close
  
88 o5·=·/tmp/M2-3696286-0/088 o5·=·/tmp/M2-2677183-0/0
  
89 o5·:·File</code></pre>89 o5·:·File</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i6·:·readlink·q92 <td>··············<pre><code·class="language-macaulay2">i6·:·readlink·q
  
93 o6·=·/tmp/M2-3696286-0/0</code></pre>93 o6·=·/tmp/M2-2677183-0/0</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i7·:·realpath·q96 <td>··············<pre><code·class="language-macaulay2">i7·:·realpath·q
  
97 o7·=·/tmp/M2-3696286-0/0</code></pre>97 o7·=·/tmp/M2-2677183-0/0</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i8·:·removeFile·p</code></pre>100 <td>··············<pre><code·class="language-macaulay2">i8·:·removeFile·p</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i9·:·removeFile·q</code></pre>103 <td>··············<pre><code·class="language-macaulay2">i9·:·removeFile·q</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ········</table>105 ········</table>
106 ········<p>The·empty·string·is·interpreted·as·a·reference·to·the·current·directory.</p>106 ········<p>The·empty·string·is·interpreted·as·a·reference·to·the·current·directory.</p>
107 ········<table·class="examples">107 ········<table·class="examples">
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i10·:·realpath·&quot;&quot;109 <td>··············<pre><code·class="language-macaulay2">i10·:·realpath·&quot;&quot;
  
110 o10·=·/tmp/M2-3680509-0/83-rundir/</code></pre>110 o10·=·/tmp/M2-2656573-0/83-rundir/</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ········</table>112 ········</table>
113 ······</div>113 ······</div>
114 ······<div>114 ······<div>
115 ········<h2>Caveat</h2>115 ········<h2>Caveat</h2>
116 Every·component·of·the·path·must·exist·in·the·file·system·and·be·accessible·to·the·user.·Terminal·slashes·will·be·dropped.··Warning:·under·most·operating·systems,·the·value·returned·is·an·absolute·path·(one·starting·at·the·root·of·the·file·system),·but·under·Solaris,·this·system·call·may,·in·certain·circumstances,·return·a·relative·path·when·given·a·relative·path.······</div>116 Every·component·of·the·path·must·exist·in·the·file·system·and·be·accessible·to·the·user.·Terminal·slashes·will·be·dropped.··Warning:·under·most·operating·systems,·the·value·returned·is·an·absolute·path·(one·starting·at·the·root·of·the·file·system),·but·under·Solaris,·this·system·call·may,·in·certain·circumstances,·return·a·relative·path·when·given·a·relative·path.······</div>
117 ······<div>117 ······<div>
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13 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename,·or·path·to·a·file13 ··········o·fn,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename,·or·path·to·a·file
14 ····*·Outputs:14 ····*·Outputs:
15 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·pathname·to·fn·passing·through·no·symbolic·links,·and15 ··········o·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·pathname·to·fn·passing·through·no·symbolic·links,·and
16 ············ending·with·a·slash·if·fn·refers·to·a·directory16 ············ending·with·a·slash·if·fn·refers·to·a·directory
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 i1·:·realpath·"."18 i1·:·realpath·"."
  
19 o1·=·/tmp/M2-3680509-0/83-rundir/19 o1·=·/tmp/M2-2656573-0/83-rundir/
20 i2·:·p·=·temporaryFileName()20 i2·:·p·=·temporaryFileName()
  
21 o2·=·/tmp/M2-3696286-0/021 o2·=·/tmp/M2-2677183-0/0
22 i3·:·q·=·temporaryFileName()22 i3·:·q·=·temporaryFileName()
  
23 o3·=·/tmp/M2-3696286-0/123 o3·=·/tmp/M2-2677183-0/1
24 i4·:·symlinkFile(p,q)24 i4·:·symlinkFile(p,q)
25 i5·:·p·<<·close25 i5·:·p·<<·close
  
26 o5·=·/tmp/M2-3696286-0/026 o5·=·/tmp/M2-2677183-0/0
  
27 o5·:·File27 o5·:·File
28 i6·:·readlink·q28 i6·:·readlink·q
  
29 o6·=·/tmp/M2-3696286-0/029 o6·=·/tmp/M2-2677183-0/0
30 i7·:·realpath·q30 i7·:·realpath·q
  
31 o7·=·/tmp/M2-3696286-0/031 o7·=·/tmp/M2-2677183-0/0
32 i8·:·removeFile·p32 i8·:·removeFile·p
33 i9·:·removeFile·q33 i9·:·removeFile·q
34 The·empty·string·is·interpreted·as·a·reference·to·the·current·directory.34 The·empty·string·is·interpreted·as·a·reference·to·the·current·directory.
35 i10·:·realpath·""35 i10·:·realpath·""
  
36 o10·=·/tmp/M2-3680509-0/83-rundir/36 o10·=·/tmp/M2-2656573-0/83-rundir/
37 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
38 Every·component·of·the·path·must·exist·in·the·file·system·and·be·accessible·to38 Every·component·of·the·path·must·exist·in·the·file·system·and·be·accessible·to
39 the·user.·Terminal·slashes·will·be·dropped.·Warning:·under·most·operating39 the·user.·Terminal·slashes·will·be·dropped.·Warning:·under·most·operating
40 systems,·the·value·returned·is·an·absolute·path·(one·starting·at·the·root·of40 systems,·the·value·returned·is·an·absolute·path·(one·starting·at·the·root·of
41 the·file·system),·but·under·Solaris,·this·system·call·may,·in·certain41 the·file·system),·but·under·Solaris,·this·system·call·may,·in·certain
42 circumstances,·return·a·relative·path·when·given·a·relative·path.42 circumstances,·return·a·relative·path·when·given·a·relative·path.
43 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*43 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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73 ········<h2>Description</h2>73 ········<h2>Description</h2>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·&quot;--·finalizing·sequence·#&quot;|i|&quot;·--&quot;))</code></pre>76 <td>··············<pre><code·class="language-macaulay2">i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·&quot;--·finalizing·sequence·#&quot;|i|&quot;·--&quot;))</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·collectGarbage()·79 <td>··············<pre><code·class="language-macaulay2">i2·:·collectGarbage()·
80 --finalization:·(1)[5]:·--·finalizing·sequence·#6·-- 
81 --finalization:·(2)[0]:·--·finalizing·sequence·#1·-- 
82 --finalization:·(3)[3]:·--·finalizing·sequence·#4·--80 --finalization:·(1)[3]:·--·finalizing·sequence·#4·--
83 --finalization:·(4)[4]:·--·finalizing·sequence·#5·--81 --finalization:·(2)[4]:·--·finalizing·sequence·#5·--
 82 --finalization:·(3)[6]:·--·finalizing·sequence·#7·--
84 --finalization:·(5)[7]:·--·finalizing·sequence·#8·--83 --finalization:·(4)[7]:·--·finalizing·sequence·#8·--
 84 --finalization:·(5)[1]:·--·finalizing·sequence·#2·--
85 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--85 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--
86 --finalization:·(7)[1]:·--·finalizing·sequence·#2·--86 --finalization:·(7)[0]:·--·finalizing·sequence·#1·--
87 --finalization:·(8)[6]:·--·finalizing·sequence·#7·--</code></pre>87 --finalization:·(8)[5]:·--·finalizing·sequence·#6·--</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
90 ······</div>90 ······</div>
91 ······<div>91 ······<div>
92 ········<h2>Caveat</h2>92 ········<h2>Caveat</h2>
93 This·function·should·mainly·be·used·for·debugging.··Having·a·large·number·of·finalizers·might·degrade·the·performance·of·the·program.··Moreover,·registering·two·or·more·objects·that·are·members·of·a·circular·chain·of·pointers·for·finalization·will·result·in·a·memory·leak,·with·none·of·the·objects·in·the·chain·being·freed,·even·if·nothing·else·points·to·any·of·them.······</div>93 This·function·should·mainly·be·used·for·debugging.··Having·a·large·number·of·finalizers·might·degrade·the·performance·of·the·program.··Moreover,·registering·two·or·more·objects·that·are·members·of·a·circular·chain·of·pointers·for·finalization·will·result·in·a·memory·leak,·with·none·of·the·objects·in·the·chain·being·freed,·even·if·nothing·else·points·to·any·of·them.······</div>
94 ······<div>94 ······<div>
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15 ····*·Consequences:15 ····*·Consequences:
16 ··········o·A·finalizer·is·registered·with·the·garbage·collector·to·print·a16 ··········o·A·finalizer·is·registered·with·the·garbage·collector·to·print·a
17 ············string·when·that·object·is·collected·as·garbage17 ············string·when·that·object·is·collected·as·garbage
18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*18 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
19 i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·"-19 i1·:·for·i·from·1·to·9·do·(x·:=·0·..·10000·;·registerFinalizer(x,·"-
20 -·finalizing·sequence·#"|i|"·--"))20 -·finalizing·sequence·#"|i|"·--"))
21 i2·:·collectGarbage()21 i2·:·collectGarbage()
22 --finalization:·(1)[5]:·--·finalizing·sequence·#6·-- 
23 --finalization:·(2)[0]:·--·finalizing·sequence·#1·-- 
24 --finalization:·(3)[3]:·--·finalizing·sequence·#4·--22 --finalization:·(1)[3]:·--·finalizing·sequence·#4·--
25 --finalization:·(4)[4]:·--·finalizing·sequence·#5·--23 --finalization:·(2)[4]:·--·finalizing·sequence·#5·--
 24 --finalization:·(3)[6]:·--·finalizing·sequence·#7·--
26 --finalization:·(5)[7]:·--·finalizing·sequence·#8·--25 --finalization:·(4)[7]:·--·finalizing·sequence·#8·--
 26 --finalization:·(5)[1]:·--·finalizing·sequence·#2·--
27 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--27 --finalization:·(6)[2]:·--·finalizing·sequence·#3·--
28 --finalization:·(7)[1]:·--·finalizing·sequence·#2·--28 --finalization:·(7)[0]:·--·finalizing·sequence·#1·--
29 --finalization:·(8)[6]:·--·finalizing·sequence·#7·--29 --finalization:·(8)[5]:·--·finalizing·sequence·#6·--
30 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
31 This·function·should·mainly·be·used·for·debugging.·Having·a·large·number·of31 This·function·should·mainly·be·used·for·debugging.·Having·a·large·number·of
32 finalizers·might·degrade·the·performance·of·the·program.·Moreover,·registering32 finalizers·might·degrade·the·performance·of·the·program.·Moreover,·registering
33 two·or·more·objects·that·are·members·of·a·circular·chain·of·pointers·for33 two·or·more·objects·that·are·members·of·a·circular·chain·of·pointers·for
34 finalization·will·result·in·a·memory·leak,·with·none·of·the·objects·in·the34 finalization·will·result·in·a·memory·leak,·with·none·of·the·objects·in·the
35 chain·being·freed,·even·if·nothing·else·points·to·any·of·them.35 chain·being·freed,·even·if·nothing·else·points·to·any·of·them.
36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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69 ······</div>69 ······</div>
70 ······<div>70 ······<div>
71 ········<h2>Description</h2>71 ········<h2>Description</h2>
72 ········<table·class="examples">72 ········<table·class="examples">
73 ··········<tr>73 ··········<tr>
74 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()74 <td>··············<pre><code·class="language-macaulay2">i1·:·dir·=·temporaryFileName()
  
75 o1·=·/tmp/M2-3681736-0/0</code></pre>75 o1·=·/tmp/M2-2661573-0/0</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir78 <td>··············<pre><code·class="language-macaulay2">i2·:·makeDirectory·dir
  
79 o2·=·/tmp/M2-3681736-0/0</code></pre>79 o2·=·/tmp/M2-2661573-0/0</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·readDirectory·dir82 <td>··············<pre><code·class="language-macaulay2">i3·:·readDirectory·dir
  
83 o3·=·{..,·.}83 o3·=·{.,·..}
  
84 o3·:·List</code></pre>84 o3·:·List</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·removeDirectory·dir</code></pre>87 <td>··············<pre><code·class="language-macaulay2">i4·:·removeDirectory·dir</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
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11 ····*·Inputs:11 ····*·Inputs:
12 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·directory12 ··········o·dir,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g,·a·filename·or·path·to·a·directory
13 ····*·Consequences:13 ····*·Consequences:
14 ··········o·the·directory·is·removed14 ··········o·the·directory·is·removed
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·dir·=·temporaryFileName()16 i1·:·dir·=·temporaryFileName()
  
17 o1·=·/tmp/M2-3681736-0/017 o1·=·/tmp/M2-2661573-0/0
18 i2·:·makeDirectory·dir18 i2·:·makeDirectory·dir
  
19 o2·=·/tmp/M2-3681736-0/019 o2·=·/tmp/M2-2661573-0/0
20 i3·:·readDirectory·dir20 i3·:·readDirectory·dir
  
21 o3·=·{..,·.}21 o3·=·{.,·..}
  
22 o3·:·List22 o3·:·List
23 i4·:·removeDirectory·dir23 i4·:·removeDirectory·dir
24 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
25 ····*·_\x8r_\x8e_\x8a_\x8d_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·read·the·contents·of·a·directory25 ····*·_\x8r_\x8e_\x8a_\x8d_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·read·the·contents·of·a·directory
26 ····*·_\x8m_\x8a_\x8k_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·make·a·directory26 ····*·_\x8m_\x8a_\x8k_\x8e_\x8D_\x8i_\x8r_\x8e_\x8c_\x8t_\x8o_\x8r_\x8y·--·make·a·directory
27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<p>This·string·may·be·concatenated·with·an·absolute·path·to·get·one·understandable·by·external·programs.·Currently,·this·makes·a·difference·only·under·Microsoft·Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that·are·not·part·of·Cygwin.··Fortunately,·programs·compiled·under·Cygwin·know·were·to·look·for·files·whose·paths·start·with·something·like·<span·class="tt">C:/</span>,·so·it·is·safe·always·to·concatenate·with·the·value·of·<a·href="_root__Path.html">rootPath</a>,·even·when·it·is·unknown·whether·the·external·program·has·been·compiled·under·Cygwin.</p>64 ········<p>This·string·may·be·concatenated·with·an·absolute·path·to·get·one·understandable·by·external·programs.·Currently,·this·makes·a·difference·only·under·Microsoft·Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that·are·not·part·of·Cygwin.··Fortunately,·programs·compiled·under·Cygwin·know·were·to·look·for·files·whose·paths·start·with·something·like·<span·class="tt">C:/</span>,·so·it·is·safe·always·to·concatenate·with·the·value·of·<a·href="_root__Path.html">rootPath</a>,·even·when·it·is·unknown·whether·the·external·program·has·been·compiled·under·Cygwin.</p>
65 ········<table·class="examples">65 ········<table·class="examples">
66 ··········<tr>66 ··········<tr>
67 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()67 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
68 o1·=·/tmp/M2-3680619-0/0</code></pre>68 o1·=·/tmp/M2-2656700-0/0</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ··········<tr>70 ··········<tr>
71 <td>··············<pre><code·class="language-macaulay2">i2·:·rootPath·|·fn71 <td>··············<pre><code·class="language-macaulay2">i2·:·rootPath·|·fn
  
72 o2·=·/tmp/M2-3680619-0/0</code></pre>72 o2·=·/tmp/M2-2656700-0/0</code></pre>
73 </td>··········</tr>73 </td>··········</tr>
74 ········</table>74 ········</table>
75 ······</div>75 ······</div>
76 ······<div>76 ······<div>
77 ········<h2>See·also</h2>77 ········<h2>See·also</h2>
78 ········<ul>78 ········<ul>
79 ··········<li>79 ··········<li>
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16 Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that16 Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that
17 are·not·part·of·Cygwin.·Fortunately,·programs·compiled·under·Cygwin·know·were17 are·not·part·of·Cygwin.·Fortunately,·programs·compiled·under·Cygwin·know·were
18 to·look·for·files·whose·paths·start·with·something·like·C:/,·so·it·is·safe18 to·look·for·files·whose·paths·start·with·something·like·C:/,·so·it·is·safe
19 always·to·concatenate·with·the·value·of·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h,·even·when·it·is·unknown19 always·to·concatenate·with·the·value·of·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h,·even·when·it·is·unknown
20 whether·the·external·program·has·been·compiled·under·Cygwin.20 whether·the·external·program·has·been·compiled·under·Cygwin.
21 i1·:·fn·=·temporaryFileName()21 i1·:·fn·=·temporaryFileName()
  
22 o1·=·/tmp/M2-3680619-0/022 o1·=·/tmp/M2-2656700-0/0
23 i2·:·rootPath·|·fn23 i2·:·rootPath·|·fn
  
24 o2·=·/tmp/M2-3680619-0/024 o2·=·/tmp/M2-2656700-0/0
25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
26 ····*·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I26 ····*·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I
27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
28 The·object·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h·is·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g.28 The·object·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h·is·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g.
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62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 ········<p>This·string·may·be·concatenated·with·an·absolute·path·to·get·one·understandable·by·an·external·browser.·Currently,·this·makes·a·difference·only·under·Microsoft·Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that·are·not·part·of·Cygwin.··Fortunately,·programs·compiled·under·Cygwin·know·were·to·look·for·files·whose·paths·start·with·something·like·<span·class="tt">C:/</span>,·so·it·is·safe·always·to·concatenate·with·the·value·of·<a·href="_root__Path.html">rootPath</a>,·even·when·it·is·unknown·whether·the·external·program·has·been·compiled·under·Cygwin.</p>64 ········<p>This·string·may·be·concatenated·with·an·absolute·path·to·get·one·understandable·by·an·external·browser.·Currently,·this·makes·a·difference·only·under·Microsoft·Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that·are·not·part·of·Cygwin.··Fortunately,·programs·compiled·under·Cygwin·know·were·to·look·for·files·whose·paths·start·with·something·like·<span·class="tt">C:/</span>,·so·it·is·safe·always·to·concatenate·with·the·value·of·<a·href="_root__Path.html">rootPath</a>,·even·when·it·is·unknown·whether·the·external·program·has·been·compiled·under·Cygwin.</p>
65 ········<table·class="examples">65 ········<table·class="examples">
66 ··········<tr>66 ··········<tr>
67 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()67 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
68 o1·=·/tmp/M2-3695159-0/0</code></pre>68 o1·=·/tmp/M2-2676330-0/0</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ··········<tr>70 ··········<tr>
71 <td>··············<pre><code·class="language-macaulay2">i2·:·rootURI·|·fn71 <td>··············<pre><code·class="language-macaulay2">i2·:·rootURI·|·fn
  
72 o2·=·file:///tmp/M2-3695159-0/0</code></pre>72 o2·=·file:///tmp/M2-2676330-0/0</code></pre>
73 </td>··········</tr>73 </td>··········</tr>
74 ········</table>74 ········</table>
75 ······</div>75 ······</div>
76 ······<div>76 ······<div>
77 ········<h2>See·also</h2>77 ········<h2>See·also</h2>
78 ········<ul>78 ········<ul>
79 ··········<li>79 ··········<li>
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16 Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that16 Windows·with·Cygwin,·but·there·it's·crucial·for·those·external·programs·that
17 are·not·part·of·Cygwin.·Fortunately,·programs·compiled·under·Cygwin·know·were17 are·not·part·of·Cygwin.·Fortunately,·programs·compiled·under·Cygwin·know·were
18 to·look·for·files·whose·paths·start·with·something·like·C:/,·so·it·is·safe18 to·look·for·files·whose·paths·start·with·something·like·C:/,·so·it·is·safe
19 always·to·concatenate·with·the·value·of·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h,·even·when·it·is·unknown19 always·to·concatenate·with·the·value·of·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h,·even·when·it·is·unknown
20 whether·the·external·program·has·been·compiled·under·Cygwin.20 whether·the·external·program·has·been·compiled·under·Cygwin.
21 i1·:·fn·=·temporaryFileName()21 i1·:·fn·=·temporaryFileName()
  
22 o1·=·/tmp/M2-3695159-0/022 o1·=·/tmp/M2-2676330-0/0
23 i2·:·rootURI·|·fn23 i2·:·rootURI·|·fn
  
24 o2·=·file:///tmp/M2-3695159-0/024 o2·=·file:///tmp/M2-2676330-0/0
25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
26 ····*·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h26 ····*·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h
27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
28 The·object·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I·is·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g.28 The·object·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I·is·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g.
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74 ·····························174 ·····························1
75 o4·:·R-module,·submodule·of·R</code></pre>75 o4·:·R-module,·submodule·of·R</code></pre>
76 </td>··········</tr>76 </td>··········</tr>
77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i5·:·f·=·temporaryFileName()78 <td>··············<pre><code·class="language-macaulay2">i5·:·f·=·temporaryFileName()
  
79 o5·=·/tmp/M2-3692110-0/0</code></pre>79 o5·=·/tmp/M2-2674564-0/0</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i6·:·f·&lt;&lt;·toString·(p,m,M)·&lt;&lt;·close82 <td>··············<pre><code·class="language-macaulay2">i6·:·f·&lt;&lt;·toString·(p,m,M)·&lt;&lt;·close
  
83 o6·=·/tmp/M2-3692110-0/083 o6·=·/tmp/M2-2674564-0/0
  
84 o6·:·File</code></pre>84 o6·:·File</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i7·:·get·f87 <td>··············<pre><code·class="language-macaulay2">i7·:·get·f
  
88 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,88 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,
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28 o4·=·image·|·x2·x2-y2·xyz7·|28 o4·=·image·|·x2·x2-y2·xyz7·|
  
29 ·····························129 ·····························1
30 o4·:·R-module,·submodule·of·R30 o4·:·R-module,·submodule·of·R
31 i5·:·f·=·temporaryFileName()31 i5·:·f·=·temporaryFileName()
  
32 o5·=·/tmp/M2-3692110-0/032 o5·=·/tmp/M2-2674564-0/0
33 i6·:·f·<<·toString·(p,m,M)·<<·close33 i6·:·f·<<·toString·(p,m,M)·<<·close
  
34 o6·=·/tmp/M2-3692110-0/034 o6·=·/tmp/M2-2674564-0/0
  
35 o6·:·File35 o6·:·File
36 i7·:·get·f36 i7·:·get·f
  
37 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,37 o7·=·(x^3-3*x^2*y+3*x*y^2-y^3-3*x^2+6*x*y-3*y^2+3*x-3*y-1,matrix·{{x^2,
38 ·····x^2-y^2,·x*y*z^7}},image·matrix·{{x^2,·x^2-y^2,·x*y*z^7}})38 ·····x^2-y^2,·x*y*z^7}},image·matrix·{{x^2,·x^2-y^2,·x*y*z^7}})
39 i8·:·(p',m',M')·=·value·get·f39 i8·:·(p',m',M')·=·value·get·f
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320 <td>··············<pre><code·class="language-macaulay2">i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;</code></pre>320 <td>··············<pre><code·class="language-macaulay2">i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;</code></pre>
321 </td>··········</tr>321 </td>··········</tr>
322 ··········<tr>322 ··········<tr>
323 <td>··············<pre><code·class="language-macaulay2">i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;</code></pre>323 <td>··············<pre><code·class="language-macaulay2">i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;</code></pre>
324 </td>··········</tr>324 </td>··········</tr>
325 ··········<tr>325 ··········<tr>
326 <td>··············<pre><code·class="language-macaulay2">i30·:·time·X·=·solve(A,B);326 <td>··············<pre><code·class="language-macaulay2">i30·:·time·X·=·solve(A,B);
327 ·--·used·0.000122043s·(cpu);·0.000111128s·(thread);·0s·(gc)</code></pre>327 ·--·used·0.000701215s·(cpu);·0.00069291s·(thread);·0s·(gc)</code></pre>
328 </td>··········</tr>328 </td>··········</tr>
329 ··········<tr>329 ··········<tr>
330 <td>··············<pre><code·class="language-macaulay2">i31·:·time·X·=·solve(A,B,·MaximalRank=>true);330 <td>··············<pre><code·class="language-macaulay2">i31·:·time·X·=·solve(A,B,·MaximalRank=>true);
331 ·--·used·7.7687e-05s·(cpu);·7.7697e-05s·(thread);·0s·(gc)</code></pre>331 ·--·used·0.000144901s·(cpu);·0.000144972s·(thread);·0s·(gc)</code></pre>
332 </td>··········</tr>332 </td>··········</tr>
333 ··········<tr>333 ··········<tr>
334 <td>··············<pre><code·class="language-macaulay2">i32·:·norm(A*X-B)334 <td>··············<pre><code·class="language-macaulay2">i32·:·norm(A*X-B)
  
335 o32·=·5.111850690840453e-15335 o32·=·5.111850690840453e-15
  
336 o32·:·RR·(of·precision·53)</code></pre>336 o32·:·RR·(of·precision·53)</code></pre>
Offset 353, 19 lines modifiedOffset 353, 19 lines modified
353 <td>··············<pre><code·class="language-macaulay2">i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;</code></pre>353 <td>··············<pre><code·class="language-macaulay2">i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;</code></pre>
354 </td>··········</tr>354 </td>··········</tr>
355 ··········<tr>355 ··········<tr>
356 <td>··············<pre><code·class="language-macaulay2">i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;</code></pre>356 <td>··············<pre><code·class="language-macaulay2">i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;</code></pre>
357 </td>··········</tr>357 </td>··········</tr>
358 ··········<tr>358 ··········<tr>
359 <td>··············<pre><code·class="language-macaulay2">i38·:·time·X·=·solve(A,B);359 <td>··············<pre><code·class="language-macaulay2">i38·:·time·X·=·solve(A,B);
360 ·--·used·0.124122s·(cpu);·0.124129s·(thread);·0s·(gc)</code></pre>360 ·--·used·0.184679s·(cpu);·0.18469s·(thread);·0s·(gc)</code></pre>
361 </td>··········</tr>361 </td>··········</tr>
362 ··········<tr>362 ··········<tr>
363 <td>··············<pre><code·class="language-macaulay2">i39·:·time·X·=·solve(A,B,·MaximalRank=>true);363 <td>··············<pre><code·class="language-macaulay2">i39·:·time·X·=·solve(A,B,·MaximalRank=>true);
364 ·--·used·0.262736s·(cpu);·0.134172s·(thread);·0s·(gc)</code></pre>364 ·--·used·0.183642s·(cpu);·0.183647s·(thread);·0s·(gc)</code></pre>
365 </td>··········</tr>365 </td>··········</tr>
366 ··········<tr>366 ··········<tr>
367 <td>··············<pre><code·class="language-macaulay2">i40·:·norm(A*X-B)367 <td>··············<pre><code·class="language-macaulay2">i40·:·norm(A*X-B)
  
368 o40·=·1.491578274689709814082355885932e-28368 o40·=·1.491578274689709814082355885932e-28
  
369 o40·:·RR·(of·precision·100)</code></pre>369 o40·:·RR·(of·precision·100)</code></pre>
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195 i24·:·printingPrecision·=·4;195 i24·:·printingPrecision·=·4;
196 i25·:·N·=·40196 i25·:·N·=·40
  
197 o25·=·40197 o25·=·40
198 i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;198 i26·:·A·=·mutableMatrix(CC_53,·N,·N);·fillMatrix·A;
199 i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;199 i28·:·B·=·mutableMatrix(CC_53,·N,·2);·fillMatrix·B;
200 i30·:·time·X·=·solve(A,B);200 i30·:·time·X·=·solve(A,B);
201 ·--·used·0.000122043s·(cpu);·0.000111128s·(thread);·0s·(gc)201 ·--·used·0.000701215s·(cpu);·0.00069291s·(thread);·0s·(gc)
202 i31·:·time·X·=·solve(A,B,·MaximalRank=>true);202 i31·:·time·X·=·solve(A,B,·MaximalRank=>true);
203 ·--·used·7.7687e-05s·(cpu);·7.7697e-05s·(thread);·0s·(gc)203 ·--·used·0.000144901s·(cpu);·0.000144972s·(thread);·0s·(gc)
204 i32·:·norm(A*X-B)204 i32·:·norm(A*X-B)
  
205 o32·=·5.111850690840453e-15205 o32·=·5.111850690840453e-15
  
206 o32·:·RR·(of·precision·53)206 o32·:·RR·(of·precision·53)
207 Over·higher·precision·RR·or·CC,·these·routines·will·be·much·slower·than·the207 Over·higher·precision·RR·or·CC,·these·routines·will·be·much·slower·than·the
208 lower·precision·lapack·routines.208 lower·precision·lapack·routines.
209 i33·:·N·=·100209 i33·:·N·=·100
  
210 o33·=·100210 o33·=·100
211 i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;211 i34·:·A·=·mutableMatrix(CC_100,·N,·N);·fillMatrix·A;
212 i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;212 i36·:·B·=·mutableMatrix(CC_100,·N,·2);·fillMatrix·B;
213 i38·:·time·X·=·solve(A,B);213 i38·:·time·X·=·solve(A,B);
214 ·--·used·0.124122s·(cpu);·0.124129s·(thread);·0s·(gc)214 ·--·used·0.184679s·(cpu);·0.18469s·(thread);·0s·(gc)
215 i39·:·time·X·=·solve(A,B,·MaximalRank=>true);215 i39·:·time·X·=·solve(A,B,·MaximalRank=>true);
216 ·--·used·0.262736s·(cpu);·0.134172s·(thread);·0s·(gc)216 ·--·used·0.183642s·(cpu);·0.183647s·(thread);·0s·(gc)
217 i40·:·norm(A*X-B)217 i40·:·norm(A*X-B)
  
218 o40·=·1.491578274689709814082355885932e-28218 o40·=·1.491578274689709814082355885932e-28
  
219 o40·:·RR·(of·precision·100)219 o40·:·RR·(of·precision·100)
220 Giving·the·option·ClosestFit=>true,·in·the·case·when·the·field·is·RR·or·CC,220 Giving·the·option·ClosestFit=>true,·in·the·case·when·the·field·is·RR·or·CC,
221 uses·a·least·squares·algorithm·to·find·a·best·fit·solution.221 uses·a·least·squares·algorithm·to·find·a·best·fit·solution.
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53 <span><a·title="an·optional·argument"·href="___Syzygies.html">Syzygies</a>·--·an·optional·argument</span>··········</li>53 <span><a·title="an·optional·argument"·href="___Syzygies.html">Syzygies</a>·--·an·optional·argument</span>··········</li>
54 ········</ul>54 ········</ul>
55 ········<h4>Symbols·used·as·an·option·name</h4>55 ········<h4>Symbols·used·as·an·option·name</h4>
56 ········<ul>56 ········<ul>
57 ··········<li>57 ··········<li>
58 <span><a·title="an·optional·argument"·href="___Threads.html">Threads</a>·--·an·optional·argument</span>··········</li>58 <span><a·title="an·optional·argument"·href="___Threads.html">Threads</a>·--·an·optional·argument</span>··········</li>
59 ··········<li>59 ··········<li>
60 <span><a·title="an·optional·argument"·href="___Strategy.html">Strategy</a>·--·an·optional·argument</span>··········</li> 
61 ··········<li> 
62 <span><a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a>·--·an·optional·argument</span>··········</li>60 <span><a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a>·--·an·optional·argument</span>··········</li>
63 ··········<li>61 ··········<li>
64 <span><a·title="an·optional·argument"·href="___Generic.html">Generic</a>·--·an·optional·argument</span>··········</li>62 <span><a·title="an·optional·argument"·href="___Strategy.html">Strategy</a>·--·an·optional·argument</span>··········</li>
65 ··········<li>63 ··········<li>
66 <span><a·title="an·optional·argument"·href="___Basis__Element__Limit.html">BasisElementLimit</a>·--·an·optional·argument</span>··········</li>64 <span><a·title="an·optional·argument"·href="___Codimension__Limit.html">CodimensionLimit</a>·--·an·optional·argument</span>··········</li>
67 ··········<li>65 ··········<li>
68 <span><a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a>·--·an·optional·argument</span>··········</li>66 <span><a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a>·--·an·optional·argument</span>··········</li>
69 ··········<li>67 ··········<li>
70 <span><a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a>·--·an·optional·argument</span>··········</li>68 <span><a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a>·--·an·optional·argument</span>··········</li>
71 ··········<li>69 ··········<li>
72 <span><a·title="an·optional·argument"·href="___Variables.html">Variables</a>·--·an·optional·argument</span>··········</li>70 <span><a·title="an·optional·argument"·href="___Variables.html">Variables</a>·--·an·optional·argument</span>··········</li>
73 ··········<li>71 ··········<li>
74 <span><a·title="an·optional·argument"·href="___Degree__Lift.html">DegreeLift</a>·--·an·optional·argument</span>··········</li>72 <span><a·title="an·optional·argument"·href="___Degree__Lift.html">DegreeLift</a>·--·an·optional·argument</span>··········</li>
75 ··········<li>73 ··········<li>
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92 <span><a·title="an·optional·argument"·href="___Constants.html">Constants</a>·--·an·optional·argument</span>··········</li>90 <span><a·title="an·optional·argument"·href="___Constants.html">Constants</a>·--·an·optional·argument</span>··········</li>
93 ··········<li>91 ··········<li>
94 <span><a·title="an·optional·argument"·href="___Skew__Commutative.html">SkewCommutative</a>·--·an·optional·argument</span>··········</li>92 <span><a·title="an·optional·argument"·href="___Skew__Commutative.html">SkewCommutative</a>·--·an·optional·argument</span>··········</li>
95 ··········<li>93 ··········<li>
96 <span><a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a>·--·an·optional·argument</span>··········</li>94 <span><a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a>·--·an·optional·argument</span>··········</li>
97 ··········<li>95 ··········<li>
98 <span><a·title="an·optional·argument"·href="___Verify.html">Verify</a>·--·an·optional·argument</span>··········</li>96 <span><a·title="an·optional·argument"·href="___Generic.html">Generic</a>·--·an·optional·argument</span>··········</li>
99 ··········<li>97 ··········<li>
100 <span><a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a>·--·an·optional·argument</span>··········</li>98 <span><a·title="an·optional·argument"·href="___Basis__Element__Limit.html">BasisElementLimit</a>·--·an·optional·argument</span>··········</li>
101 ··········<li>99 ··········<li>
102 <span><a·title="an·optional·argument"·href="___Codimension__Limit.html">CodimensionLimit</a>·--·an·optional·argument</span>··········</li>100 <span><a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a>·--·an·optional·argument</span>··········</li>
 101 ··········<li>
 102 <span><a·title="an·optional·argument"·href="___Verify.html">Verify</a>·--·an·optional·argument</span>··········</li>
103 ··········<li>103 ··········<li>
104 <span><a·title="an·optional·argument"·href="___Coefficient__Ring.html">CoefficientRing</a>·--·an·optional·argument</span>··········</li>104 <span><a·title="an·optional·argument"·href="___Coefficient__Ring.html">CoefficientRing</a>·--·an·optional·argument</span>··········</li>
105 ··········<li>105 ··········<li>
106 <span><a·title="an·optional·argument"·href="___Follow__Links.html">FollowLinks</a>·--·an·optional·argument</span>··········</li>106 <span><a·title="an·optional·argument"·href="___Follow__Links.html">FollowLinks</a>·--·an·optional·argument</span>··········</li>
107 ··········<li>107 ··········<li>
108 <span><a·title="an·optional·argument"·href="___Exclude.html">Exclude</a>·--·an·optional·argument</span>··········</li>108 <span><a·title="an·optional·argument"·href="___Exclude.html">Exclude</a>·--·an·optional·argument</span>··········</li>
109 ··········<li>109 ··········<li>
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./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
    
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85 ······</div>85 ······</div>
86 ······<div>86 ······<div>
87 ········<h2>Description</h2>87 ········<h2>Description</h2>
88 ········<table·class="examples">88 ········<table·class="examples">
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()·|·&quot;/&quot;90 <td>··············<pre><code·class="language-macaulay2">i1·:·src·=·temporaryFileName()·|·&quot;/&quot;
  
91 o1·=·/tmp/M2-3685802-0/0/</code></pre>91 o1·=·/tmp/M2-2665493-0/0/</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()·|·&quot;/&quot;94 <td>··············<pre><code·class="language-macaulay2">i2·:·dst·=·temporaryFileName()·|·&quot;/&quot;
  
95 o2·=·/tmp/M2-3685802-0/1/</code></pre>95 o2·=·/tmp/M2-2665493-0/1/</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i3·:·makeDirectory·(src|&quot;a/&quot;)98 <td>··············<pre><code·class="language-macaulay2">i3·:·makeDirectory·(src|&quot;a/&quot;)
  
99 o3·=·/tmp/M2-3685802-0/0/a/</code></pre>99 o3·=·/tmp/M2-2665493-0/0/a/</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i4·:·makeDirectory·(src|&quot;b/&quot;)102 <td>··············<pre><code·class="language-macaulay2">i4·:·makeDirectory·(src|&quot;b/&quot;)
  
103 o4·=·/tmp/M2-3685802-0/0/b/</code></pre>103 o4·=·/tmp/M2-2665493-0/0/b/</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·makeDirectory·(src|&quot;b/c/&quot;)106 <td>··············<pre><code·class="language-macaulay2">i5·:·makeDirectory·(src|&quot;b/c/&quot;)
  
107 o5·=·/tmp/M2-3685802-0/0/b/c/</code></pre>107 o5·=·/tmp/M2-2665493-0/0/b/c/</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i6·:·src|&quot;a/f&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close110 <td>··············<pre><code·class="language-macaulay2">i6·:·src|&quot;a/f&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
111 o6·=·/tmp/M2-3685802-0/0/a/f111 o6·=·/tmp/M2-2665493-0/0/a/f
  
112 o6·:·File</code></pre>112 o6·:·File</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i7·:·src|&quot;a/g&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close115 <td>··············<pre><code·class="language-macaulay2">i7·:·src|&quot;a/g&quot;·&lt;&lt;·&quot;hi·there&quot;·&lt;&lt;·close
  
116 o7·=·/tmp/M2-3685802-0/0/a/g116 o7·=·/tmp/M2-2665493-0/0/a/g
  
117 o7·:·File</code></pre>117 o7·:·File</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i8·:·src|&quot;b/c/g&quot;·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close120 <td>··············<pre><code·class="language-macaulay2">i8·:·src|&quot;b/c/g&quot;·&lt;&lt;·&quot;ho·there&quot;·&lt;&lt;·close
  
121 o8·=·/tmp/M2-3685802-0/0/b/c/g121 o8·=·/tmp/M2-2665493-0/0/b/c/g
  
122 o8·:·File</code></pre>122 o8·:·File</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i9·:·symlinkDirectory(src,dst,Verbose=>true)125 <td>··············<pre><code·class="language-macaulay2">i9·:·symlinkDirectory(src,dst,Verbose=>true)
126 --symlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g126 --symlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
127 --symlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f127 --symlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
128 --symlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g</code></pre>128 --symlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i10·:·get·(dst|&quot;b/c/g&quot;)131 <td>··············<pre><code·class="language-macaulay2">i10·:·get·(dst|&quot;b/c/g&quot;)
  
132 o10·=·ho·there</code></pre>132 o10·=·ho·there</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ··········<tr>134 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)135 <td>··············<pre><code·class="language-macaulay2">i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
136 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g136 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
137 --unsymlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f137 --unsymlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
138 --unsymlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g</code></pre>138 --unsymlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ········</table>140 ········</table>
141 Now·we·remove·the·files·and·directories·we·created.········<table·class="examples">141 Now·we·remove·the·files·and·directories·we·created.········<table·class="examples">
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d143 <td>··············<pre><code·class="language-macaulay2">i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d
  
144 o12·=·rm144 o12·=·rm
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31 ··········o·The·directory·tree·rooted·at·src·is·duplicated·by·a·directory·tree31 ··········o·The·directory·tree·rooted·at·src·is·duplicated·by·a·directory·tree
32 ············rooted·at·dst.·The·files·in·the·source·tree·are·represented·by32 ············rooted·at·dst.·The·files·in·the·source·tree·are·represented·by
33 ············relative·symbolic·links·in·the·destination·tree·to·the·original33 ············relative·symbolic·links·in·the·destination·tree·to·the·original
34 ············files·in·the·source·tree.34 ············files·in·the·source·tree.
35 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*35 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
36 i1·:·src·=·temporaryFileName()·|·"/"36 i1·:·src·=·temporaryFileName()·|·"/"
  
37 o1·=·/tmp/M2-3685802-0/0/37 o1·=·/tmp/M2-2665493-0/0/
38 i2·:·dst·=·temporaryFileName()·|·"/"38 i2·:·dst·=·temporaryFileName()·|·"/"
  
39 o2·=·/tmp/M2-3685802-0/1/39 o2·=·/tmp/M2-2665493-0/1/
40 i3·:·makeDirectory·(src|"a/")40 i3·:·makeDirectory·(src|"a/")
  
41 o3·=·/tmp/M2-3685802-0/0/a/41 o3·=·/tmp/M2-2665493-0/0/a/
42 i4·:·makeDirectory·(src|"b/")42 i4·:·makeDirectory·(src|"b/")
  
43 o4·=·/tmp/M2-3685802-0/0/b/43 o4·=·/tmp/M2-2665493-0/0/b/
44 i5·:·makeDirectory·(src|"b/c/")44 i5·:·makeDirectory·(src|"b/c/")
  
45 o5·=·/tmp/M2-3685802-0/0/b/c/45 o5·=·/tmp/M2-2665493-0/0/b/c/
46 i6·:·src|"a/f"·<<·"hi·there"·<<·close46 i6·:·src|"a/f"·<<·"hi·there"·<<·close
  
47 o6·=·/tmp/M2-3685802-0/0/a/f47 o6·=·/tmp/M2-2665493-0/0/a/f
  
48 o6·:·File48 o6·:·File
49 i7·:·src|"a/g"·<<·"hi·there"·<<·close49 i7·:·src|"a/g"·<<·"hi·there"·<<·close
  
50 o7·=·/tmp/M2-3685802-0/0/a/g50 o7·=·/tmp/M2-2665493-0/0/a/g
  
51 o7·:·File51 o7·:·File
52 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close52 i8·:·src|"b/c/g"·<<·"ho·there"·<<·close
  
53 o8·=·/tmp/M2-3685802-0/0/b/c/g53 o8·=·/tmp/M2-2665493-0/0/b/c/g
  
54 o8·:·File54 o8·:·File
55 i9·:·symlinkDirectory(src,dst,Verbose=>true)55 i9·:·symlinkDirectory(src,dst,Verbose=>true)
56 --symlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g56 --symlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
57 --symlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f 
58 --symlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g57 --symlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
 58 --symlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f
59 i10·:·get·(dst|"b/c/g")59 i10·:·get·(dst|"b/c/g")
  
60 o10·=·ho·there60 o10·=·ho·there
61 i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)61 i11·:·symlinkDirectory(src,dst,Verbose=>true,Undo=>true)
62 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-3685802-0/1/b/c/g62 --unsymlinking:·../../../0/b/c/g·->·/tmp/M2-2665493-0/1/b/c/g
63 --unsymlinking:·../../0/a/f·->·/tmp/M2-3685802-0/1/a/f 
64 --unsymlinking:·../../0/a/g·->·/tmp/M2-3685802-0/1/a/g63 --unsymlinking:·../../0/a/g·->·/tmp/M2-2665493-0/1/a/g
 64 --unsymlinking:·../../0/a/f·->·/tmp/M2-2665493-0/1/a/f
65 Now·we·remove·the·files·and·directories·we·created.65 Now·we·remove·the·files·and·directories·we·created.
66 i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d66 i12·:·rm·=·d·->·if·isDirectory·d·then·removeDirectory·d·else·removeFile·d
  
67 o12·=·rm67 o12·=·rm
  
68 o12·:·FunctionClosure68 o12·:·FunctionClosure
69 i13·:·scan(reverse·findFiles·src,·rm)69 i13·:·scan(reverse·findFiles·src,·rm)
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71 ······</div>71 ······</div>
72 ······<div>72 ······<div>
73 ········<h2>Description</h2>73 ········<h2>Description</h2>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()76 <td>··············<pre><code·class="language-macaulay2">i1·:·fn·=·temporaryFileName()
  
77 o1·=·/tmp/M2-3689376-0/0</code></pre>77 o1·=·/tmp/M2-2667794-0/0</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i2·:·symlinkFile(&quot;qwert&quot;,·fn)</code></pre>80 <td>··············<pre><code·class="language-macaulay2">i2·:·symlinkFile(&quot;qwert&quot;,·fn)</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileExists·fn83 <td>··············<pre><code·class="language-macaulay2">i3·:·fileExists·fn
  
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13 ··········o·dst,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g13 ··········o·dst,·a·_\x8s_\x8t_\x8r_\x8i_\x8n_\x8g
14 ····*·Consequences:14 ····*·Consequences:
15 ··········o·a·symbolic·link·at·the·location·in·the·directory·tree·specified·by15 ··········o·a·symbolic·link·at·the·location·in·the·directory·tree·specified·by
16 ············dst·is·created,·pointing·to·src16 ············dst·is·created,·pointing·to·src
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 i1·:·fn·=·temporaryFileName()18 i1·:·fn·=·temporaryFileName()
  
19 o1·=·/tmp/M2-3689376-0/019 o1·=·/tmp/M2-2667794-0/0
20 i2·:·symlinkFile("qwert",·fn)20 i2·:·symlinkFile("qwert",·fn)
21 i3·:·fileExists·fn21 i3·:·fileExists·fn
  
22 o3·=·false22 o3·=·false
23 i4·:·readlink·fn23 i4·:·readlink·fn
  
24 o4·=·qwert24 o4·=·qwert
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61 ······</div>61 ······</div>
62 ······<div>62 ······<div>
63 ········<h2>Description</h2>63 ········<h2>Description</h2>
64 The·file·name·is·so·unique·that·even·with·various·suffixes·appended,·no·collision·with·existing·files·will·occur.··The·files·will·be·removed·when·the·program·terminates,·unless·it·terminates·as·the·result·of·an·error.········<table·class="examples">64 The·file·name·is·so·unique·that·even·with·various·suffixes·appended,·no·collision·with·existing·files·will·occur.··The·files·will·be·removed·when·the·program·terminates,·unless·it·terminates·as·the·result·of·an·error.········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·temporaryFileName·()·|·&quot;.tex&quot;66 <td>··············<pre><code·class="language-macaulay2">i1·:·temporaryFileName·()·|·&quot;.tex&quot;
  
67 o1·=·/tmp/M2-3697216-0/0.tex</code></pre>67 o1·=·/tmp/M2-2679119-0/0.tex</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ··········<tr>69 ··········<tr>
70 <td>··············<pre><code·class="language-macaulay2">i2·:·temporaryFileName·()·|·&quot;.html&quot;70 <td>··············<pre><code·class="language-macaulay2">i2·:·temporaryFileName·()·|·&quot;.html&quot;
  
71 o2·=·/tmp/M2-3697216-0/1.html</code></pre>71 o2·=·/tmp/M2-2679119-0/1.html</code></pre>
72 </td>··········</tr>72 </td>··········</tr>
73 ········</table>73 ········</table>
74 ········<p>This·function·will·work·under·Unix,·and·also·under·Windows·if·you·have·a·directory·on·the·same·drive·called·<span·class="tt">/tmp</span>.</p>74 ········<p>This·function·will·work·under·Unix,·and·also·under·Windows·if·you·have·a·directory·on·the·same·drive·called·<span·class="tt">/tmp</span>.</p>
75 ········<p>If·the·name·of·the·temporary·file·will·be·given·to·an·external·program,·it·may·be·necessary·to·concatenate·it·with·<a·href="_root__Path.html">rootPath</a>·or·<a·href="_root__U__R__I.html">rootURI</a>·to·enable·the·external·program·to·find·the·file.</p>75 ········<p>If·the·name·of·the·temporary·file·will·be·given·to·an·external·program,·it·may·be·necessary·to·concatenate·it·with·<a·href="_root__Path.html">rootPath</a>·or·<a·href="_root__U__R__I.html">rootURI</a>·to·enable·the·external·program·to·find·the·file.</p>
76 ········<p>The·temporary·file·name·is·derived·from·the·value·of·the·environment·variable·<span·class="tt">TMPDIR</span>,·if·it·has·one.</p>76 ········<p>The·temporary·file·name·is·derived·from·the·value·of·the·environment·variable·<span·class="tt">TMPDIR</span>,·if·it·has·one.</p>
77 ········<p>If·<a·title="fork·the·process"·href="_fork.html">fork</a>·is·used,·then·the·parent·and·child·Macaulay2·processes·will·each·remove·their·own·temporary·files·upon·termination,·with·the·parent·removing·any·files·created·before·<a·title="fork·the·process"·href="_fork.html">fork</a>·was·called.</p>77 ········<p>If·<a·title="fork·the·process"·href="_fork.html">fork</a>·is·used,·then·the·parent·and·child·Macaulay2·processes·will·each·remove·their·own·temporary·files·upon·termination,·with·the·parent·removing·any·files·created·before·<a·title="fork·the·process"·href="_fork.html">fork</a>·was·called.</p>
78 ······</div>78 ······</div>
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12 ··········o·a·unique·temporary·file·name.12 ··········o·a·unique·temporary·file·name.
13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*13 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
14 The·file·name·is·so·unique·that·even·with·various·suffixes·appended,·no14 The·file·name·is·so·unique·that·even·with·various·suffixes·appended,·no
15 collision·with·existing·files·will·occur.·The·files·will·be·removed·when·the15 collision·with·existing·files·will·occur.·The·files·will·be·removed·when·the
16 program·terminates,·unless·it·terminates·as·the·result·of·an·error.16 program·terminates,·unless·it·terminates·as·the·result·of·an·error.
17 i1·:·temporaryFileName·()·|·".tex"17 i1·:·temporaryFileName·()·|·".tex"
  
18 o1·=·/tmp/M2-3697216-0/0.tex18 o1·=·/tmp/M2-2679119-0/0.tex
19 i2·:·temporaryFileName·()·|·".html"19 i2·:·temporaryFileName·()·|·".html"
  
20 o2·=·/tmp/M2-3697216-0/1.html20 o2·=·/tmp/M2-2679119-0/1.html
21 This·function·will·work·under·Unix,·and·also·under·Windows·if·you·have·a21 This·function·will·work·under·Unix,·and·also·under·Windows·if·you·have·a
22 directory·on·the·same·drive·called·/tmp.22 directory·on·the·same·drive·called·/tmp.
23 If·the·name·of·the·temporary·file·will·be·given·to·an·external·program,·it·may23 If·the·name·of·the·temporary·file·will·be·given·to·an·external·program,·it·may
24 be·necessary·to·concatenate·it·with·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h·or·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I·to·enable·the·external24 be·necessary·to·concatenate·it·with·_\x8r_\x8o_\x8o_\x8t_\x8P_\x8a_\x8t_\x8h·or·_\x8r_\x8o_\x8o_\x8t_\x8U_\x8R_\x8I·to·enable·the·external
25 program·to·find·the·file.25 program·to·find·the·file.
26 The·temporary·file·name·is·derived·from·the·value·of·the·environment·variable26 The·temporary·file·name·is·derived·from·the·value·of·the·environment·variable
27 TMPDIR,·if·it·has·one.27 TMPDIR,·if·it·has·one.
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54 ········</ul>54 ········</ul>
55 ······</div>55 ······</div>
56 ······<div>56 ······<div>
57 ········<h2>Description</h2>57 ········<h2>Description</h2>
58 <span·class="tt">time·e</span>·evaluates·<span·class="tt">e</span>,·prints·the·amount·of·cpu·time·used,·and·returns·the·value·of·<span·class="tt">e</span>.··The·time·used·by·the·the·current·thread·and·garbage·collection·during·the·evaluation·of·<span·class="tt">e</span>·is·also·shown.········<table·class="examples">58 <span·class="tt">time·e</span>·evaluates·<span·class="tt">e</span>,·prints·the·amount·of·cpu·time·used,·and·returns·the·value·of·<span·class="tt">e</span>.··The·time·used·by·the·the·current·thread·and·garbage·collection·during·the·evaluation·of·<span·class="tt">e</span>·is·also·shown.········<table·class="examples">
59 ··········<tr>59 ··········<tr>
60 <td>··············<pre><code·class="language-macaulay2">i1·:·time·3^3060 <td>··············<pre><code·class="language-macaulay2">i1·:·time·3^30
61 ·--·used·1.7386e-05s·(cpu);·4.627e-06s·(thread);·0s·(gc)61 ·--·used·1.7433e-05s·(cpu);·4.648e-06s·(thread);·0s·(gc)
  
62 o1·=·205891132094649</code></pre>62 o1·=·205891132094649</code></pre>
63 </td>··········</tr>63 </td>··········</tr>
64 ········</table>64 ········</table>
65 ······</div>65 ······</div>
66 ······<div>66 ······<div>
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8 ····*···Usage:8 ····*···Usage:
9 ············time·e9 ············time·e
10 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*10 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
11 time·e·evaluates·e,·prints·the·amount·of·cpu·time·used,·and·returns·the·value11 time·e·evaluates·e,·prints·the·amount·of·cpu·time·used,·and·returns·the·value
12 of·e.·The·time·used·by·the·the·current·thread·and·garbage·collection·during·the12 of·e.·The·time·used·by·the·the·current·thread·and·garbage·collection·during·the
13 evaluation·of·e·is·also·shown.13 evaluation·of·e·is·also·shown.
14 i1·:·time·3^3014 i1·:·time·3^30
15 ·--·used·1.7386e-05s·(cpu);·4.627e-06s·(thread);·0s·(gc)15 ·--·used·1.7433e-05s·(cpu);·4.648e-06s·(thread);·0s·(gc)
  
16 o1·=·20589113209464916 o1·=·205891132094649
17 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
18 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation18 ····*·_\x8t_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation
19 ····*·_\x8c_\x8p_\x8u_\x8T_\x8i_\x8m_\x8e·--·seconds·of·cpu·time·used·since·Macaulay2·began19 ····*·_\x8c_\x8p_\x8u_\x8T_\x8i_\x8m_\x8e·--·seconds·of·cpu·time·used·since·Macaulay2·began
20 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation·using·time·elapsed20 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8i_\x8n_\x8g·--·time·a·computation·using·time·elapsed
21 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed21 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed
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46 ········<h2>Description</h2>46 ········<h2>Description</h2>
47 <span·class="tt">timing·e</span>·evaluates·<span·class="tt">e</span>·and·returns·a·list·of·type·<a·title="the·class·of·all·timing·results"·href="___Time.html">Time</a>·of·the·form·<span·class="tt">{t,v}</span>,·where·<span·class="tt">t</span>·is·the·number·of·seconds·of·cpu·timing·used,·and·<span·class="tt">v</span>·is·the·value·of·the·expression.········<p></p>47 <span·class="tt">timing·e</span>·evaluates·<span·class="tt">e</span>·and·returns·a·list·of·type·<a·title="the·class·of·all·timing·results"·href="___Time.html">Time</a>·of·the·form·<span·class="tt">{t,v}</span>,·where·<span·class="tt">t</span>·is·the·number·of·seconds·of·cpu·timing·used,·and·<span·class="tt">v</span>·is·the·value·of·the·expression.········<p></p>
48 The·default·method·for·printing·such·timing·results·is·to·display·the·timing·separately·in·a·comment·below·the·computed·value.········<table·class="examples">48 The·default·method·for·printing·such·timing·results·is·to·display·the·timing·separately·in·a·comment·below·the·computed·value.········<table·class="examples">
49 ··········<tr>49 ··········<tr>
50 <td>··············<pre><code·class="language-macaulay2">i1·:·timing·3^3050 <td>··············<pre><code·class="language-macaulay2">i1·:·timing·3^30
  
51 o1·=·20589113209464951 o1·=·205891132094649
52 ·····--·.00001293·seconds52 ·····--·.000019527·seconds
  
53 o1·:·Time</code></pre>53 o1·:·Time</code></pre>
54 </td>··········</tr>54 </td>··········</tr>
55 ··········<tr>55 ··········<tr>
56 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·oo56 <td>··············<pre><code·class="language-macaulay2">i2·:·peek·oo
  
57 o2·=·Time{.00001293,·205891132094649}</code></pre>57 o2·=·Time{.000019527,·205891132094649}</code></pre>
58 </td>··········</tr>58 </td>··········</tr>
59 ········</table>59 ········</table>
60 ······</div>60 ······</div>
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62 ········<h2>See·also</h2>62 ········<h2>See·also</h2>
63 ········<ul>63 ········<ul>
64 ··········<li>64 ··········<li>
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9 is·the·number·of·seconds·of·cpu·timing·used,·and·v·is·the·value·of·the9 is·the·number·of·seconds·of·cpu·timing·used,·and·v·is·the·value·of·the
10 expression.10 expression.
11 The·default·method·for·printing·such·timing·results·is·to·display·the·timing11 The·default·method·for·printing·such·timing·results·is·to·display·the·timing
12 separately·in·a·comment·below·the·computed·value.12 separately·in·a·comment·below·the·computed·value.
13 i1·:·timing·3^3013 i1·:·timing·3^30
  
14 o1·=·20589113209464914 o1·=·205891132094649
15 ·····--·.00001293·seconds15 ·····--·.000019527·seconds
  
16 o1·:·Time16 o1·:·Time
17 i2·:·peek·oo17 i2·:·peek·oo
  
18 o2·=·Time{.00001293,·205891132094649}18 o2·=·Time{.000019527,·205891132094649}
19 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*19 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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24 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed24 ····*·_\x8e_\x8l_\x8a_\x8p_\x8s_\x8e_\x8d_\x8T_\x8i_\x8m_\x8e·--·time·a·computation·including·time·elapsed
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7406 <span><a·title="an·optional·argument"·href="___Strategy.html">Strategy</a>·--·an·optional·argument</span>··············</div> 
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7409 ··············<div> 
7410 <span><a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a>·--·an·optional·argument</span>··············</div>7406 <span><a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a>·--·an·optional·argument</span>··············</div>
7411 ············</li>7407 ············</li>
7412 ············<li>7408 ············<li>
7413 ··············<div>7409 ··············<div>
7414 <span><a·title="an·optional·argument"·href="___Generic.html">Generic</a>·--·an·optional·argument</span>··············</div>7410 <span><a·title="an·optional·argument"·href="___Strategy.html">Strategy</a>·--·an·optional·argument</span>··············</div>
7415 ············</li>7411 ············</li>
7416 ············<li>7412 ············<li>
7417 ··············<div>7413 ··············<div>
7418 <span><a·title="an·optional·argument"·href="___Basis__Element__Limit.html">BasisElementLimit</a>·--·an·optional·argument</span>··············</div>7414 <span><a·title="an·optional·argument"·href="___Codimension__Limit.html">CodimensionLimit</a>·--·an·optional·argument</span>··············</div>
7419 ············</li>7415 ············</li>
7420 ············<li>7416 ············<li>
7421 ··············<div>7417 ··············<div>
7422 <span><a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a>·--·an·optional·argument</span>··············</div>7418 <span><a·title="an·optional·argument"·href="___Degree__Limit.html">DegreeLimit</a>·--·an·optional·argument</span>··············</div>
7423 ············</li>7419 ············</li>
7424 ············<li>7420 ············<li>
7425 ··············<div>7421 ··············<div>
7426 <span><a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a>·--·an·optional·argument</span>··············</div>7422 <span><a·title="an·optional·argument"·href="___Degree__Group.html">DegreeGroup</a>·--·an·optional·argument</span>··············</div>
7427 ············</li>7423 ············</li>
7428 ············<li>7424 ············<li>
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7477 ··············<div>7473 ··············<div>
7478 <span><a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a>·--·an·optional·argument</span>··············</div>7474 <span><a·title="an·optional·argument"·href="___Degree__Map.html">DegreeMap</a>·--·an·optional·argument</span>··············</div>
7479 ············</li>7475 ············</li>
7480 ············<li>7476 ············<li>
7481 ··············<div>7477 ··············<div>
7482 <span><a·title="an·optional·argument"·href="___Verify.html">Verify</a>·--·an·optional·argument</span>··············</div>7478 <span><a·title="an·optional·argument"·href="___Generic.html">Generic</a>·--·an·optional·argument</span>··············</div>
7483 ············</li>7479 ············</li>
7484 ············<li>7480 ············<li>
7485 ··············<div>7481 ··············<div>
7486 <span><a·title="an·optional·argument"·href="___Verbosity.html">Verbosity</a>·--·an·optional·argument</span>··············</div>7482 <span><a·title="an·optional·argument"·href="___Basis__Element__Limit.html">BasisElementLimit</a>·--·an·optional·argument</span>··············</div>
7487 ············</li>7483 ············</li>
7488 ············<li>7484 ············<li>
7489 ··············<div>7485 ··············<div>
7490 <span><a·title="an·optional·argument"·href="___Codimension__Limit.html">CodimensionLimit</a>·--·an·optional·argument</span>··············</div>7486 <span><a·title="an·optional·argument"·href="___Pair__Limit.html">PairLimit</a>·--·an·optional·argument</span>··············</div>
 7487 ············</li>
 7488 ············<li>
 7489 ··············<div>
 7490 <span><a·title="an·optional·argument"·href="___Verify.html">Verify</a>·--·an·optional·argument</span>··············</div>
7491 ············</li>7491 ············</li>
7492 ············<li>7492 ············<li>
7493 ··············<div>7493 ··············<div>
7494 <span><a·title="an·optional·argument"·href="___Coefficient__Ring.html">CoefficientRing</a>·--·an·optional·argument</span>··············</div>7494 <span><a·title="an·optional·argument"·href="___Coefficient__Ring.html">CoefficientRing</a>·--·an·optional·argument</span>··············</div>
7495 ············</li>7495 ············</li>
7496 ············<li>7496 ············<li>
7497 ··············<div>7497 ··············<div>
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2077 ··········o·_\x8F_\x8o_\x8l_\x8l_\x8o_\x8w_\x8L_\x8i_\x8n_\x8k_\x8s·--·an·optional·argument2077 ··········o·_\x8F_\x8o_\x8l_\x8l_\x8o_\x8w_\x8L_\x8i_\x8n_\x8k_\x8s·--·an·optional·argument
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2080 ··········o·_\x8S_\x8y_\x8z_\x8y_\x8g_\x8y_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·an·optional·argument2080 ··········o·_\x8S_\x8y_\x8z_\x8y_\x8g_\x8y_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·an·optional·argument
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./usr/share/doc/Macaulay2/Markov/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
    
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70 ·····|···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2|···1,2,2,2·2,2,2,1····1,2,2,1·2,2,2,2|70 ·····|···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2|···1,2,2,2·2,2,2,1····1,2,2,1·2,2,2,2|
71 ·····+-------------------------------------+-------------------------------------+71 ·····+-------------------------------------+-------------------------------------+
72 ·····|-·p·······p········+·p·······p·······|-·p·······p········+·p·······p·······|72 ·····|-·p·······p········+·p·······p·······|-·p·······p········+·p·······p·······|
73 ·····|···1,1,2,1·1,2,1,1····1,1,1,1·1,2,2,1|···1,1,2,2·1,2,1,2····1,1,1,2·1,2,2,2|73 ·····|···1,1,2,1·1,2,1,1····1,1,1,1·1,2,2,1|···1,1,2,2·1,2,1,2····1,1,1,2·1,2,2,2|
74 ·····+-------------------------------------+-------------------------------------+74 ·····+-------------------------------------+-------------------------------------+
  
75 i8·:·time·netList·primaryDecomposition·J75 i8·:·time·netList·primaryDecomposition·J
76 ·--·used·2.45435s·(cpu);·1.39639s·(thread);·0s·(gc)76 ·--·used·3.06052s·(cpu);·1.57126s·(thread);·0s·(gc)
  
77 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+77 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
78 o8·=·|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|78 o8·=·|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|
79 ·····|········1,2,2,2···1,2,2,1···1,2,1,2···1,2,1,1···1,1,2,2·2,1,2,1····1,1,2,1·2,1,2,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|79 ·····|········1,2,2,2···1,2,2,1···1,2,1,2···1,2,1,1···1,1,2,2·2,1,2,1····1,1,2,1·2,1,2,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|
80 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+80 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
81 ·····|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|81 ·····|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|
82 ·····|········1,2,2,2···1,2,2,1···1,1,2,2···1,1,2,1···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|82 ·····|········1,2,2,2···1,2,2,1···1,1,2,2···1,1,2,1···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|
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./usr/share/doc/Macaulay2/Markov/html/index.html
    
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139 ········</table>139 ········</table>
140 ········<div>140 ········<div>
141 ··········<p>This·ideal·has·5·primary·components.··The·first·is·the·one·that·has·statistical·significance.·The·significance·of·the·other·components·is·still·poorly·understood.</p>141 ··········<p>This·ideal·has·5·primary·components.··The·first·is·the·one·that·has·statistical·significance.·The·significance·of·the·other·components·is·still·poorly·understood.</p>
142 ········</div>142 ········</div>
143 ········<table·class="examples">143 ········<table·class="examples">
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i8·:·time·netList·primaryDecomposition·J145 <td>··············<pre><code·class="language-macaulay2">i8·:·time·netList·primaryDecomposition·J
146 ·--·used·2.45435s·(cpu);·1.39639s·(thread);·0s·(gc)146 ·--·used·3.06052s·(cpu);·1.57126s·(thread);·0s·(gc)
  
147 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+147 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
148 o8·=·|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|148 o8·=·|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|
149 ·····|········1,2,2,2···1,2,2,1···1,2,1,2···1,2,1,1···1,1,2,2·2,1,2,1····1,1,2,1·2,1,2,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|149 ·····|········1,2,2,2···1,2,2,1···1,2,1,2···1,2,1,1···1,1,2,2·2,1,2,1····1,1,2,1·2,1,2,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|
150 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+150 ·····+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
151 ·····|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|151 ·····|ideal·(p·······,·p·······,·p·······,·p·······,·p·······p········-·p·······p·······,·p·······p········-·p·······p·······)·························································································································································································································································································································································································································································································································································································································································································································································|
152 ·····|········1,2,2,2···1,2,2,1···1,1,2,2···1,1,2,1···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|152 ·····|········1,2,2,2···1,2,2,1···1,1,2,2···1,1,2,1···1,2,1,2·2,2,1,1····1,2,1,1·2,2,1,2···1,1,1,2·2,1,1,1····1,1,1,1·2,1,1,2··························································································································································································································································································································································································································································································································································································································································································································································|
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102 1,2,2,2|102 1,2,2,2|
103 ·····+-------------------------------------+-----------------------------------103 ·····+-------------------------------------+-----------------------------------
104 --+104 --+
105 This·ideal·has·5·primary·components.·The·first·is·the·one·that·has·statistical105 This·ideal·has·5·primary·components.·The·first·is·the·one·that·has·statistical
106 significance.·The·significance·of·the·other·components·is·still·poorly106 significance.·The·significance·of·the·other·components·is·still·poorly
107 understood.107 understood.
108 i8·:·time·netList·primaryDecomposition·J108 i8·:·time·netList·primaryDecomposition·J
109 ·--·used·2.45435s·(cpu);·1.39639s·(thread);·0s·(gc)109 ·--·used·3.06052s·(cpu);·1.57126s·(thread);·0s·(gc)
  
110 ·····+-------------------------------------------------------------------------110 ·····+-------------------------------------------------------------------------
111 -------------------------------------------------------------------------------111 -------------------------------------------------------------------------------
112 -------------------------------------------------------------------------------112 -------------------------------------------------------------------------------
113 -------------------------------------------------------------------------------113 -------------------------------------------------------------------------------
114 -------------------------------------------------------------------------------114 -------------------------------------------------------------------------------
115 -------------------------------------------------------------------------------115 -------------------------------------------------------------------------------
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9 #:len=14619 #:len=1461
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTA2MSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTA2MSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM
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212 ······|·1·-1·1·|212 ······|·1·-1·1·|
213 ······|·0·1··0·|213 ······|·0·1··0·|
  
214 ···············3·······3214 ···············3·······3
215 o22·:·Matrix·ZZ··<--·ZZ215 o22·:·Matrix·ZZ··<--·ZZ
  
216 i23·:·time·schubertRegularity·B216 i23·:·time·schubertRegularity·B
217 ·--·used·0.0624238s·(cpu);·0.0240059s·(thread);·0s·(gc)217 ·--·used·0.0788768s·(cpu);·0.0355843s·(thread);·0s·(gc)
  
218 o23·=·1218 o23·=·1
  
219 i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B219 i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B
220 ·--·used·0.0115237s·(cpu);·0.0125926s·(thread);·0s·(gc)220 ·--·used·0.011523s·(cpu);·0.0137505s·(thread);·0s·(gc)
  
221 o24·=·1221 o24·=·1
  
222 i25·:·222 i25·:·
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166 ······z···z···z···,·z···z···z····-·z···z···,·z···z···z····-·z···z···)166 ······z···z···z···,·z···z···z····-·z···z···,·z···z···z····-·z···z···)
167 ·······1,2·1,3·2,4···1,2·1,4·2,2····1,2·2,4···1,2·1,3·2,2····1,2·2,3167 ·······1,2·1,3·2,4···1,2·1,4·2,2····1,2·2,4···1,2·1,3·2,2····1,2·2,3
  
168 o14·:·Ideal·of·QQ[z···..z···]168 o14·:·Ideal·of·QQ[z···..z···]
169 ···················1,1···5,5169 ···················1,1···5,5
  
170 i15·:·time·schubertRegularity·p170 i15·:·time·schubertRegularity·p
171 ·--·used·0.000350127s·(cpu);·0.00023921s·(thread);·0s·(gc)171 ·--·used·0.000269535s·(cpu);·0.000258334s·(thread);·0s·(gc)
  
172 o15·=·5172 o15·=·5
  
173 i16·:·time·regularity·comodule·I173 i16·:·time·regularity·comodule·I
174 ·--·used·0.0114064s·(cpu);·0.012244s·(thread);·0s·(gc)174 ·--·used·0.0122112s·(cpu);·0.013756s·(thread);·0s·(gc)
  
175 o16·=·5175 o16·=·5
  
176 i17·:·176 i17·:·
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3 i1·:·w·=·{2,1,4,3}3 i1·:·w·=·{2,1,4,3}
  
4 o1·=·{2,·1,·4,·3}4 o1·=·{2,·1,·4,·3}
  
5 o1·:·List5 o1·:·List
  
6 i2·:·time·grothendieckPolynomial·w6 i2·:·time·grothendieckPolynomial·w
7 ·--·used·0.00399827s·(cpu);·0.00333733s·(thread);·0s·(gc)7 ·--·used·0.00400174s·(cpu);·0.00384269s·(thread);·0s·(gc)
  
8 ······2········2······2···············28 ······2········2······2···············2
9 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x9 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
10 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·310 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
11 o2·:·QQ[x·..x·]11 o2·:·QQ[x·..x·]
12 ·········1···412 ·········1···4
  
13 i3·:·time·grothendieckPolynomial·(w,Algorithm=>"PipeDream")13 i3·:·time·grothendieckPolynomial·(w,Algorithm=>"PipeDream")
14 ·--·used·0.00105651s·(cpu);·0.00160304s·(thread);·0s·(gc)14 ·--·used·0.00177028s·(cpu);·0.00170159s·(thread);·0s·(gc)
  
15 ······2········2······2···············215 ······2········2······2···············2
16 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x16 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
17 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·317 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
18 o3·:·QQ[x·..x·]18 o3·:·QQ[x·..x·]
19 ·········1···419 ·········1···4
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333 ········</table>333 ········</table>
334 ········<div>334 ········<div>
335 ··········<p>Additionally,·this·package·facilitates·the·investigating·homological·invariants·of·ASM·ideals·efficiently·by·computing·the·associated·invariants·for·their·antidiagonal·initial·ideals,·which·are·known·to·be·squarefree·by·[Wei17].·Therefore·the·extremal·Betti·numbers·(which·encode·regularity,·depth,·and·projective·dimension)·of·ASM·ideals·coincide·with·those·of·their·antidiagonal·initial·ideals·by·[CV20].</p>335 ··········<p>Additionally,·this·package·facilitates·the·investigating·homological·invariants·of·ASM·ideals·efficiently·by·computing·the·associated·invariants·for·their·antidiagonal·initial·ideals,·which·are·known·to·be·squarefree·by·[Wei17].·Therefore·the·extremal·Betti·numbers·(which·encode·regularity,·depth,·and·projective·dimension)·of·ASM·ideals·coincide·with·those·of·their·antidiagonal·initial·ideals·by·[CV20].</p>
336 ········</div>336 ········</div>
337 ········<table·class="examples">337 ········<table·class="examples">
338 ··········<tr>338 ··········<tr>
339 <td>··············<pre><code·class="language-macaulay2">i23·:·time·schubertRegularity·B339 <td>··············<pre><code·class="language-macaulay2">i23·:·time·schubertRegularity·B
340 ·--·used·0.0624238s·(cpu);·0.0240059s·(thread);·0s·(gc)340 ·--·used·0.0788768s·(cpu);·0.0355843s·(thread);·0s·(gc)
  
341 o23·=·1</code></pre>341 o23·=·1</code></pre>
342 </td>··········</tr>342 </td>··········</tr>
343 ··········<tr>343 ··········<tr>
344 <td>··············<pre><code·class="language-macaulay2">i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B344 <td>··············<pre><code·class="language-macaulay2">i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B
345 ·--·used·0.0115237s·(cpu);·0.0125926s·(thread);·0s·(gc)345 ·--·used·0.011523s·(cpu);·0.0137505s·(thread);·0s·(gc)
  
346 o24·=·1</code></pre>346 o24·=·1</code></pre>
347 </td>··········</tr>347 </td>··········</tr>
348 ········</table>348 ········</table>
349 ········<div>349 ········<div>
350 ··········<h2>Functions·for·investigating·ASM·varieties</h2>350 ··········<h2>Functions·for·investigating·ASM·varieties</h2>
351 ········</div>351 ········</div>
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241 Additionally,·this·package·facilitates·the·investigating·homological·invariants241 Additionally,·this·package·facilitates·the·investigating·homological·invariants
242 of·ASM·ideals·efficiently·by·computing·the·associated·invariants·for·their242 of·ASM·ideals·efficiently·by·computing·the·associated·invariants·for·their
243 antidiagonal·initial·ideals,·which·are·known·to·be·squarefree·by·[Wei17].243 antidiagonal·initial·ideals,·which·are·known·to·be·squarefree·by·[Wei17].
244 Therefore·the·extremal·Betti·numbers·(which·encode·regularity,·depth,·and244 Therefore·the·extremal·Betti·numbers·(which·encode·regularity,·depth,·and
245 projective·dimension)·of·ASM·ideals·coincide·with·those·of·their·antidiagonal245 projective·dimension)·of·ASM·ideals·coincide·with·those·of·their·antidiagonal
246 initial·ideals·by·[CV20].246 initial·ideals·by·[CV20].
247 i23·:·time·schubertRegularity·B247 i23·:·time·schubertRegularity·B
248 ·--·used·0.0624238s·(cpu);·0.0240059s·(thread);·0s·(gc)248 ·--·used·0.0788768s·(cpu);·0.0355843s·(thread);·0s·(gc)
  
249 o23·=·1249 o23·=·1
250 i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B250 i24·:·time·regularity·comodule·schubertDeterminantalIdeal·B
251 ·--·used·0.0115237s·(cpu);·0.0125926s·(thread);·0s·(gc)251 ·--·used·0.011523s·(cpu);·0.0137505s·(thread);·0s·(gc)
  
252 o24·=·1252 o24·=·1
253 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·f\x8fo\x8or\x8r·i\x8in\x8nv\x8ve\x8es\x8st\x8ti\x8ig\x8ga\x8at\x8ti\x8in\x8ng\x8g·A\x8AS\x8SM\x8M·v\x8va\x8ar\x8ri\x8ie\x8et\x8ti\x8ie\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*253 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·f\x8fo\x8or\x8r·i\x8in\x8nv\x8ve\x8es\x8st\x8ti\x8ig\x8ga\x8at\x8ti\x8in\x8ng\x8g·A\x8AS\x8SM\x8M·v\x8va\x8ar\x8ri\x8ie\x8et\x8ti\x8ie\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
254 ····*·_\x8i_\x8s_\x8P_\x8a_\x8r_\x8t_\x8i_\x8a_\x8l_\x8A_\x8S_\x8M_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·whether·a·matrix·is·a·partial·alternating·sign254 ····*·_\x8i_\x8s_\x8P_\x8a_\x8r_\x8t_\x8i_\x8a_\x8l_\x8A_\x8S_\x8M_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·whether·a·matrix·is·a·partial·alternating·sign
255 ······matrix255 ······matrix
256 ····*·_\x8p_\x8a_\x8r_\x8t_\x8i_\x8a_\x8l_\x8A_\x8S_\x8M_\x8T_\x8o_\x8A_\x8S_\x8M_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·extend·a·partial·alternating·sign·matrix·to·an256 ····*·_\x8p_\x8a_\x8r_\x8t_\x8i_\x8a_\x8l_\x8A_\x8S_\x8M_\x8T_\x8o_\x8A_\x8S_\x8M_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·extend·a·partial·alternating·sign·matrix·to·an
257 ······alternating·sign·matrix257 ······alternating·sign·matrix
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268 ········</table>268 ········</table>
269 ········<div>269 ········<div>
270 ··········<p>Finally,·this·package·contains·functions·for·investigating·homological·invariants·of·matrix·Schubert·varieties·efficiently·through·combinatorial·algorithms·produced·in·[PSW21].</p>270 ··········<p>Finally,·this·package·contains·functions·for·investigating·homological·invariants·of·matrix·Schubert·varieties·efficiently·through·combinatorial·algorithms·produced·in·[PSW21].</p>
271 ········</div>271 ········</div>
272 ········<table·class="examples">272 ········<table·class="examples">
273 ··········<tr>273 ··········<tr>
274 <td>··············<pre><code·class="language-macaulay2">i15·:·time·schubertRegularity·p274 <td>··············<pre><code·class="language-macaulay2">i15·:·time·schubertRegularity·p
275 ·--·used·0.000350127s·(cpu);·0.00023921s·(thread);·0s·(gc)275 ·--·used·0.000269535s·(cpu);·0.000258334s·(thread);·0s·(gc)
  
276 o15·=·5</code></pre>276 o15·=·5</code></pre>
277 </td>··········</tr>277 </td>··········</tr>
278 ··········<tr>278 ··········<tr>
279 <td>··············<pre><code·class="language-macaulay2">i16·:·time·regularity·comodule·I279 <td>··············<pre><code·class="language-macaulay2">i16·:·time·regularity·comodule·I
280 ·--·used·0.0114064s·(cpu);·0.012244s·(thread);·0s·(gc)280 ·--·used·0.0122112s·(cpu);·0.013756s·(thread);·0s·(gc)
  
281 o16·=·5</code></pre>281 o16·=·5</code></pre>
282 </td>··········</tr>282 </td>··········</tr>
283 ········</table>283 ········</table>
284 ········<div>284 ········<div>
285 ··········<h2>Functions·for·investigating·matrix·Schubert·varieties</h2>285 ··········<h2>Functions·for·investigating·matrix·Schubert·varieties</h2>
286 ········</div>286 ········</div>
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1.09 KB
html2text {}
    
Offset 532, 19 lines modifiedOffset 532, 19 lines modified
  
532 o14·:·Ideal·of·QQ[z···..z···]532 o14·:·Ideal·of·QQ[z···..z···]
533 ···················1,1···5,5533 ···················1,1···5,5
534 Finally,·this·package·contains·functions·for·investigating·homological534 Finally,·this·package·contains·functions·for·investigating·homological
535 invariants·of·matrix·Schubert·varieties·efficiently·through·combinatorial535 invariants·of·matrix·Schubert·varieties·efficiently·through·combinatorial
536 algorithms·produced·in·[PSW21].536 algorithms·produced·in·[PSW21].
537 i15·:·time·schubertRegularity·p537 i15·:·time·schubertRegularity·p
538 ·--·used·0.000350127s·(cpu);·0.00023921s·(thread);·0s·(gc)538 ·--·used·0.000269535s·(cpu);·0.000258334s·(thread);·0s·(gc)
  
539 o15·=·5539 o15·=·5
540 i16·:·time·regularity·comodule·I540 i16·:·time·regularity·comodule·I
541 ·--·used·0.0114064s·(cpu);·0.012244s·(thread);·0s·(gc)541 ·--·used·0.0122112s·(cpu);·0.013756s·(thread);·0s·(gc)
  
542 o16·=·5542 o16·=·5
543 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·f\x8fo\x8or\x8r·i\x8in\x8nv\x8ve\x8es\x8st\x8ti\x8ig\x8ga\x8at\x8ti\x8in\x8ng\x8g·m\x8ma\x8at\x8tr\x8ri\x8ix\x8x·S\x8Sc\x8ch\x8hu\x8ub\x8be\x8er\x8rt\x8t·v\x8va\x8ar\x8ri\x8ie\x8et\x8ti\x8ie\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*543 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·f\x8fo\x8or\x8r·i\x8in\x8nv\x8ve\x8es\x8st\x8ti\x8ig\x8ga\x8at\x8ti\x8in\x8ng\x8g·m\x8ma\x8at\x8tr\x8ri\x8ix\x8x·S\x8Sc\x8ch\x8hu\x8ub\x8be\x8er\x8rt\x8t·v\x8va\x8ar\x8ri\x8ie\x8et\x8ti\x8ie\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
544 ····*·_\x8a_\x8n_\x8t_\x8i_\x8D_\x8i_\x8a_\x8g_\x8I_\x8n_\x8i_\x8t_\x8(_\x8L_\x8i_\x8s_\x8t_\x8)·--·compute·the·(unique)·antidiagonal·initial·ideal·of544 ····*·_\x8a_\x8n_\x8t_\x8i_\x8D_\x8i_\x8a_\x8g_\x8I_\x8n_\x8i_\x8t_\x8(_\x8L_\x8i_\x8s_\x8t_\x8)·--·compute·the·(unique)·antidiagonal·initial·ideal·of
545 ······an·ASM·ideal545 ······an·ASM·ideal
546 ····*·_\x8r_\x8a_\x8n_\x8k_\x8T_\x8a_\x8b_\x8l_\x8e_\x8(_\x8L_\x8i_\x8s_\x8t_\x8)·--·compute·a·table·of·rank·conditions·that·determines·a546 ····*·_\x8r_\x8a_\x8n_\x8k_\x8T_\x8a_\x8b_\x8l_\x8e_\x8(_\x8L_\x8i_\x8s_\x8t_\x8)·--·compute·a·table·of·rank·conditions·that·determines·a
547 ······Schubert·determinantal·ideal·or,·more·generally,·an·alternating·sign547 ······Schubert·determinantal·ideal·or,·more·generally,·an·alternating·sign
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./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
    
Offset 77, 26 lines modifiedOffset 77, 26 lines modified
  
77 o1·=·{2,·1,·4,·3}77 o1·=·{2,·1,·4,·3}
  
78 o1·:·List</code></pre>78 o1·:·List</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·time·grothendieckPolynomial·w81 <td>··············<pre><code·class="language-macaulay2">i2·:·time·grothendieckPolynomial·w
82 ·--·used·0.00399827s·(cpu);·0.00333733s·(thread);·0s·(gc)82 ·--·used·0.00400174s·(cpu);·0.00384269s·(thread);·0s·(gc)
  
83 ······2········2······2···············283 ······2········2······2···············2
84 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x84 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
85 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·385 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
86 o2·:·QQ[x·..x·]86 o2·:·QQ[x·..x·]
87 ·········1···4</code></pre>87 ·········1···4</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i3·:·time·grothendieckPolynomial·(w,Algorithm=>&quot;PipeDream&quot;)90 <td>··············<pre><code·class="language-macaulay2">i3·:·time·grothendieckPolynomial·(w,Algorithm=>&quot;PipeDream&quot;)
91 ·--·used·0.00105651s·(cpu);·0.00160304s·(thread);·0s·(gc)91 ·--·used·0.00177028s·(cpu);·0.00170159s·(thread);·0s·(gc)
  
92 ······2········2······2···············292 ······2········2······2···············2
93 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x93 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
94 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·394 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
95 o3·:·QQ[x·..x·]95 o3·:·QQ[x·..x·]
96 ·········1···4</code></pre>96 ·········1···4</code></pre>
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Offset 19, 24 lines modifiedOffset 19, 24 lines modified
19 PipeDream.19 PipeDream.
20 i1·:·w·=·{2,1,4,3}20 i1·:·w·=·{2,1,4,3}
  
21 o1·=·{2,·1,·4,·3}21 o1·=·{2,·1,·4,·3}
  
22 o1·:·List22 o1·:·List
23 i2·:·time·grothendieckPolynomial·w23 i2·:·time·grothendieckPolynomial·w
24 ·--·used·0.00399827s·(cpu);·0.00333733s·(thread);·0s·(gc)24 ·--·used·0.00400174s·(cpu);·0.00384269s·(thread);·0s·(gc)
  
25 ······2········2······2···············225 ······2········2······2···············2
26 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x26 o2·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
27 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·327 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
28 o2·:·QQ[x·..x·]28 o2·:·QQ[x·..x·]
29 ·········1···429 ·········1···4
30 i3·:·time·grothendieckPolynomial·(w,Algorithm=>"PipeDream")30 i3·:·time·grothendieckPolynomial·(w,Algorithm=>"PipeDream")
31 ·--·used·0.00105651s·(cpu);·0.00160304s·(thread);·0s·(gc)31 ·--·used·0.00177028s·(cpu);·0.00170159s·(thread);·0s·(gc)
  
32 ······2········2······2···············232 ······2········2······2···············2
33 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x33 o3·=·x·x·x··-·x·x··-·x·x··-·x·x·x··+·x··+·x·x··+·x·x
34 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·334 ······1·2·3····1·2····1·3····1·2·3····1····1·2····1·3
  
35 o3·:·QQ[x·..x·]35 o3·:·QQ[x·..x·]
36 ·········1···436 ·········1···4
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./usr/share/doc/Macaulay2/Matroids/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=247 #:len=24
8 ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp8 ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
9 #:len=2559 #:len=255
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF011 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF0
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./usr/share/doc/Macaulay2/Matroids/example-output/___Matroid.out
    
Offset 51, 20 lines modifiedOffset 51, 20 lines modified
51 i9·:·keys·R10.cache51 i9·:·keys·R10.cache
  
52 o9·=·{groundSet,·rankFunction,·storedRepresentation}52 o9·=·{groundSet,·rankFunction,·storedRepresentation}
  
53 o9·:·List53 o9·:·List
  
54 i10·:·time·isWellDefined·R1054 i10·:·time·isWellDefined·R10
55 ·--·used·0.0975723s·(cpu);·0.0614429s·(thread);·0s·(gc)55 ·--·used·0.113366s·(cpu);·0.0580415s·(thread);·0s·(gc)
  
56 o10·=·true56 o10·=·true
  
57 i11·:·time·fVector·R1057 i11·:·time·fVector·R10
58 ·--·used·0.0975614s·(cpu);·0.0503785s·(thread);·0s·(gc)58 ·--·used·0.120757s·(cpu);·0.0497311s·(thread);·0s·(gc)
  
59 o11·=·HashTable{0·=>·1·}59 o11·=·HashTable{0·=>·1·}
60 ················1·=>·1060 ················1·=>·10
61 ················2·=>·4561 ················2·=>·45
62 ················3·=>·7562 ················3·=>·75
63 ················4·=>·3063 ················4·=>·30
64 ················5·=>·164 ················5·=>·1
Offset 76, 15 lines modifiedOffset 76, 15 lines modified
76 o12·=·{hyperplanes,·flatsRelationsTable,·rankFunction,·ideal,·ranks,·flats,76 o12·=·{hyperplanes,·flatsRelationsTable,·rankFunction,·ideal,·ranks,·flats,
77 ······-----------------------------------------------------------------------77 ······-----------------------------------------------------------------------
78 ······groundSet,·dual,·storedRepresentation}78 ······groundSet,·dual,·storedRepresentation}
  
79 o12·:·List79 o12·:·List
  
80 i13·:·time·fVector·R1080 i13·:·time·fVector·R10
81 ·--·used·0.000343096s·(cpu);·0.00017902s·(thread);·0s·(gc)81 ·--·used·0.000310613s·(cpu);·0.000166893s·(thread);·0s·(gc)
  
82 o13·=·HashTable{0·=>·1·}82 o13·=·HashTable{0·=>·1·}
83 ················1·=>·1083 ················1·=>·10
84 ················2·=>·4584 ················2·=>·45
85 ················3·=>·7585 ················3·=>·75
86 ················4·=>·3086 ················4·=>·30
87 ················5·=>·187 ················5·=>·1
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378 B
./usr/share/doc/Macaulay2/Matroids/example-output/_all__Minors.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i2·:·U25·=·uniformMatroid(2,5)9 i2·:·U25·=·uniformMatroid(2,5)
  
10 o2·=·a·"matroid"·of·rank·2·on·5·elements10 o2·=·a·"matroid"·of·rank·2·on·5·elements
  
11 o2·:·Matroid11 o2·:·Matroid
  
12 i3·:·elapsedTime·L·=·allMinors(V,·U25);12 i3·:·elapsedTime·L·=·allMinors(V,·U25);
13 ·--·.164547s·elapsed13 ·--·.271408s·elapsed
  
14 i4·:·#L14 i4·:·#L
  
15 o4·=·6415 o4·=·64
  
16 i5·:·netList·L_{0..4}16 i5·:·netList·L_{0..4}
  
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428 B
./usr/share/doc/Macaulay2/Matroids/example-output/_get__Isos.out
    
Offset 33, 14 lines modifiedOffset 33, 14 lines modified
33 i6·:·F7·=·specificMatroid·"fano"33 i6·:·F7·=·specificMatroid·"fano"
  
34 o6·=·a·"matroid"·of·rank·3·on·7·elements34 o6·=·a·"matroid"·of·rank·3·on·7·elements
  
35 o6·:·Matroid35 o6·:·Matroid
  
36 i7·:·time·autF7·=·getIsos(F7,·F7);36 i7·:·time·autF7·=·getIsos(F7,·F7);
37 ·--·used·0.148316s·(cpu);·0.0635807s·(thread);·0s·(gc)37 ·--·used·0.175128s·(cpu);·0.0796063s·(thread);·0s·(gc)
  
38 i8·:·#autF738 i8·:·#autF7
  
39 o8·=·16839 o8·=·168
  
40 i9·:·40 i9·:·
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378 B
./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
    
Offset 9, 12 lines modifiedOffset 9, 12 lines modified
9 o1·:·Sequence9 o1·:·Sequence
  
10 i2·:·hasMinor(M4,·uniformMatroid(2,4))10 i2·:·hasMinor(M4,·uniformMatroid(2,4))
  
11 o2·=·false11 o2·=·false
  
12 i3·:·time·hasMinor(M6,·M5)12 i3·:·time·hasMinor(M6,·M5)
13 ·--·used·1.26401s·(cpu);·0.970971s·(thread);·0s·(gc)13 ·--·used·1.46937s·(cpu);·1.03387s·(thread);·0s·(gc)
  
14 o3·=·true14 o3·=·true
  
15 i4·:·15 i4·:·
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995 B
./usr/share/doc/Macaulay2/Matroids/example-output/_isomorphism_lp__Matroid_cm__Matroid_rp.out
    
Offset 19, 15 lines modifiedOffset 19, 15 lines modified
19 i4·:·minorM6·=·minor(M6,·set{8},·set{4,5,6,7})19 i4·:·minorM6·=·minor(M6,·set{8},·set{4,5,6,7})
  
20 o4·=·a·"matroid"·of·rank·4·on·10·elements20 o4·=·a·"matroid"·of·rank·4·on·10·elements
  
21 o4·:·Matroid21 o4·:·Matroid
  
22 i5·:·time·isomorphism(M5,·minorM6)22 i5·:·time·isomorphism(M5,·minorM6)
23 ·--·used·0.0104909s·(cpu);·0.0110456s·(thread);·0s·(gc)23 ·--·used·0.0127777s·(cpu);·0.0116455s·(thread);·0s·(gc)
  
24 o5·=·HashTable{0·=>·1}24 o5·=·HashTable{0·=>·1}
25 ···············1·=>·025 ···············1·=>·0
26 ···············2·=>·326 ···············2·=>·3
27 ···············3·=>·227 ···············3·=>·2
28 ···············4·=>·628 ···············4·=>·6
29 ···············5·=>·529 ···············5·=>·5
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 i7·:·N·=·relabel·M656 i7·:·N·=·relabel·M6
  
57 o7·=·a·"matroid"·of·rank·5·on·15·elements57 o7·=·a·"matroid"·of·rank·5·on·15·elements
  
58 o7·:·Matroid58 o7·:·Matroid
  
59 i8·:·time·phi·=·isomorphism(N,M6)59 i8·:·time·phi·=·isomorphism(N,M6)
60 ·--·used·3.62287s·(cpu);·2.40814s·(thread);·0s·(gc)60 ·--·used·4.2942s·(cpu);·2.50229s·(thread);·0s·(gc)
  
61 o8·=·HashTable{0·=>·11·}61 o8·=·HashTable{0·=>·11·}
62 ···············1·=>·062 ···············1·=>·0
63 ···············2·=>·163 ···············2·=>·1
64 ···············3·=>·664 ···············3·=>·6
65 ···············4·=>·965 ···············4·=>·9
66 ···············5·=>·866 ···············5·=>·8
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497 B
./usr/share/doc/Macaulay2/Matroids/example-output/_quick__Isomorphism__Test.out
    
Offset 37, 15 lines modifiedOffset 37, 15 lines modified
37 o7·:·Matroid37 o7·:·Matroid
  
38 i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)38 i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)
  
39 o9·=·true39 o9·=·true
  
40 i10·:·time·quickIsomorphismTest(M0,·M1)40 i10·:·time·quickIsomorphismTest(M0,·M1)
41 ·--·used·0.00290971s·(cpu);·0.000378069s·(thread);·0s·(gc)41 ·--·used·0.00104445s·(cpu);·0.000418094s·(thread);·0s·(gc)
  
42 o10·=·false42 o10·=·false
  
43 i11·:·value·oo·===·false43 i11·:·value·oo·===·false
  
44 o11·=·true44 o11·=·true
  
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455 B
./usr/share/doc/Macaulay2/Matroids/example-output/_set__Representation.out
    
Offset 35, 15 lines modifiedOffset 35, 15 lines modified
35 i5·:·keys·M.cache35 i5·:·keys·M.cache
  
36 o5·=·{groundSet,·rankFunction,·storedRepresentation}36 o5·=·{groundSet,·rankFunction,·storedRepresentation}
  
37 o5·:·List37 o5·:·List
  
38 i6·:·elapsedTime·fVector·M38 i6·:·elapsedTime·fVector·M
39 ·--·.0801178s·elapsed39 ·--·.0853871s·elapsed
  
40 o6·=·HashTable{0·=>·1·}40 o6·=·HashTable{0·=>·1·}
41 ···············1·=>·641 ···············1·=>·6
42 ···············2·=>·1542 ···············2·=>·15
43 ···············3·=>·2043 ···············3·=>·20
44 ···············4·=>·144 ···············4·=>·1
  
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2.5 KB
./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
    
Offset 122, 21 lines modifiedOffset 122, 21 lines modified
  
122 o9·=·{groundSet,·rankFunction,·storedRepresentation}122 o9·=·{groundSet,·rankFunction,·storedRepresentation}
  
123 o9·:·List</code></pre>123 o9·:·List</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isWellDefined·R10126 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isWellDefined·R10
127 ·--·used·0.0975723s·(cpu);·0.0614429s·(thread);·0s·(gc)127 ·--·used·0.113366s·(cpu);·0.0580415s·(thread);·0s·(gc)
  
128 o10·=·true</code></pre>128 o10·=·true</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i11·:·time·fVector·R10131 <td>··············<pre><code·class="language-macaulay2">i11·:·time·fVector·R10
132 ·--·used·0.0975614s·(cpu);·0.0503785s·(thread);·0s·(gc)132 ·--·used·0.120757s·(cpu);·0.0497311s·(thread);·0s·(gc)
  
133 o11·=·HashTable{0·=>·1·}133 o11·=·HashTable{0·=>·1·}
134 ················1·=>·10134 ················1·=>·10
135 ················2·=>·45135 ················2·=>·45
136 ················3·=>·75136 ················3·=>·75
137 ················4·=>·30137 ················4·=>·30
138 ················5·=>·1138 ················5·=>·1
Offset 150, 15 lines modifiedOffset 150, 15 lines modified
150 ······-----------------------------------------------------------------------150 ······-----------------------------------------------------------------------
151 ······groundSet,·dual,·storedRepresentation}151 ······groundSet,·dual,·storedRepresentation}
  
152 o12·:·List</code></pre>152 o12·:·List</code></pre>
153 </td>··········</tr>153 </td>··········</tr>
154 ··········<tr>154 ··········<tr>
155 <td>··············<pre><code·class="language-macaulay2">i13·:·time·fVector·R10155 <td>··············<pre><code·class="language-macaulay2">i13·:·time·fVector·R10
156 ·--·used·0.000343096s·(cpu);·0.00017902s·(thread);·0s·(gc)156 ·--·used·0.000310613s·(cpu);·0.000166893s·(thread);·0s·(gc)
  
157 o13·=·HashTable{0·=>·1·}157 o13·=·HashTable{0·=>·1·}
158 ················1·=>·10158 ················1·=>·10
159 ················2·=>·45159 ················2·=>·45
160 ················3·=>·75160 ················3·=>·75
161 ················4·=>·30161 ················4·=>·30
162 ················5·=>·1162 ················5·=>·1
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71 o8·:·Matroid71 o8·:·Matroid
72 i9·:·keys·R10.cache72 i9·:·keys·R10.cache
  
73 o9·=·{groundSet,·rankFunction,·storedRepresentation}73 o9·=·{groundSet,·rankFunction,·storedRepresentation}
  
74 o9·:·List74 o9·:·List
75 i10·:·time·isWellDefined·R1075 i10·:·time·isWellDefined·R10
76 ·--·used·0.0975723s·(cpu);·0.0614429s·(thread);·0s·(gc)76 ·--·used·0.113366s·(cpu);·0.0580415s·(thread);·0s·(gc)
  
77 o10·=·true77 o10·=·true
78 i11·:·time·fVector·R1078 i11·:·time·fVector·R10
79 ·--·used·0.0975614s·(cpu);·0.0503785s·(thread);·0s·(gc)79 ·--·used·0.120757s·(cpu);·0.0497311s·(thread);·0s·(gc)
  
80 o11·=·HashTable{0·=>·1·}80 o11·=·HashTable{0·=>·1·}
81 ················1·=>·1081 ················1·=>·10
82 ················2·=>·4582 ················2·=>·45
83 ················3·=>·7583 ················3·=>·75
84 ················4·=>·3084 ················4·=>·30
85 ················5·=>·185 ················5·=>·1
Offset 93, 15 lines modifiedOffset 93, 15 lines modified
  
93 o12·=·{hyperplanes,·flatsRelationsTable,·rankFunction,·ideal,·ranks,·flats,93 o12·=·{hyperplanes,·flatsRelationsTable,·rankFunction,·ideal,·ranks,·flats,
94 ······-----------------------------------------------------------------------94 ······-----------------------------------------------------------------------
95 ······groundSet,·dual,·storedRepresentation}95 ······groundSet,·dual,·storedRepresentation}
  
96 o12·:·List96 o12·:·List
97 i13·:·time·fVector·R1097 i13·:·time·fVector·R10
98 ·--·used·0.000343096s·(cpu);·0.00017902s·(thread);·0s·(gc)98 ·--·used·0.000310613s·(cpu);·0.000166893s·(thread);·0s·(gc)
  
99 o13·=·HashTable{0·=>·1·}99 o13·=·HashTable{0·=>·1·}
100 ················1·=>·10100 ················1·=>·10
101 ················2·=>·45101 ················2·=>·45
102 ················3·=>·75102 ················3·=>·75
103 ················4·=>·30103 ················4·=>·30
104 ················5·=>·1104 ················5·=>·1
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88 o2·=·a·&quot;matroid&quot;·of·rank·2·on·5·elements88 o2·=·a·&quot;matroid&quot;·of·rank·2·on·5·elements
  
89 o2·:·Matroid</code></pre>89 o2·:·Matroid</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·L·=·allMinors(V,·U25);92 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·L·=·allMinors(V,·U25);
93 ·--·.164547s·elapsed</code></pre>93 ·--·.271408s·elapsed</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i4·:·#L96 <td>··············<pre><code·class="language-macaulay2">i4·:·#L
  
97 o4·=·64</code></pre>97 o4·=·64</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
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28 o1·:·Matroid28 o1·:·Matroid
29 i2·:·U25·=·uniformMatroid(2,5)29 i2·:·U25·=·uniformMatroid(2,5)
  
30 o2·=·a·"matroid"·of·rank·2·on·5·elements30 o2·=·a·"matroid"·of·rank·2·on·5·elements
  
31 o2·:·Matroid31 o2·:·Matroid
32 i3·:·elapsedTime·L·=·allMinors(V,·U25);32 i3·:·elapsedTime·L·=·allMinors(V,·U25);
33 ·--·.164547s·elapsed33 ·--·.271408s·elapsed
34 i4·:·#L34 i4·:·#L
  
35 o4·=·6435 o4·=·64
36 i5·:·netList·L_{0..4}36 i5·:·netList·L_{0..4}
  
37 ·····+----------+-------+37 ·····+----------+-------+
38 o5·=·|set·{5,·3}|set·{2}|38 o5·=·|set·{5,·3}|set·{2}|
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Offset 123, 15 lines modifiedOffset 123, 15 lines modified
  
123 o6·=·a·&quot;matroid&quot;·of·rank·3·on·7·elements123 o6·=·a·&quot;matroid&quot;·of·rank·3·on·7·elements
  
124 o6·:·Matroid</code></pre>124 o6·:·Matroid</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ··········<tr>126 ··········<tr>
127 <td>··············<pre><code·class="language-macaulay2">i7·:·time·autF7·=·getIsos(F7,·F7);127 <td>··············<pre><code·class="language-macaulay2">i7·:·time·autF7·=·getIsos(F7,·F7);
128 ·--·used·0.148316s·(cpu);·0.0635807s·(thread);·0s·(gc)</code></pre>128 ·--·used·0.175128s·(cpu);·0.0796063s·(thread);·0s·(gc)</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i8·:·#autF7131 <td>··············<pre><code·class="language-macaulay2">i8·:·#autF7
  
132 o8·=·168</code></pre>132 o8·=·168</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ········</table>134 ········</table>
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52 symmetric·group·S_7:52 symmetric·group·S_7:
53 i6·:·F7·=·specificMatroid·"fano"53 i6·:·F7·=·specificMatroid·"fano"
  
54 o6·=·a·"matroid"·of·rank·3·on·7·elements54 o6·=·a·"matroid"·of·rank·3·on·7·elements
  
55 o6·:·Matroid55 o6·:·Matroid
56 i7·:·time·autF7·=·getIsos(F7,·F7);56 i7·:·time·autF7·=·getIsos(F7,·F7);
57 ·--·used·0.148316s·(cpu);·0.0635807s·(thread);·0s·(gc)57 ·--·used·0.175128s·(cpu);·0.0796063s·(thread);·0s·(gc)
58 i8·:·#autF758 i8·:·#autF7
  
59 o8·=·16859 o8·=·168
60 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*60 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
61 ····*·_\x8i_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8,_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8)·--·computes·an·isomorphism·between61 ····*·_\x8i_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8,_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8)·--·computes·an·isomorphism·between
62 ······isomorphic·matroids62 ······isomorphic·matroids
63 ····*·_\x8q_\x8u_\x8i_\x8c_\x8k_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8T_\x8e_\x8s_\x8t·--·quick·checks·for·isomorphism·between·matroids63 ····*·_\x8q_\x8u_\x8i_\x8c_\x8k_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8T_\x8e_\x8s_\x8t·--·quick·checks·for·isomorphism·between·matroids
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94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i2·:·hasMinor(M4,·uniformMatroid(2,4))95 <td>··············<pre><code·class="language-macaulay2">i2·:·hasMinor(M4,·uniformMatroid(2,4))
  
96 o2·=·false</code></pre>96 o2·=·false</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·hasMinor(M6,·M5)99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·hasMinor(M6,·M5)
100 ·--·used·1.26401s·(cpu);·0.970971s·(thread);·0s·(gc)100 ·--·used·1.46937s·(cpu);·1.03387s·(thread);·0s·(gc)
  
101 o3·=·true</code></pre>101 o3·=·true</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ········</table>103 ········</table>
104 ······</div>104 ······</div>
105 ······<div>105 ······<div>
106 ········<h2>See·also</h2>106 ········<h2>See·also</h2>
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35 ·····elements,·a·"matroid"·of·rank·5·on·15·elements)35 ·····elements,·a·"matroid"·of·rank·5·on·15·elements)
  
36 o1·:·Sequence36 o1·:·Sequence
37 i2·:·hasMinor(M4,·uniformMatroid(2,4))37 i2·:·hasMinor(M4,·uniformMatroid(2,4))
  
38 o2·=·false38 o2·=·false
39 i3·:·time·hasMinor(M6,·M5)39 i3·:·time·hasMinor(M6,·M5)
40 ·--·used·1.26401s·(cpu);·0.970971s·(thread);·0s·(gc)40 ·--·used·1.46937s·(cpu);·1.03387s·(thread);·0s·(gc)
  
41 o3·=·true41 o3·=·true
42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
43 ····*·_\x8m_\x8i_\x8n_\x8o_\x8r·--·minor·of·matroid43 ····*·_\x8m_\x8i_\x8n_\x8o_\x8r·--·minor·of·matroid
44 ····*·_\x8i_\x8s_\x8B_\x8i_\x8n_\x8a_\x8r_\x8y·--·whether·a·matroid·is·representable·over·F_244 ····*·_\x8i_\x8s_\x8B_\x8i_\x8n_\x8a_\x8r_\x8y·--·whether·a·matroid·is·representable·over·F_2
45 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ha\x8as\x8sM\x8Mi\x8in\x8no\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*45 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8ha\x8as\x8sM\x8Mi\x8in\x8no\x8or\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
46 ····*·hasMinor(Matroid,Matroid)46 ····*·hasMinor(Matroid,Matroid)
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102 o4·=·a·&quot;matroid&quot;·of·rank·4·on·10·elements102 o4·=·a·&quot;matroid&quot;·of·rank·4·on·10·elements
  
103 o4·:·Matroid</code></pre>103 o4·:·Matroid</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isomorphism(M5,·minorM6)106 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isomorphism(M5,·minorM6)
107 ·--·used·0.0104909s·(cpu);·0.0110456s·(thread);·0s·(gc)107 ·--·used·0.0127777s·(cpu);·0.0116455s·(thread);·0s·(gc)
  
108 o5·=·HashTable{0·=>·1}108 o5·=·HashTable{0·=>·1}
109 ···············1·=>·0109 ···············1·=>·0
110 ···············2·=>·3110 ···············2·=>·3
111 ···············3·=>·2111 ···············3·=>·2
112 ···············4·=>·6112 ···············4·=>·6
113 ···············5·=>·5113 ···············5·=>·5
Offset 142, 15 lines modifiedOffset 142, 15 lines modified
  
142 o7·=·a·&quot;matroid&quot;·of·rank·5·on·15·elements142 o7·=·a·&quot;matroid&quot;·of·rank·5·on·15·elements
  
143 o7·:·Matroid</code></pre>143 o7·:·Matroid</code></pre>
144 </td>··········</tr>144 </td>··········</tr>
145 ··········<tr>145 ··········<tr>
146 <td>··············<pre><code·class="language-macaulay2">i8·:·time·phi·=·isomorphism(N,M6)146 <td>··············<pre><code·class="language-macaulay2">i8·:·time·phi·=·isomorphism(N,M6)
147 ·--·used·3.62287s·(cpu);·2.40814s·(thread);·0s·(gc)147 ·--·used·4.2942s·(cpu);·2.50229s·(thread);·0s·(gc)
  
148 o8·=·HashTable{0·=>·11·}148 o8·=·HashTable{0·=>·11·}
149 ···············1·=>·0149 ···············1·=>·0
150 ···············2·=>·1150 ···············2·=>·1
151 ···············3·=>·6151 ···············3·=>·6
152 ···············4·=>·9152 ···············4·=>·9
153 ···············5·=>·8153 ···············5·=>·8
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35 o3·:·Sequence35 o3·:·Sequence
36 i4·:·minorM6·=·minor(M6,·set{8},·set{4,5,6,7})36 i4·:·minorM6·=·minor(M6,·set{8},·set{4,5,6,7})
  
37 o4·=·a·"matroid"·of·rank·4·on·10·elements37 o4·=·a·"matroid"·of·rank·4·on·10·elements
  
38 o4·:·Matroid38 o4·:·Matroid
39 i5·:·time·isomorphism(M5,·minorM6)39 i5·:·time·isomorphism(M5,·minorM6)
40 ·--·used·0.0104909s·(cpu);·0.0110456s·(thread);·0s·(gc)40 ·--·used·0.0127777s·(cpu);·0.0116455s·(thread);·0s·(gc)
  
41 o5·=·HashTable{0·=>·1}41 o5·=·HashTable{0·=>·1}
42 ···············1·=>·042 ···············1·=>·0
43 ···············2·=>·343 ···············2·=>·3
44 ···············3·=>·244 ···············3·=>·2
45 ···············4·=>·645 ···············4·=>·6
46 ···············5·=>·546 ···············5·=>·5
Offset 69, 15 lines modifiedOffset 69, 15 lines modified
69 o6·:·HashTable69 o6·:·HashTable
70 i7·:·N·=·relabel·M670 i7·:·N·=·relabel·M6
  
71 o7·=·a·"matroid"·of·rank·5·on·15·elements71 o7·=·a·"matroid"·of·rank·5·on·15·elements
  
72 o7·:·Matroid72 o7·:·Matroid
73 i8·:·time·phi·=·isomorphism(N,M6)73 i8·:·time·phi·=·isomorphism(N,M6)
74 ·--·used·3.62287s·(cpu);·2.40814s·(thread);·0s·(gc)74 ·--·used·4.2942s·(cpu);·2.50229s·(thread);·0s·(gc)
  
75 o8·=·HashTable{0·=>·11·}75 o8·=·HashTable{0·=>·11·}
76 ···············1·=>·076 ···············1·=>·0
77 ···············2·=>·177 ···············2·=>·1
78 ···············3·=>·678 ···············3·=>·6
79 ···············4·=>·979 ···············4·=>·9
80 ···············5·=>·880 ···············5·=>·8
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121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)122 <td>··············<pre><code·class="language-macaulay2">i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)
  
123 o9·=·true</code></pre>123 o9·=·true</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i10·:·time·quickIsomorphismTest(M0,·M1)126 <td>··············<pre><code·class="language-macaulay2">i10·:·time·quickIsomorphismTest(M0,·M1)
127 ·--·used·0.00290971s·(cpu);·0.000378069s·(thread);·0s·(gc)127 ·--·used·0.00104445s·(cpu);·0.000418094s·(thread);·0s·(gc)
  
128 o10·=·false</code></pre>128 o10·=·false</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i11·:·value·oo·===·false131 <td>··············<pre><code·class="language-macaulay2">i11·:·value·oo·===·false
  
132 o11·=·true</code></pre>132 o11·=·true</code></pre>
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Offset 52, 15 lines modifiedOffset 52, 15 lines modified
52 o7·=·a·"matroid"·of·rank·7·on·11·elements52 o7·=·a·"matroid"·of·rank·7·on·11·elements
  
53 o7·:·Matroid53 o7·:·Matroid
54 i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)54 i8·:·R·=·ZZ[x,y];·tuttePolynomial(M0,·R)·==·tuttePolynomial(M1,·R)
  
55 o9·=·true55 o9·=·true
56 i10·:·time·quickIsomorphismTest(M0,·M1)56 i10·:·time·quickIsomorphismTest(M0,·M1)
57 ·--·used·0.00290971s·(cpu);·0.000378069s·(thread);·0s·(gc)57 ·--·used·0.00104445s·(cpu);·0.000418094s·(thread);·0s·(gc)
  
58 o10·=·false58 o10·=·false
59 i11·:·value·oo·===·false59 i11·:·value·oo·===·false
  
60 o11·=·true60 o11·=·true
61 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*61 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
62 ····*·_\x8i_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8,_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8)·--·computes·an·isomorphism·between62 ····*·_\x8i_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8,_\x8M_\x8a_\x8t_\x8r_\x8o_\x8i_\x8d_\x8)·--·computes·an·isomorphism·between
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Offset 116, 15 lines modifiedOffset 116, 15 lines modified
  
116 o5·=·{groundSet,·rankFunction,·storedRepresentation}116 o5·=·{groundSet,·rankFunction,·storedRepresentation}
  
117 o5·:·List</code></pre>117 o5·:·List</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·fVector·M120 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·fVector·M
121 ·--·.0801178s·elapsed121 ·--·.0853871s·elapsed
  
122 o6·=·HashTable{0·=>·1·}122 o6·=·HashTable{0·=>·1·}
123 ···············1·=>·6123 ···············1·=>·6
124 ···············2·=>·15124 ···············2·=>·15
125 ···············3·=>·20125 ···············3·=>·20
126 ···············4·=>·1126 ···············4·=>·1
  
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49 o4·:·Matrix·QQ··<--·QQ49 o4·:·Matrix·QQ··<--·QQ
50 i5·:·keys·M.cache50 i5·:·keys·M.cache
  
51 o5·=·{groundSet,·rankFunction,·storedRepresentation}51 o5·=·{groundSet,·rankFunction,·storedRepresentation}
  
52 o5·:·List52 o5·:·List
53 i6·:·elapsedTime·fVector·M53 i6·:·elapsedTime·fVector·M
54 ·--·.0801178s·elapsed54 ·--·.0853871s·elapsed
  
55 o6·=·HashTable{0·=>·1·}55 o6·=·HashTable{0·=>·1·}
56 ···············1·=>·656 ···············1·=>·6
57 ···············2·=>·1557 ···············2·=>·15
58 ···············3·=>·2058 ···············3·=>·20
59 ···············4·=>·159 ···············4·=>·1
  
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl
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./usr/share/doc/Macaulay2/MinimalPrimes/example-output/___Hybrid.out
    
Offset 6, 15 lines modifiedOffset 6, 15 lines modified
  
6 i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);6 i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);
  
7 o3·:·Ideal·of·R7 o3·:·Ideal·of·R
  
8 i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid{Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);8 i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid{Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);
9 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·09 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·0
10 ··Strategy:·Birational········(time·.00782142)··#primes·=·0·#prunedViaCodim·=·010 ··Strategy:·Birational········(time·.0154607)··#primes·=·0·#prunedViaCodim·=·0
11 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·011 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·0
12 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and·selecting·minimal·primes...12 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and·selecting·minimal·primes...
13 ·--··Time·taken·:·013 ·--··Time·taken·:·0
14 ·--·.0177554s·elapsed14 ·--·.0444702s·elapsed
15 (time·0)·········#primes·=·1·#prunedViaCodim·=·015 (time·0)·········#primes·=·1·#prunedViaCodim·=·0
  
16 i5·:·16 i5·:·
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./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical.out
    
Offset 30, 21 lines modifiedOffset 30, 21 lines modified
  
30 ·············2········2···3·····230 ·············2········2···3·····2
31 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)31 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)
  
32 o5·:·Ideal·of·R32 o5·:·Ideal·of·R
  
33 i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)33 i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)
34 ·--·.000361947s·elapsed34 ·--·.00911872s·elapsed
  
35 o6·=·ideal·(a,·b,·c)35 o6·=·ideal·(a,·b,·c)
  
36 o6·:·Ideal·of·R36 o6·:·Ideal·of·R
  
37 i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)37 i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)
38 ·--·.00933667s·elapsed38 ·--·.013221s·elapsed
  
39 o7·=·ideal·(c,·b,·a)39 o7·=·ideal·(c,·b,·a)
  
40 o7·:·Ideal·of·R40 o7·:·Ideal·of·R
  
41 i8·:·41 i8·:·
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./usr/share/doc/Macaulay2/MinimalPrimes/example-output/_radical__Containment.out
    
Offset 29, 22 lines modifiedOffset 29, 22 lines modified
29 o5·=·84029 o5·=·840
  
30 i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·030 i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·0
  
31 o6·=·true31 o6·=·true
  
32 i7·:·elapsedTime·radicalContainment(x_0,·I)32 i7·:·elapsedTime·radicalContainment(x_0,·I)
33 ·--·.0513011s·elapsed33 ·--·.0936957s·elapsed
  
34 o7·=·true34 o7·=·true
  
35 i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·"Kollar")35 i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·"Kollar")
36 ·--·.00115772s·elapsed36 ·--·.00166549s·elapsed
  
37 o8·=·true37 o8·=·true
  
38 i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·"Kollar")38 i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·"Kollar")
39 ·--·.000864384s·elapsed39 ·--·.00140371s·elapsed
  
40 o9·=·false40 o9·=·false
  
41 i10·:·41 i10·:·
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59 <td>··············<pre><code·class="language-macaulay2">i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);59 <td>··············<pre><code·class="language-macaulay2">i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);
  
60 o3·:·Ideal·of·R</code></pre>60 o3·:·Ideal·of·R</code></pre>
61 </td>··········</tr>61 </td>··········</tr>
62 ··········<tr>62 ··········<tr>
63 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid{Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);63 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid{Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);
64 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·064 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·0
65 ··Strategy:·Birational········(time·.00782142)··#primes·=·0·#prunedViaCodim·=·065 ··Strategy:·Birational········(time·.0154607)··#primes·=·0·#prunedViaCodim·=·0
66 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·066 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·0
67 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and·selecting·minimal·primes...67 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and·selecting·minimal·primes...
68 ·--··Time·taken·:·068 ·--··Time·taken·:·0
69 ·--·.0177554s·elapsed69 ·--·.0444702s·elapsed
70 (time·0)·········#primes·=·1·#prunedViaCodim·=·0</code></pre>70 (time·0)·········#primes·=·1·#prunedViaCodim·=·0</code></pre>
71 </td>··········</tr>71 </td>··········</tr>
72 ········</table>72 ········</table>
73 ······</div>73 ······</div>
74 ······<div>74 ······<div>
75 ········<h2>See·also</h2>75 ········<h2>See·also</h2>
76 ········<ul>76 ········<ul>
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12 i2·:·R·=·ZZ/101[w..z];12 i2·:·R·=·ZZ/101[w..z];
13 i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);13 i3·:·I·=·ideal(w*x^2-42*y*z,·x^6+12*w*y+x^3*z,·w^2-47*x^4*z-47*x*z^2);
  
14 o3·:·Ideal·of·R14 o3·:·Ideal·of·R
15 i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid15 i4·:·elapsedTime·minimalPrimes(ideal·I_*,·Strategy·=>·Hybrid
16 {Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);16 {Linear,Birational,Factorization,DecomposeMonomials},·Verbosity·=>·2);
17 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·017 ··Strategy:·Linear············(time·0)·········#primes·=·0·#prunedViaCodim·=·0
18 ··Strategy:·Birational········(time·.00782142)··#primes·=·0·#prunedViaCodim·=·018 ··Strategy:·Birational········(time·.0154607)··#primes·=·0·#prunedViaCodim·=·0
19 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·019 ··Strategy:·Factorization·····(time·0)·········#primes·=·0·#prunedViaCodim·=·0
20 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and20 ··Strategy:·DecomposeMonomials·--·Converting·annotated·ideals·to·ideals·and
21 selecting·minimal·primes...21 selecting·minimal·primes...
22 ·--··Time·taken·:·022 ·--··Time·taken·:·0
23 ·--·.0177554s·elapsed23 ·--·.0444702s·elapsed
24 (time·0)·········#primes·=·1·#prunedViaCodim·=·024 (time·0)·········#primes·=·1·#prunedViaCodim·=·0
25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
26 ····*·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8D_\x8e_\x8c_\x8o_\x8m_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8o_\x8n_\x8(_\x8._\x8._\x8._\x8,_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y_\x8=_\x8>_\x8._\x8._\x8._\x8)26 ····*·_\x8p_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8D_\x8e_\x8c_\x8o_\x8m_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8o_\x8n_\x8(_\x8._\x8._\x8._\x8,_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y_\x8=_\x8>_\x8._\x8._\x8._\x8)
27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
28 The·object·_\x8H_\x8y_\x8b_\x8r_\x8i_\x8d·is·a·_\x8s_\x8e_\x8l_\x8f_\x8·_\x8i_\x8n_\x8i_\x8t_\x8i_\x8a_\x8l_\x8i_\x8z_\x8i_\x8n_\x8g_\x8·_\x8t_\x8y_\x8p_\x8e,·with·ancestor·classes·_\x8L_\x8i_\x8s_\x8t·<28 The·object·_\x8H_\x8y_\x8b_\x8r_\x8i_\x8d·is·a·_\x8s_\x8e_\x8l_\x8f_\x8·_\x8i_\x8n_\x8i_\x8t_\x8i_\x8a_\x8l_\x8i_\x8z_\x8i_\x8n_\x8g_\x8·_\x8t_\x8y_\x8p_\x8e,·with·ancestor·classes·_\x8L_\x8i_\x8s_\x8t·<
29 _\x8V_\x8i_\x8s_\x8i_\x8b_\x8l_\x8e_\x8L_\x8i_\x8s_\x8t·<·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t·<·_\x8T_\x8h_\x8i_\x8n_\x8g.29 _\x8V_\x8i_\x8s_\x8i_\x8b_\x8l_\x8e_\x8L_\x8i_\x8s_\x8t·<·_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t·<·_\x8T_\x8h_\x8i_\x8n_\x8g.
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123 ·············2········2···3·····2123 ·············2········2···3·····2
124 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)124 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)
  
125 o5·:·Ideal·of·R</code></pre>125 o5·:·Ideal·of·R</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)128 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)
129 ·--·.000361947s·elapsed129 ·--·.00911872s·elapsed
  
130 o6·=·ideal·(a,·b,·c)130 o6·=·ideal·(a,·b,·c)
  
131 o6·:·Ideal·of·R</code></pre>131 o6·:·Ideal·of·R</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)134 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)
135 ·--·.00933667s·elapsed135 ·--·.013221s·elapsed
  
136 o7·=·ideal·(c,·b,·a)136 o7·=·ideal·(c,·b,·a)
  
137 o7·:·Ideal·of·R</code></pre>137 o7·:·Ideal·of·R</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ········</table>139 ········</table>
140 ········<div>140 ········<div>
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Offset 63, 21 lines modifiedOffset 63, 21 lines modified
63 i5·:·I·=·intersect(ideal(a^2,b^2,c),·ideal(a,b^3,c^2))63 i5·:·I·=·intersect(ideal(a^2,b^2,c),·ideal(a,b^3,c^2))
  
64 ·············2········2···3·····264 ·············2········2···3·····2
65 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)65 o5·=·ideal·(c·,·a*c,·a·,·b·,·a*b·)
  
66 o5·:·Ideal·of·R66 o5·:·Ideal·of·R
67 i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)67 i6·:·elapsedTime·radical(ideal·I_*,·Strategy·=>·Monomial)
68 ·--·.000361947s·elapsed68 ·--·.00911872s·elapsed
  
69 o6·=·ideal·(a,·b,·c)69 o6·=·ideal·(a,·b,·c)
  
70 o6·:·Ideal·of·R70 o6·:·Ideal·of·R
71 i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)71 i7·:·elapsedTime·radical(ideal·I_*,·Unmixed·=>·true)
72 ·--·.00933667s·elapsed72 ·--·.013221s·elapsed
  
73 o7·=·ideal·(c,·b,·a)73 o7·=·ideal·(c,·b,·a)
  
74 o7·:·Ideal·of·R74 o7·:·Ideal·of·R
75 For·another·example,·see·_\x8P_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8D_\x8e_\x8c_\x8o_\x8m_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8o_\x8n.75 For·another·example,·see·_\x8P_\x8r_\x8i_\x8m_\x8a_\x8r_\x8y_\x8D_\x8e_\x8c_\x8o_\x8m_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8o_\x8n.
76 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*76 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
77 Eisenbud,·Huneke,·Vasconcelos,·Invent.·Math.·110·207-235·(1992).77 Eisenbud,·Huneke,·Vasconcelos,·Invent.·Math.·110·207-235·(1992).
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115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·0116 <td>··············<pre><code·class="language-macaulay2">i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·0
  
117 o6·=·true</code></pre>117 o6·=·true</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·radicalContainment(x_0,·I)120 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·radicalContainment(x_0,·I)
121 ·--·.0513011s·elapsed121 ·--·.0936957s·elapsed
  
122 o7·=·true</code></pre>122 o7·=·true</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·&quot;Kollar&quot;)125 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·&quot;Kollar&quot;)
126 ·--·.00115772s·elapsed126 ·--·.00166549s·elapsed
  
127 o8·=·true</code></pre>127 o8·=·true</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ··········<tr>129 ··········<tr>
130 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·&quot;Kollar&quot;)130 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·&quot;Kollar&quot;)
131 ·--·.000864384s·elapsed131 ·--·.00140371s·elapsed
  
132 o9·=·false</code></pre>132 o9·=·false</code></pre>
133 </td>··········</tr>133 </td>··········</tr>
134 ········</table>134 ········</table>
135 ······</div>135 ······</div>
136 ······<div>136 ······<div>
137 ········<h2>See·also</h2>137 ········<h2>See·also</h2>
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51 i5·:·D·=·product(I_*/degree/sum)51 i5·:·D·=·product(I_*/degree/sum)
  
52 o5·=·84052 o5·=·840
53 i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·053 i6·:·x_0^(D-1)·%·I·!=·0·and·x_0^D·%·I·==·0
  
54 o6·=·true54 o6·=·true
55 i7·:·elapsedTime·radicalContainment(x_0,·I)55 i7·:·elapsedTime·radicalContainment(x_0,·I)
56 ·--·.0513011s·elapsed56 ·--·.0936957s·elapsed
  
57 o7·=·true57 o7·=·true
58 i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·"Kollar")58 i8·:·elapsedTime·radicalContainment(x_0,·I,·Strategy·=>·"Kollar")
59 ·--·.00115772s·elapsed59 ·--·.00166549s·elapsed
  
60 o8·=·true60 o8·=·true
61 i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·"Kollar")61 i9·:·elapsedTime·radicalContainment(x_n,·I,·Strategy·=>·"Kollar")
62 ·--·.000864384s·elapsed62 ·--·.00140371s·elapsed
  
63 o9·=·false63 o9·=·false
64 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*64 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
65 ····*·_\x8r_\x8a_\x8d_\x8i_\x8c_\x8a_\x8l·--·the·radical·of·an·ideal65 ····*·_\x8r_\x8a_\x8d_\x8i_\x8c_\x8a_\x8l·--·the·radical·of·an·ideal
66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8ad\x8di\x8ic\x8ca\x8al\x8lC\x8Co\x8on\x8nt\x8ta\x8ai\x8in\x8nm\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8ad\x8di\x8ic\x8ca\x8al\x8lC\x8Co\x8on\x8nt\x8ta\x8ai\x8in\x8nm\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
67 ····*·radicalContainment(Ideal,Ideal)67 ····*·radicalContainment(Ideal,Ideal)
68 ····*·radicalContainment(RingElement,Ideal)68 ····*·radicalContainment(RingElement,Ideal)
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11 bGUgVGltZXMiLCAibGluZW51bSIgPT4gMjAwLCBJbnB1dHMgPT4ge1NQQU57VFR7IkoifSwiLCAi11 bGUgVGltZXMiLCAibGluZW51bSIgPT4gMjAwLCBJbnB1dHMgPT4ge1NQQU57VFR7IkoifSwiLCAi
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57 i9·:·J·=·ideal·vars·U57 i9·:·J·=·ideal·vars·U
  
58 o9·=·ideal·(a,·b,·c)58 o9·=·ideal·(a,·b,·c)
  
59 o9·:·Ideal·of·U59 o9·:·Ideal·of·U
  
60 i10·:·time·multiReesIdeal·J60 i10·:·time·multiReesIdeal·J
61 ·--·used·1.00657s·(cpu);·0.219058s·(thread);·0s·(gc)61 ·--·used·0.860993s·(cpu);·0.162415s·(thread);·0s·(gc)
  
62 ·············································································62 ·············································································
63 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,63 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
64 ················1······2·····1······2·····1······2·····0······2·····0······2·64 ················1······2·····1······2·····1······2·····0······2·····0······2·
65 ······-----------------------------------------------------------------------65 ······-----------------------------------------------------------------------
66 ····················2·················2···266 ····················2·················2···2
67 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)67 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
68 ·········0······2···1····0·2···0·1····2···0····1·268 ·········0······2···1····0·2···0·1····2···0····1·2
  
69 o10·:·Ideal·of·U[X·..X·]69 o10·:·Ideal·of·U[X·..X·]
70 ··················0···270 ··················0···2
  
71 i11·:·time·multiReesIdeal·(J,·a)71 i11·:·time·multiReesIdeal·(J,·a)
72 ·--·used·0.189324s·(cpu);·0.0388645s·(thread);·0s·(gc)72 ·--·used·0.210665s·(cpu);·0.0296081s·(thread);·0s·(gc)
  
73 ·············································································73 ·············································································
74 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,74 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
75 ················1······2·····1······2·····1······2·····0······2·····0······2·75 ················1······2·····1······2·····1······2·····0······2·····0······2·
76 ······-----------------------------------------------------------------------76 ······-----------------------------------------------------------------------
77 ····················2·················2···277 ····················2·················2···2
78 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)78 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
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157 o9·=·ideal·(a,·b,·c)157 o9·=·ideal·(a,·b,·c)
  
158 o9·:·Ideal·of·U</code></pre>158 o9·:·Ideal·of·U</code></pre>
159 </td>··········</tr>159 </td>··········</tr>
160 ··········<tr>160 ··········<tr>
161 <td>··············<pre><code·class="language-macaulay2">i10·:·time·multiReesIdeal·J161 <td>··············<pre><code·class="language-macaulay2">i10·:·time·multiReesIdeal·J
162 ·--·used·1.00657s·(cpu);·0.219058s·(thread);·0s·(gc)162 ·--·used·0.860993s·(cpu);·0.162415s·(thread);·0s·(gc)
  
163 ·············································································163 ·············································································
164 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,164 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
165 ················1······2·····1······2·····1······2·····0······2·····0······2·165 ················1······2·····1······2·····1······2·····0······2·····0······2·
166 ······-----------------------------------------------------------------------166 ······-----------------------------------------------------------------------
167 ····················2·················2···2167 ····················2·················2···2
168 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)168 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
169 ·········0······2···1····0·2···0·1····2···0····1·2169 ·········0······2···1····0·2···0·1····2···0····1·2
  
170 o10·:·Ideal·of·U[X·..X·]170 o10·:·Ideal·of·U[X·..X·]
171 ··················0···2</code></pre>171 ··················0···2</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i11·:·time·multiReesIdeal·(J,·a)174 <td>··············<pre><code·class="language-macaulay2">i11·:·time·multiReesIdeal·(J,·a)
175 ·--·used·0.189324s·(cpu);·0.0388645s·(thread);·0s·(gc)175 ·--·used·0.210665s·(cpu);·0.0296081s·(thread);·0s·(gc)
  
176 ·············································································176 ·············································································
177 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,177 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
178 ················1······2·····1······2·····1······2·····0······2·····0······2·178 ················1······2·····1······2·····1······2·····0······2·····0······2·
179 ······-----------------------------------------------------------------------179 ······-----------------------------------------------------------------------
180 ····················2·················2···2180 ····················2·················2···2
181 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)181 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
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80 i8·:·U·=·T/minors(2,m);80 i8·:·U·=·T/minors(2,m);
81 i9·:·J·=·ideal·vars·U81 i9·:·J·=·ideal·vars·U
  
82 o9·=·ideal·(a,·b,·c)82 o9·=·ideal·(a,·b,·c)
  
83 o9·:·Ideal·of·U83 o9·:·Ideal·of·U
84 i10·:·time·multiReesIdeal·J84 i10·:·time·multiReesIdeal·J
85 ·--·used·1.00657s·(cpu);·0.219058s·(thread);·0s·(gc)85 ·--·used·0.860993s·(cpu);·0.162415s·(thread);·0s·(gc)
  
  
86 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,86 o10·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
87 ················1······2·····1······2·····1······2·····0······2·····0······287 ················1······2·····1······2·····1······2·····0······2·····0······2
88 ······-----------------------------------------------------------------------88 ······-----------------------------------------------------------------------
89 ····················2·················2···289 ····················2·················2···2
90 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)90 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
91 ·········0······2···1····0·2···0·1····2···0····1·291 ·········0······2···1····0·2···0·1····2···0····1·2
  
92 o10·:·Ideal·of·U[X·..X·]92 o10·:·Ideal·of·U[X·..X·]
93 ··················0···293 ··················0···2
94 i11·:·time·multiReesIdeal·(J,·a)94 i11·:·time·multiReesIdeal·(J,·a)
95 ·--·used·0.189324s·(cpu);·0.0388645s·(thread);·0s·(gc)95 ·--·used·0.210665s·(cpu);·0.0296081s·(thread);·0s·(gc)
  
  
96 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,96 o11·=·ideal·(c*X··-·b*X·,·b*X··-·a*X·,·a*X··-·c*X·,·c*X··-·a*X·,·b*X··-·c*X·,
97 ················1······2·····1······2·····1······2·····0······2·····0······297 ················1······2·····1······2·····1······2·····0······2·····0······2
98 ······-----------------------------------------------------------------------98 ······-----------------------------------------------------------------------
99 ····················2·················2···299 ····················2·················2···2
100 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)100 ······a*X··-·b*X·,·X··-·X·X·,·X·X··-·X·,·X··-·X·X·)
⊟
742 B
./usr/share/doc/Macaulay2/ModuleDeformations/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=157 #:len=15
8 ZGVmb3JtTUNNTW9kdWxl8 ZGVmb3JtTUNNTW9kdWxl
9 #:len=22379 #:len=2237
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D
11 TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt11 TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt
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1.43 KB
./usr/share/doc/Macaulay2/ModuleDeformations/example-output/_deform__M__C__M__Module_lp__Module_rp.out
    
Offset 40, 15 lines modifiedOffset 40, 15 lines modified
  
40 o7·=·image·|·x2·y2·|40 o7·=·image·|·x2·y2·|
  
41 ·····························141 ·····························1
42 o7·:·R-module,·submodule·of·R42 o7·:·R-module,·submodule·of·R
  
43 i8·:·(S,N)·=·time·deformMCMModule·N0·43 i8·:·(S,N)·=·time·deformMCMModule·N0·
44 ·--·used·0.363627s·(cpu);·0.273194s·(thread);·0s·(gc)44 ·--·used·0.44072s·(cpu);·0.317646s·(thread);·0s·(gc)
  
45 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^345 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
46 ··················{8}·|·xxi_4-y+xi_3·······································46 ··················{8}·|·xxi_4-y+xi_3·······································
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
48 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)48 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)
49 ·····-x2-xxi_2-xi_1···········································|49 ·····-x2-xxi_2-xi_1···········································|
  
Offset 70, 15 lines modifiedOffset 70, 15 lines modified
70 o10·=·cokernel·|·x2·y2··|70 o10·=·cokernel·|·x2·y2··|
71 ···············|·-y·-x2·|71 ···············|·-y·-x2·|
  
72 ·····························272 ·····························2
73 o10·:·R-module,·quotient·of·R73 o10·:·R-module,·quotient·of·R
  
74 i11·:·(S',N')·=·time·deformMCMModule·N0'74 i11·:·(S',N')·=·time·deformMCMModule·N0'
75 ·--·used·0.49468s·(cpu);·0.392181s·(thread);·0s·(gc)75 ·--·used·0.621073s·(cpu);·0.467315s·(thread);·0s·(gc)
  
76 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_176 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
77 ····················|·xxi_4-y+xi_3·······································77 ····················|·xxi_4-y+xi_3·······································
78 ······-----------------------------------------------------------------------78 ······-----------------------------------------------------------------------
79 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)79 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)
80 ······-x2-xxi_2-xi_1·····························|80 ······-x2-xxi_2-xi_1·····························|
  
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2.8 KB
./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
    
Offset 134, 15 lines modifiedOffset 134, 15 lines modified
134 o7·=·image·|·x2·y2·|134 o7·=·image·|·x2·y2·|
  
135 ·····························1135 ·····························1
136 o7·:·R-module,·submodule·of·R</code></pre>136 o7·:·R-module,·submodule·of·R</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i8·:·(S,N)·=·time·deformMCMModule·N0·139 <td>··············<pre><code·class="language-macaulay2">i8·:·(S,N)·=·time·deformMCMModule·N0·
140 ·--·used·0.363627s·(cpu);·0.273194s·(thread);·0s·(gc)140 ·--·used·0.44072s·(cpu);·0.317646s·(thread);·0s·(gc)
  
141 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3141 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
142 ··················{8}·|·xxi_4-y+xi_3·······································142 ··················{8}·|·xxi_4-y+xi_3·······································
143 ·····------------------------------------------------------------------------143 ·····------------------------------------------------------------------------
144 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)144 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)
145 ·····-x2-xxi_2-xi_1···········································|145 ·····-x2-xxi_2-xi_1···········································|
  
Offset 169, 15 lines modifiedOffset 169, 15 lines modified
169 ···············|·-y·-x2·|169 ···············|·-y·-x2·|
  
170 ·····························2170 ·····························2
171 o10·:·R-module,·quotient·of·R</code></pre>171 o10·:·R-module,·quotient·of·R</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i11·:·(S',N')·=·time·deformMCMModule·N0'174 <td>··············<pre><code·class="language-macaulay2">i11·:·(S',N')·=·time·deformMCMModule·N0'
175 ·--·used·0.49468s·(cpu);·0.392181s·(thread);·0s·(gc)175 ·--·used·0.621073s·(cpu);·0.467315s·(thread);·0s·(gc)
  
176 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1176 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
177 ····················|·xxi_4-y+xi_3·······································177 ····················|·xxi_4-y+xi_3·······································
178 ······-----------------------------------------------------------------------178 ······-----------------------------------------------------------------------
179 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)179 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)
180 ······-x2-xxi_2-xi_1·····························|180 ······-x2-xxi_2-xi_1·····························|
  
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1.21 KB
html2text {}
    
Offset 71, 15 lines modifiedOffset 71, 15 lines modified
71 i7·:·N0·=·module·ideal·(x^2,y^2)71 i7·:·N0·=·module·ideal·(x^2,y^2)
  
72 o7·=·image·|·x2·y2·|72 o7·=·image·|·x2·y2·|
  
73 ·····························173 ·····························1
74 o7·:·R-module,·submodule·of·R74 o7·:·R-module,·submodule·of·R
75 i8·:·(S,N)·=·time·deformMCMModule·N075 i8·:·(S,N)·=·time·deformMCMModule·N0
76 ·--·used·0.363627s·(cpu);·0.273194s·(thread);·0s·(gc)76 ·--·used·0.44072s·(cpu);·0.317646s·(thread);·0s·(gc)
  
77 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^377 o8·=·(S,·cokernel·{6}·|·x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
78 ··················{8}·|·xxi_4-y+xi_378 ··················{8}·|·xxi_4-y+xi_3
79 ·····------------------------------------------------------------------------79 ·····------------------------------------------------------------------------
80 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)80 ·····xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2·|)
81 ·····-x2-xxi_2-xi_1···········································|81 ·····-x2-xxi_2-xi_1···········································|
  
Offset 104, 15 lines modifiedOffset 104, 15 lines modified
  
104 o10·=·cokernel·|·x2·y2··|104 o10·=·cokernel·|·x2·y2··|
105 ···············|·-y·-x2·|105 ···············|·-y·-x2·|
  
106 ·····························2106 ·····························2
107 o10·:·R-module,·quotient·of·R107 o10·:·R-module,·quotient·of·R
108 i11·:·(S',N')·=·time·deformMCMModule·N0'108 i11·:·(S',N')·=·time·deformMCMModule·N0'
109 ·--·used·0.49468s·(cpu);·0.392181s·(thread);·0s·(gc)109 ·--·used·0.621073s·(cpu);·0.467315s·(thread);·0s·(gc)
  
110 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1110 o11·=·(S',·cokernel·|·x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
111 ····················|·xxi_4-y+xi_3111 ····················|·xxi_4-y+xi_3
112 ······-----------------------------------------------------------------------112 ······-----------------------------------------------------------------------
113 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)113 ······x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2·|)
114 ······-x2-xxi_2-xi_1·····························|114 ······-x2-xxi_2-xi_1·····························|
  
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779 B
./usr/share/doc/Macaulay2/MonodromySolver/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=467 #:len=46
8 c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==8 c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==
9 #:len=3459 #:len=345
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl
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1.15 KB
./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
    
Offset 3, 27 lines modifiedOffset 3, 27 lines modified
3 i1·:·R·=·CC[a,b,c,d][x,y];3 i1·:·R·=·CC[a,b,c,d][x,y];
  
4 i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};4 i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};
  
5 i3·:·(p0,·x0)·=·createSeedPair·polys;5 i3·:·(p0,·x0)·=·createSeedPair·polys;
  
6 i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})6 i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})
7 ·--·.00297526s·elapsed 
8 ·--·.0110854s·elapsed7 ·--·.0115241s·elapsed
9 ·--·.000363749s·elapsed8 ·--·.00349479s·elapsed
10 ·--·.00293305s·elapsed 
11 ·--·.0741836s·elapsed 
12 ·--·.000311972s·elapsed9 ·--·.000395211s·elapsed
13 ·--·.00234819s·elapsed10 ·--·.00330616s·elapsed
14 ·--·.0106965s·elapsed11 ·--·.0610964s·elapsed
 12 ·--·.000439194s·elapsed
15 ·--·.00024558s·elapsed13 ·--·.00784521s·elapsed
16 ·--·.00652893s·elapsed14 ·--·.00761673s·elapsed
17 ·--·.00925233s·elapsed 
18 ·--·.000324199s·elapsed15 ·--·.000312495s·elapsed
 16 ·--·.00365501s·elapsed
 17 ·--·.00372107s·elapsed
 18 ·--·.000379782s·elapsed
19 --backup·directory·created:·/tmp/M2-3861463-0/119 --backup·directory·created:·/tmp/M2-2830557-0/1
20 ··H01:·120 ··H01:·1
21 ··H10:·121 ··H10:·1
22 number·of·paths·tracked:·222 number·of·paths·tracked:·2
23 found·1·points·in·the·fiber·so·far23 found·1·points·in·the·fiber·so·far
24 ··H01:·124 ··H01:·1
25 ··H10:·125 ··H10:·1
26 number·of·paths·tracked:·426 number·of·paths·tracked:·4
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13.7 KB
./usr/share/doc/Macaulay2/MonodromySolver/example-output/_monodromy__Group.out
    
Offset 14, 128 lines modifiedOffset 14, 128 lines modified
  
14 i7·:·dLoss·=·diff(varMatrix,·gateMatrix{{loss}});14 i7·:·dLoss·=·diff(varMatrix,·gateMatrix{{loss}});
  
15 i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);15 i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);
  
16 i9·:·monodromyGroup(G,"msOptions"·=>·{NumberOfEdges=>10})16 i9·:·monodromyGroup(G,"msOptions"·=>·{NumberOfEdges=>10})
  
17 o9·=·{{1,·6,·3,·19,·16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,17 o9·=·{{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,·15,·6,·9,
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ·····7},·{7,·1,·9,·0,·12,·17,·2,·3,·4,·8,·10,·15,·6,·11,·14,·13,·5,·16,·19,19 ·····4},·{0,·9,·2,·3,·10,·5,·14,·7,·8,·1,·4,·11,·12,·13,·6,·15,·16,·17,·18,
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ·····20,·18},·{3,·1,·6,·19,·0,·2,·13,·8,·9,·16,·10,·12,·4,·15,·14,·20,·17,21 ·····19,·20},·{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,
22 ·····------------------------------------------------------------------------22 ·····------------------------------------------------------------------------
23 ·····18,·11,·5,·7},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,23 ·····15,·6,·9,·4},·{16,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·0,·14,·3,
24 ·····------------------------------------------------------------------------24 ·····------------------------------------------------------------------------
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26 ·····------------------------------------------------------------------------26 ·····------------------------------------------------------------------------
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28 ·····------------------------------------------------------------------------28 ·····------------------------------------------------------------------------
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30 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
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34 ·····------------------------------------------------------------------------34 ·····------------------------------------------------------------------------
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36 ·····------------------------------------------------------------------------36 ·····------------------------------------------------------------------------
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48 ·····------------------------------------------------------------------------48 ·····------------------------------------------------------------------------
49 ·····10,·8,·13,·6,·20,·9,·18,·12,·4,·14,·19,·2,·5,·1,·16,·7,·0,·3,·15,·17},49 ·····16,·14,·19,·4,·3,·6,·15,·1,·9,·0,·17,·10,·2,·13,·20,·8,·18,·7,·11,·5},
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78 ·····------------------------------------------------------------------------78 ·····------------------------------------------------------------------------
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110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
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112 ·····------------------------------------------------------------------------112 ·····------------------------------------------------------------------------
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118 ·····------------------------------------------------------------------------118 ·····------------------------------------------------------------------------
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120 ·····------------------------------------------------------------------------120 ·····------------------------------------------------------------------------
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122 ·····------------------------------------------------------------------------122 ·····------------------------------------------------------------------------
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132 ·····------------------------------------------------------------------------132 ·····------------------------------------------------------------------------
133 ·····16,·17,·18,·19,·20}}133 ·····5,·20,·14,·18,·3,·15,·17}}
  
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./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
    
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96 <td>··············<pre><code·class="language-macaulay2">i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};</code></pre>96 <td>··············<pre><code·class="language-macaulay2">i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·(p0,·x0)·=·createSeedPair·polys;</code></pre>99 <td>··············<pre><code·class="language-macaulay2">i3·:·(p0,·x0)·=·createSeedPair·polys;</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})102 <td>··············<pre><code·class="language-macaulay2">i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})
103 ·--·.00297526s·elapsed 
104 ·--·.0110854s·elapsed103 ·--·.0115241s·elapsed
105 ·--·.000363749s·elapsed104 ·--·.00349479s·elapsed
106 ·--·.00293305s·elapsed 
107 ·--·.0741836s·elapsed 
108 ·--·.000311972s·elapsed105 ·--·.000395211s·elapsed
109 ·--·.00234819s·elapsed106 ·--·.00330616s·elapsed
110 ·--·.0106965s·elapsed107 ·--·.0610964s·elapsed
 108 ·--·.000439194s·elapsed
111 ·--·.00024558s·elapsed109 ·--·.00784521s·elapsed
112 ·--·.00652893s·elapsed110 ·--·.00761673s·elapsed
113 ·--·.00925233s·elapsed 
114 ·--·.000324199s·elapsed111 ·--·.000312495s·elapsed
 112 ·--·.00365501s·elapsed
 113 ·--·.00372107s·elapsed
 114 ·--·.000379782s·elapsed
115 --backup·directory·created:·/tmp/M2-3861463-0/1115 --backup·directory·created:·/tmp/M2-2830557-0/1
116 ··H01:·1116 ··H01:·1
117 ··H10:·1117 ··H10:·1
118 number·of·paths·tracked:·2118 number·of·paths·tracked:·2
119 found·1·points·in·the·fiber·so·far119 found·1·points·in·the·fiber·so·far
120 ··H01:·1120 ··H01:·1
121 ··H10:·1121 ··H10:·1
122 number·of·paths·tracked:·4122 number·of·paths·tracked:·4
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html2text {}
    
Offset 23, 27 lines modifiedOffset 23, 27 lines modified
23 ··········o·npaths,·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,23 ··········o·npaths,·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,
24 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*24 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
25 Output·is·verbose.·For·other·dynamic·strategies,·see·_\x8M_\x8o_\x8n_\x8o_\x8d_\x8r_\x8o_\x8m_\x8y_\x8S_\x8o_\x8l_\x8v_\x8e_\x8r_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.25 Output·is·verbose.·For·other·dynamic·strategies,·see·_\x8M_\x8o_\x8n_\x8o_\x8d_\x8r_\x8o_\x8m_\x8y_\x8S_\x8o_\x8l_\x8v_\x8e_\x8r_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
26 i1·:·R·=·CC[a,b,c,d][x,y];26 i1·:·R·=·CC[a,b,c,d][x,y];
27 i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};27 i2·:·polys·=·polySystem·{a*x+b*y^2,c*x*y+d};
28 i3·:·(p0,·x0)·=·createSeedPair·polys;28 i3·:·(p0,·x0)·=·createSeedPair·polys;
29 i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})29 i4·:·(L,·npaths)·=·dynamicFlowerSolve(polys.PolyMap,p0,{x0})
30 ·--·.00297526s·elapsed 
31 ·--·.0110854s·elapsed30 ·--·.0115241s·elapsed
32 ·--·.000363749s·elapsed31 ·--·.00349479s·elapsed
33 ·--·.00293305s·elapsed 
34 ·--·.0741836s·elapsed 
35 ·--·.000311972s·elapsed32 ·--·.000395211s·elapsed
36 ·--·.00234819s·elapsed33 ·--·.00330616s·elapsed
37 ·--·.0106965s·elapsed34 ·--·.0610964s·elapsed
 35 ·--·.000439194s·elapsed
38 ·--·.00024558s·elapsed36 ·--·.00784521s·elapsed
39 ·--·.00652893s·elapsed37 ·--·.00761673s·elapsed
40 ·--·.00925233s·elapsed 
41 ·--·.000324199s·elapsed38 ·--·.000312495s·elapsed
 39 ·--·.00365501s·elapsed
 40 ·--·.00372107s·elapsed
 41 ·--·.000379782s·elapsed
42 --backup·directory·created:·/tmp/M2-3861463-0/142 --backup·directory·created:·/tmp/M2-2830557-0/1
43 ··H01:·143 ··H01:·1
44 ··H10:·144 ··H10:·1
45 number·of·paths·tracked:·245 number·of·paths·tracked:·2
46 found·1·points·in·the·fiber·so·far46 found·1·points·in·the·fiber·so·far
47 ··H01:·147 ··H01:·1
48 ··H10:·148 ··H10:·1
49 number·of·paths·tracked:·449 number·of·paths·tracked:·4
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./usr/share/doc/Macaulay2/MonodromySolver/html/_monodromy__Group.html
    
Offset 104, 131 lines modifiedOffset 104, 131 lines modified
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);</code></pre>106 <td>··············<pre><code·class="language-macaulay2">i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i9·:·monodromyGroup(G,&quot;msOptions&quot;·=>·{NumberOfEdges=>10})109 <td>··············<pre><code·class="language-macaulay2">i9·:·monodromyGroup(G,&quot;msOptions&quot;·=>·{NumberOfEdges=>10})
  
110 o9·=·{{1,·6,·3,·19,·16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,110 o9·=·{{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,·15,·6,·9,
111 ·····------------------------------------------------------------------------111 ·····------------------------------------------------------------------------
112 ·····7},·{7,·1,·9,·0,·12,·17,·2,·3,·4,·8,·10,·15,·6,·11,·14,·13,·5,·16,·19,112 ·····4},·{0,·9,·2,·3,·10,·5,·14,·7,·8,·1,·4,·11,·12,·13,·6,·15,·16,·17,·18,
113 ·····------------------------------------------------------------------------113 ·····------------------------------------------------------------------------
114 ·····20,·18},·{3,·1,·6,·19,·0,·2,·13,·8,·9,·16,·10,·12,·4,·15,·14,·20,·17,114 ·····19,·20},·{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,
115 ·····------------------------------------------------------------------------115 ·····------------------------------------------------------------------------
116 ·····18,·11,·5,·7},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,116 ·····15,·6,·9,·4},·{16,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·0,·14,·3,
117 ·····------------------------------------------------------------------------117 ·····------------------------------------------------------------------------
118 ·····0,·3,·19,·20,·18},·{1,·6,·3,·19,·16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,118 ·····13,·15,·6,·9,·4},·{0,·11,·2,·3,·10,·4,·14,·6,·8,·1,·5,·9,·12,·13,·7,·15,
119 ·····------------------------------------------------------------------------119 ·····------------------------------------------------------------------------
120 ·····20,·14,·18,·11,·5,·7},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,120 ·····16,·17,·18,·19,·20},·{0,·1,·2,·3,·4,·11,·6,·5,·8,·9,·10,·7,·12,·13,·14,
121 ·····------------------------------------------------------------------------121 ·····------------------------------------------------------------------------
122 ·····14,·15,·16,·17,·18,·19,·20},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,122 ·····15,·16,·17,·18,·19,·20},·{12,·1,·7,·17,·16,·9,·0,·4,·11,·13,·10,·6,·5,
123 ·····------------------------------------------------------------------------123 ·····------------------------------------------------------------------------
124 ·····2,·16,·1,·17,·0,·3,·19,·20,·18},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,124 ·····2,·14,·3,·8,·15,·20,·18,·19},·{0,·9,·2,·3,·10,·5,·14,·7,·8,·1,·4,·11,
125 ·····------------------------------------------------------------------------125 ·····------------------------------------------------------------------------
126 ·····5,·2,·16,·1,·17,·0,·3,·19,·20,·18},·{10,·9,·7,·0,·12,·17,·2,·3,·11,·8,126 ·····12,·13,·6,·15,·16,·17,·18,·19,·20},·{0,·1,·2,·3,·9,·11,·4,·5,·8,·6,·10,
127 ·····------------------------------------------------------------------------127 ·····------------------------------------------------------------------------
128 ·····4,·15,·5,·14,·6,·13,·1,·16,·19,·20,·18},·{11,·10,·12,·13,·6,·20,·9,·18,128 ·····7,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{12,·1,·3,·19,·0,·11,·13,·5,·15,
129 ·····------------------------------------------------------------------------129 ·····------------------------------------------------------------------------
130 ·····2,·4,·14,·19,·8,·5,·1,·16,·7,·0,·3,·15,·17},·{13,·10,·19,·2,·6,·5,·9,·7,130 ·····16,·10,·7,·17,·2,·14,·20,·8,·18,·9,·4,·6},·{0,·1,·3,·19,·12,·11,·2,·5,
131 ·····------------------------------------------------------------------------131 ·····------------------------------------------------------------------------
132 ·····20,·4,·14,·11,·18,·16,·1,·8,·0,·12,·17,·3,·15},·{6,·10,·12,·13,·11,·20,132 ·····15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{12,·16,·14,·17,·4,·7,
133 ·····------------------------------------------------------------------------133 ·····------------------------------------------------------------------------
134 ·····5,·18,·2,·7,·14,·19,·8,·9,·1,·16,·4,·0,·3,·15,·17},·{13,·10,·19,·2,·6,134 ·····6,·11,·1,·9,·0,·5,·10,·2,·13,·3,·8,·15,·20,·18,·19},·{0,·1,·3,·19,·12,
135 ·····------------------------------------------------------------------------135 ·····------------------------------------------------------------------------
136 ·····5,·9,·7,·20,·4,·14,·11,·18,·16,·1,·8,·0,·12,·17,·3,·15},·{13,·1,·19,·2,136 ·····11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{0,·1,·3,·19,
137 ·····------------------------------------------------------------------------137 ·····------------------------------------------------------------------------
138 ·····6,·5,·9,·7,·20,·4,·10,·11,·18,·16,·14,·8,·0,·12,·17,·3,·15},·{6,·10,·7,138 ·····12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{17,·16,
139 ·····------------------------------------------------------------------------139 ·····------------------------------------------------------------------------
140 ·····13,·2,·20,·8,·18,·11,·12,·14,·19,·5,·9,·1,·16,·4,·0,·3,·15,·17},·{11,140 ·····14,·9,·8,·7,·12,·11,·1,·2,·0,·5,·10,·3,·13,·4,·15,·6,·20,·18,·19},·{12,
141 ·····------------------------------------------------------------------------141 ·····------------------------------------------------------------------------
142 ·····10,·8,·13,·6,·20,·9,·18,·12,·4,·14,·19,·2,·5,·1,·16,·7,·0,·3,·15,·17},142 ·····16,·14,·19,·4,·3,·6,·15,·1,·9,·0,·17,·10,·2,·13,·20,·8,·18,·7,·11,·5},
143 ·····------------------------------------------------------------------------143 ·····------------------------------------------------------------------------
144 ·····{10,·0,·5,·2,·6,·19,·9,·20,·7,·4,·13,·18,·11,·14,·16,·8,·1,·12,·17,·3,144 ·····{12,·16,·3,·19,·10,·11,·14,·5,·15,·1,·0,·7,·17,·2,·13,·20,·8,·18,·9,·4,
145 ·····------------------------------------------------------------------------145 ·····------------------------------------------------------------------------
146 ·····15},·{6,·10,·12,·13,·11,·20,·5,·18,·2,·7,·14,·19,·8,·9,·1,·16,·4,·0,·3,146 ·····6},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,
147 ·····------------------------------------------------------------------------147 ·····------------------------------------------------------------------------
148 ·····15,·17},·{7,·10,·8,·13,·6,·20,·9,·18,·12,·4,·14,·19,·2,·11,·1,·16,·5,·0,148 ·····4,·6},·{12,·16,·14,·19,·4,·7,·6,·11,·1,·9,·0,·5,·10,·2,·13,·20,·8,·18,
149 ·····------------------------------------------------------------------------149 ·····------------------------------------------------------------------------
150 ·····3,·15,·17},·{0,·9,·15,·20,·10,·8,·14,·12,·17,·1,·4,·2,·3,·13,·6,·18,·16,150 ·····3,·15,·17},·{12,·16,·7,·17,·10,·9,·14,·4,·11,·1,·0,·6,·5,·2,·13,·3,·8,
151 ·····------------------------------------------------------------------------151 ·····------------------------------------------------------------------------
152 ·····19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,·14,·6,·18,152 ·····15,·20,·18,·19},·{2,·0,·9,·3,·10,·5,·14,·7,·4,·1,·13,·11,·6,·8,·16,·15,
153 ·····------------------------------------------------------------------------153 ·····------------------------------------------------------------------------
154 ·····1,·19,·5,·7,·11},·{0,·1,·6,·3,·7,·2,·11,·8,·9,·5,·10,·12,·4,·13,·14,·15,154 ·····12,·17,·18,·19,·20},·{3,·16,·12,·4,·10,·11,·14,·5,·2,·1,·0,·7,·8,·15,
155 ·····------------------------------------------------------------------------155 ·····------------------------------------------------------------------------
156 ·····16,·17,·18,·19,·20},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,·14,156 ·····13,·6,·17,·9,·18,·19,·20},·{2,·0,·9,·3,·10,·5,·14,·7,·4,·1,·13,·11,·6,
157 ·····------------------------------------------------------------------------157 ·····------------------------------------------------------------------------
158 ·····6,·18,·1,·19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,158 ·····8,·16,·15,·12,·17,·18,·19,·20},·{0,·1,·2,·20,·6,·5,·9,·7,·8,·4,·10,·11,
159 ·····------------------------------------------------------------------------159 ·····------------------------------------------------------------------------
160 ·····14,·6,·18,·1,·19,·5,·7,·11},·{3,·1,·6,·5,·16,·2,·0,·8,·9,·13,·10,·12,·4,160 ·····12,·13,·14,·18,·16,·19,·15,·17,·3},·{12,·16,·7,·17,·10,·9,·14,·4,·11,·1,
161 ·····------------------------------------------------------------------------161 ·····------------------------------------------------------------------------
162 ·····15,·14,·7,·17,·11,·18,·19,·20},·{0,·1,·6,·20,·7,·15,·11,·17,·9,·5,·10,162 ·····0,·6,·5,·2,·13,·3,·8,·15,·20,·18,·19},·{0,·1,·2,·20,·6,·5,·9,·7,·8,·4,
163 ·····------------------------------------------------------------------------163 ·····------------------------------------------------------------------------
164 ·····3,·4,·13,·14,·18,·16,·19,·2,·8,·12},·{0,·1,·15,·20,·4,·8,·6,·12,·17,·9,164 ·····10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{0,·1,·2,·20,·6,·5,·9,·7,·8,
165 ·····------------------------------------------------------------------------165 ·····------------------------------------------------------------------------
166 ·····10,·2,·3,·13,·14,·18,·16,·19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,166 ·····4,·10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{2,·0,·9,·3,·10,·5,·14,
167 ·····------------------------------------------------------------------------167 ·····------------------------------------------------------------------------
168 ·····17,·16,·4,·2,·3,·14,·6,·18,·1,·19,·5,·7,·11},·{0,·1,·6,·20,·7,·2,·11,·8,168 ·····7,·4,·1,·13,·11,·6,·8,·16,·15,·12,·17,·20,·18,·19},·{1,·16,·12,·19,·11,
169 ·····------------------------------------------------------------------------169 ·····------------------------------------------------------------------------
170 ·····9,·5,·10,·12,·4,·13,·14,·18,·16,·19,·15,·17,·3},·{13,·10,·5,·15,·4,·12,170 ·····15,·5,·17,·2,·7,·0,·3,·8,·10,·13,·20,·14,·18,·9,·4,·6},·{12,·16,·7,·17,
171 ·····------------------------------------------------------------------------171 ·····------------------------------------------------------------------------
172 ·····6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·19,·20,·18},·{15,·1,·0,·7,·4,·8,172 ·····10,·9,·14,·4,·11,·1,·0,·6,·5,·2,·13,·3,·8,·15,·18,·19,·20},·{1,·16,·12,
173 ·····------------------------------------------------------------------------173 ·····------------------------------------------------------------------------
174 ·····6,·12,·13,·9,·10,·2,·16,·17,·14,·11,·3,·5,·19,·20,·18},·{13,·10,·5,·15,174 ·····19,·11,·15,·5,·17,·2,·7,·0,·3,·8,·10,·13,·20,·14,·18,·9,·4,·6},·{2,·0,
175 ·····------------------------------------------------------------------------175 ·····------------------------------------------------------------------------
176 ·····4,·12,·6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·19,·20,·18},·{0,·1,·2,176 ·····11,·3,·10,·4,·14,·6,·5,·1,·13,·9,·7,·8,·16,·15,·12,·17,·18,·19,·20},·{0,
177 ·····------------------------------------------------------------------------177 ·····------------------------------------------------------------------------
178 ·····20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{0,·1,178 ·····1,·2,·20,·6,·5,·9,·7,·8,·4,·10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},
179 ·····------------------------------------------------------------------------179 ·····------------------------------------------------------------------------
180 ·····2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{0,180 ·····{3,·1,·12,·4,·0,·11,·13,·5,·2,·16,·10,·7,·8,·15,·14,·6,·17,·9,·18,·19,
181 ·····------------------------------------------------------------------------181 ·····------------------------------------------------------------------------
182 ·····1,·2,·20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,·3},182 ·····20},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,
183 ·····------------------------------------------------------------------------183 ·····------------------------------------------------------------------------
184 ·····{0,·1,·2,·20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,184 ·····19,·20},·{0,·14,·12,·19,·11,·15,·5,·17,·2,·7,·1,·3,·8,·13,·10,·20,·16,
185 ·····------------------------------------------------------------------------185 ·····------------------------------------------------------------------------
186 ·····3},·{13,·10,·5,·15,·4,·12,·6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·18,186 ·····18,·9,·4,·6},·{2,·1,·0,·3,·4,·5,·6,·7,·13,·9,·10,·11,·16,·8,·14,·15,·12,
187 ·····------------------------------------------------------------------------187 ·····------------------------------------------------------------------------
188 ·····19,·20},·{16,·6,·3,·19,·10,·2,·14,·8,·15,·1,·9,·12,·17,·0,·4,·20,·13,188 ·····17,·18,·19,·20},·{0,·1,·3,·19,·12,·5,·2,·7,·15,·8,·10,·11,·17,·13,·14,
189 ·····------------------------------------------------------------------------189 ·····------------------------------------------------------------------------
190 ·····18,·11,·5,·7},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,190 ·····20,·16,·18,·9,·4,·6},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,
191 ·····------------------------------------------------------------------------191 ·····------------------------------------------------------------------------
192 ·····16,·17,·19,·20,·18},·{16,·6,·3,·19,·10,·2,·14,·8,·15,·1,·9,·12,·17,·0,192 ·····15,·16,·17,·18,·19,·20},·{16,·14,·17,·18,·8,·7,·12,·11,·3,·2,·1,·5,·15,
193 ·····------------------------------------------------------------------------193 ·····------------------------------------------------------------------------
194 ·····4,·20,·13,·18,·11,·5,·7},·{13,·10,·8,·15,·4,·7,·6,·11,·12,·9,·14,·5,·2,194 ·····0,·10,·19,·13,·20,·6,·9,·4},·{4,·1,·2,·3,·0,·5,·13,·7,·8,·16,·10,·11,
195 ·····------------------------------------------------------------------------195 ·····------------------------------------------------------------------------
196 ·····16,·1,·17,·0,·3,·19,·20,·18},·{0,·1,·2,·20,·4,·7,·6,·11,·8,·9,·10,·5,196 ·····12,·6,·14,·15,·9,·17,·18,·19,·20},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,
197 ·····------------------------------------------------------------------------197 ·····------------------------------------------------------------------------
198 ·····12,·13,·14,·18,·16,·19,·15,·17,·3},·{15,·9,·0,·7,·10,·8,·14,·12,·13,·1,198 ·····10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{3,·16,·14,·4,·5,·2,·7,·8,·1,
199 ·····------------------------------------------------------------------------199 ·····------------------------------------------------------------------------
200 ·····4,·2,·16,·17,·6,·11,·3,·5,·19,·20,·18},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,200 ·····11,·0,·12,·10,·15,·13,·6,·17,·9,·18,·19,·20},·{0,·1,·3,·19,·12,·11,·2,
201 ·····------------------------------------------------------------------------201 ·····------------------------------------------------------------------------
202 ·····10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{1,·6,·3,·19,·16,·2,·0,·8,202 ·····5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{4,·16,·14,·3,·5,·2,
203 ·····------------------------------------------------------------------------203 ·····------------------------------------------------------------------------
204 ·····15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,·7},·{16,·1,·3,·19,·9,·2,·4,204 ·····7,·8,·1,·11,·0,·12,·10,·6,·13,·15,·9,·17,·18,·19,·20},·{16,·14,·17,·18,
205 ·····------------------------------------------------------------------------205 ·····------------------------------------------------------------------------
206 ·····8,·15,·6,·10,·12,·17,·0,·14,·20,·13,·18,·11,·5,·7},·{3,·6,·1,·19,·16,·2,206 ·····8,·7,·12,·11,·3,·2,·1,·5,·15,·0,·10,·19,·13,·20,·6,·9,·4},·{0,·1,·2,·3,
207 ·····------------------------------------------------------------------------207 ·····------------------------------------------------------------------------
208 ·····0,·8,·10,·13,·9,·12,·14,·15,·4,·20,·17,·18,·11,·5,·7},·{3,·1,·2,·19,·4,208 ·····4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{16,·7,
209 ·····------------------------------------------------------------------------209 ·····------------------------------------------------------------------------
210 ·····16,·6,·0,·8,·9,·10,·13,·12,·15,·14,·20,·17,·18,·11,·5,·7},·{1,·6,·3,·19,210 ·····12,·19,·9,·1,·4,·10,·2,·6,·11,·14,·8,·0,·5,·20,·13,·18,·3,·15,·17},·{2,
211 ·····------------------------------------------------------------------------211 ·····------------------------------------------------------------------------
212 ·····16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,·7},·{16,·6,·3,212 ·····0,·1,·3,·6,·5,·9,·7,·10,·4,·13,·11,·14,·8,·16,·15,·12,·17,·18,·19,·20},
213 ·····------------------------------------------------------------------------213 ·····------------------------------------------------------------------------
214 ·····19,·1,·2,·10,·8,·15,·14,·9,·12,·17,·0,·4,·20,·13,·18,·11,·5,·7},·{13,214 ·····{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,
215 ·····------------------------------------------------------------------------215 ·····------------------------------------------------------------------------
216 ·····10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,·0,·3,·19,·20,·18},216 ·····6},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,
217 ·····------------------------------------------------------------------------217 ·····------------------------------------------------------------------------
218 ·····{19,·14,·4,·16,·2,·3,·8,·15,·6,·12,·1,·17,·9,·20,·10,·0,·18,·13,·11,·5,218 ·····19,·20},·{16,·14,·17,·18,·8,·7,·12,·11,·3,·2,·1,·5,·15,·0,·10,·19,·13,
219 ·····------------------------------------------------------------------------219 ·····------------------------------------------------------------------------
220 ·····7},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,·0,·3,·19,220 ·····20,·6,·9,·4},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,
221 ·····------------------------------------------------------------------------221 ·····------------------------------------------------------------------------
222 ·····20,·18},·{16,·14,·4,·19,·2,·3,·8,·15,·6,·12,·1,·17,·9,·0,·10,·20,·13,222 ·····16,·18,·9,·4,·6},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,
223 ·····------------------------------------------------------------------------223 ·····------------------------------------------------------------------------
224 ·····18,·11,·5,·7},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,224 ·····20,·16,·18,·9,·4,·6},·{1,·7,·12,·19,·16,·4,·0,·6,·2,·13,·11,·9,·8,·10,
225 ·····------------------------------------------------------------------------225 ·····------------------------------------------------------------------------
226 ·····16,·17,·18,·19,·20}}226 ·····5,·20,·14,·18,·3,·15,·17}}
  
227 o9·:·List</code></pre>227 o9·:·List</code></pre>
228 </td>··········</tr>228 </td>··········</tr>
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Offset 32, 131 lines modifiedOffset 32, 131 lines modified
32 i4·:·varMatrix·=·gateMatrix{{t_1,t_2}};32 i4·:·varMatrix·=·gateMatrix{{t_1,t_2}};
33 i5·:·phi·=·transpose·gateMatrix{{t_1^3,·t_1^2*t_2,·t_1*t_2^2,·t_2^3}};33 i5·:·phi·=·transpose·gateMatrix{{t_1^3,·t_1^2*t_2,·t_1*t_2^2,·t_2^3}};
34 i6·:·loss·=·sum·for·i·from·0·to·3·list·(u_i·-·phi_(i,0))^2;34 i6·:·loss·=·sum·for·i·from·0·to·3·list·(u_i·-·phi_(i,0))^2;
35 i7·:·dLoss·=·diff(varMatrix,·gateMatrix{{loss}});35 i7·:·dLoss·=·diff(varMatrix,·gateMatrix{{loss}});
36 i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);36 i8·:·G·=·gateSystem(paramMatrix,varMatrix,transpose·dLoss);
37 i9·:·monodromyGroup(G,"msOptions"·=>·{NumberOfEdges=>10})37 i9·:·monodromyGroup(G,"msOptions"·=>·{NumberOfEdges=>10})
  
38 o9·=·{{1,·6,·3,·19,·16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,38 o9·=·{{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,·15,·6,·9,
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ·····7},·{7,·1,·9,·0,·12,·17,·2,·3,·4,·8,·10,·15,·6,·11,·14,·13,·5,·16,·19,40 ·····4},·{0,·9,·2,·3,·10,·5,·14,·7,·8,·1,·4,·11,·12,·13,·6,·15,·16,·17,·18,
41 ·····------------------------------------------------------------------------41 ·····------------------------------------------------------------------------
42 ·····20,·18},·{3,·1,·6,·19,·0,·2,·13,·8,·9,·16,·10,·12,·4,·15,·14,·20,·17,42 ·····19,·20},·{0,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·13,·14,·3,·16,
43 ·····------------------------------------------------------------------------43 ·····------------------------------------------------------------------------
44 ·····18,·11,·5,·7},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,44 ·····15,·6,·9,·4},·{16,·1,·7,·17,·12,·18,·2,·19,·11,·8,·10,·20,·5,·0,·14,·3,
45 ·····------------------------------------------------------------------------45 ·····------------------------------------------------------------------------
46 ·····0,·3,·19,·20,·18},·{1,·6,·3,·19,·16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,46 ·····13,·15,·6,·9,·4},·{0,·11,·2,·3,·10,·4,·14,·6,·8,·1,·5,·9,·12,·13,·7,·15,
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
48 ·····20,·14,·18,·11,·5,·7},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,48 ·····16,·17,·18,·19,·20},·{0,·1,·2,·3,·4,·11,·6,·5,·8,·9,·10,·7,·12,·13,·14,
49 ·····------------------------------------------------------------------------49 ·····------------------------------------------------------------------------
50 ·····14,·15,·16,·17,·18,·19,·20},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,50 ·····15,·16,·17,·18,·19,·20},·{12,·1,·7,·17,·16,·9,·0,·4,·11,·13,·10,·6,·5,
51 ·····------------------------------------------------------------------------51 ·····------------------------------------------------------------------------
52 ·····2,·16,·1,·17,·0,·3,·19,·20,·18},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,52 ·····2,·14,·3,·8,·15,·20,·18,·19},·{0,·9,·2,·3,·10,·5,·14,·7,·8,·1,·4,·11,
53 ·····------------------------------------------------------------------------53 ·····------------------------------------------------------------------------
54 ·····5,·2,·16,·1,·17,·0,·3,·19,·20,·18},·{10,·9,·7,·0,·12,·17,·2,·3,·11,·8,54 ·····12,·13,·6,·15,·16,·17,·18,·19,·20},·{0,·1,·2,·3,·9,·11,·4,·5,·8,·6,·10,
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ·····4,·15,·5,·14,·6,·13,·1,·16,·19,·20,·18},·{11,·10,·12,·13,·6,·20,·9,·18,56 ·····7,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{12,·1,·3,·19,·0,·11,·13,·5,·15,
57 ·····------------------------------------------------------------------------57 ·····------------------------------------------------------------------------
58 ·····2,·4,·14,·19,·8,·5,·1,·16,·7,·0,·3,·15,·17},·{13,·10,·19,·2,·6,·5,·9,·7,58 ·····16,·10,·7,·17,·2,·14,·20,·8,·18,·9,·4,·6},·{0,·1,·3,·19,·12,·11,·2,·5,
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····20,·4,·14,·11,·18,·16,·1,·8,·0,·12,·17,·3,·15},·{6,·10,·12,·13,·11,·20,60 ·····15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{12,·16,·14,·17,·4,·7,
61 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
62 ·····5,·18,·2,·7,·14,·19,·8,·9,·1,·16,·4,·0,·3,·15,·17},·{13,·10,·19,·2,·6,62 ·····6,·11,·1,·9,·0,·5,·10,·2,·13,·3,·8,·15,·20,·18,·19},·{0,·1,·3,·19,·12,
63 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
64 ·····5,·9,·7,·20,·4,·14,·11,·18,·16,·1,·8,·0,·12,·17,·3,·15},·{13,·1,·19,·2,64 ·····11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{0,·1,·3,·19,
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ·····6,·5,·9,·7,·20,·4,·10,·11,·18,·16,·14,·8,·0,·12,·17,·3,·15},·{6,·10,·7,66 ·····12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{17,·16,
67 ·····------------------------------------------------------------------------67 ·····------------------------------------------------------------------------
68 ·····13,·2,·20,·8,·18,·11,·12,·14,·19,·5,·9,·1,·16,·4,·0,·3,·15,·17},·{11,68 ·····14,·9,·8,·7,·12,·11,·1,·2,·0,·5,·10,·3,·13,·4,·15,·6,·20,·18,·19},·{12,
69 ·····------------------------------------------------------------------------69 ·····------------------------------------------------------------------------
70 ·····10,·8,·13,·6,·20,·9,·18,·12,·4,·14,·19,·2,·5,·1,·16,·7,·0,·3,·15,·17},70 ·····16,·14,·19,·4,·3,·6,·15,·1,·9,·0,·17,·10,·2,·13,·20,·8,·18,·7,·11,·5},
71 ·····------------------------------------------------------------------------71 ·····------------------------------------------------------------------------
72 ·····{10,·0,·5,·2,·6,·19,·9,·20,·7,·4,·13,·18,·11,·14,·16,·8,·1,·12,·17,·3,72 ·····{12,·16,·3,·19,·10,·11,·14,·5,·15,·1,·0,·7,·17,·2,·13,·20,·8,·18,·9,·4,
73 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
74 ·····15},·{6,·10,·12,·13,·11,·20,·5,·18,·2,·7,·14,·19,·8,·9,·1,·16,·4,·0,·3,74 ·····6},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,
75 ·····------------------------------------------------------------------------75 ·····------------------------------------------------------------------------
76 ·····15,·17},·{7,·10,·8,·13,·6,·20,·9,·18,·12,·4,·14,·19,·2,·11,·1,·16,·5,·0,76 ·····4,·6},·{12,·16,·14,·19,·4,·7,·6,·11,·1,·9,·0,·5,·10,·2,·13,·20,·8,·18,
77 ·····------------------------------------------------------------------------77 ·····------------------------------------------------------------------------
78 ·····3,·15,·17},·{0,·9,·15,·20,·10,·8,·14,·12,·17,·1,·4,·2,·3,·13,·6,·18,·16,78 ·····3,·15,·17},·{12,·16,·7,·17,·10,·9,·14,·4,·11,·1,·0,·6,·5,·2,·13,·3,·8,
79 ·····------------------------------------------------------------------------79 ·····------------------------------------------------------------------------
80 ·····19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,·14,·6,·18,80 ·····15,·20,·18,·19},·{2,·0,·9,·3,·10,·5,·14,·7,·4,·1,·13,·11,·6,·8,·16,·15,
81 ·····------------------------------------------------------------------------81 ·····------------------------------------------------------------------------
82 ·····1,·19,·5,·7,·11},·{0,·1,·6,·3,·7,·2,·11,·8,·9,·5,·10,·12,·4,·13,·14,·15,82 ·····12,·17,·18,·19,·20},·{3,·16,·12,·4,·10,·11,·14,·5,·2,·1,·0,·7,·8,·15,
83 ·····------------------------------------------------------------------------83 ·····------------------------------------------------------------------------
84 ·····16,·17,·18,·19,·20},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,·14,84 ·····13,·6,·17,·9,·18,·19,·20},·{2,·0,·9,·3,·10,·5,·14,·7,·4,·1,·13,·11,·6,
85 ·····------------------------------------------------------------------------85 ·····------------------------------------------------------------------------
86 ·····6,·18,·1,·19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,·17,·16,·4,·2,·3,86 ·····8,·16,·15,·12,·17,·18,·19,·20},·{0,·1,·2,·20,·6,·5,·9,·7,·8,·4,·10,·11,
87 ·····------------------------------------------------------------------------87 ·····------------------------------------------------------------------------
88 ·····14,·6,·18,·1,·19,·5,·7,·11},·{3,·1,·6,·5,·16,·2,·0,·8,·9,·13,·10,·12,·4,88 ·····12,·13,·14,·18,·16,·19,·15,·17,·3},·{12,·16,·7,·17,·10,·9,·14,·4,·11,·1,
89 ·····------------------------------------------------------------------------89 ·····------------------------------------------------------------------------
90 ·····15,·14,·7,·17,·11,·18,·19,·20},·{0,·1,·6,·20,·7,·15,·11,·17,·9,·5,·10,90 ·····0,·6,·5,·2,·13,·3,·8,·15,·20,·18,·19},·{0,·1,·2,·20,·6,·5,·9,·7,·8,·4,
91 ·····------------------------------------------------------------------------91 ·····------------------------------------------------------------------------
92 ·····3,·4,·13,·14,·18,·16,·19,·2,·8,·12},·{0,·1,·15,·20,·4,·8,·6,·12,·17,·9,92 ·····10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{0,·1,·2,·20,·6,·5,·9,·7,·8,
93 ·····------------------------------------------------------------------------93 ·····------------------------------------------------------------------------
94 ·····10,·2,·3,·13,·14,·18,·16,·19,·5,·7,·11},·{10,·9,·15,·20,·0,·8,·13,·12,94 ·····4,·10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{2,·0,·9,·3,·10,·5,·14,
95 ·····------------------------------------------------------------------------95 ·····------------------------------------------------------------------------
96 ·····17,·16,·4,·2,·3,·14,·6,·18,·1,·19,·5,·7,·11},·{0,·1,·6,·20,·7,·2,·11,·8,96 ·····7,·4,·1,·13,·11,·6,·8,·16,·15,·12,·17,·20,·18,·19},·{1,·16,·12,·19,·11,
97 ·····------------------------------------------------------------------------97 ·····------------------------------------------------------------------------
98 ·····9,·5,·10,·12,·4,·13,·14,·18,·16,·19,·15,·17,·3},·{13,·10,·5,·15,·4,·12,98 ·····15,·5,·17,·2,·7,·0,·3,·8,·10,·13,·20,·14,·18,·9,·4,·6},·{12,·16,·7,·17,
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ·····6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·19,·20,·18},·{15,·1,·0,·7,·4,·8,100 ·····10,·9,·14,·4,·11,·1,·0,·6,·5,·2,·13,·3,·8,·15,·18,·19,·20},·{1,·16,·12,
101 ·····------------------------------------------------------------------------101 ·····------------------------------------------------------------------------
102 ·····6,·12,·13,·9,·10,·2,·16,·17,·14,·11,·3,·5,·19,·20,·18},·{13,·10,·5,·15,102 ·····19,·11,·15,·5,·17,·2,·7,·0,·3,·8,·10,·13,·20,·14,·18,·9,·4,·6},·{2,·0,
103 ·····------------------------------------------------------------------------103 ·····------------------------------------------------------------------------
104 ·····4,·12,·6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·19,·20,·18},·{0,·1,·2,104 ·····11,·3,·10,·4,·14,·6,·5,·1,·13,·9,·7,·8,·16,·15,·12,·17,·18,·19,·20},·{0,
105 ·····------------------------------------------------------------------------105 ·····------------------------------------------------------------------------
106 ·····20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,·3},·{0,·1,106 ·····1,·2,·20,·6,·5,·9,·7,·8,·4,·10,·11,·12,·13,·14,·18,·16,·19,·15,·17,·3},
107 ·····------------------------------------------------------------------------107 ·····------------------------------------------------------------------------
108 ·····2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{0,108 ·····{3,·1,·12,·4,·0,·11,·13,·5,·2,·16,·10,·7,·8,·15,·14,·6,·17,·9,·18,·19,
109 ·····------------------------------------------------------------------------109 ·····------------------------------------------------------------------------
110 ·····1,·2,·20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,·3},110 ·····20},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,
111 ·····------------------------------------------------------------------------111 ·····------------------------------------------------------------------------
112 ·····{0,·1,·2,·20,·4,·7,·6,·11,·8,·9,·10,·5,·12,·13,·14,·18,·16,·19,·15,·17,112 ·····19,·20},·{0,·14,·12,·19,·11,·15,·5,·17,·2,·7,·1,·3,·8,·13,·10,·20,·16,
113 ·····------------------------------------------------------------------------113 ·····------------------------------------------------------------------------
114 ·····3},·{13,·10,·5,·15,·4,·12,·6,·2,·7,·9,·14,·8,·11,·16,·1,·17,·0,·3,·18,114 ·····18,·9,·4,·6},·{2,·1,·0,·3,·4,·5,·6,·7,·13,·9,·10,·11,·16,·8,·14,·15,·12,
115 ·····------------------------------------------------------------------------115 ·····------------------------------------------------------------------------
116 ·····19,·20},·{16,·6,·3,·19,·10,·2,·14,·8,·15,·1,·9,·12,·17,·0,·4,·20,·13,116 ·····17,·18,·19,·20},·{0,·1,·3,·19,·12,·5,·2,·7,·15,·8,·10,·11,·17,·13,·14,
117 ·····------------------------------------------------------------------------117 ·····------------------------------------------------------------------------
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121 ·····------------------------------------------------------------------------121 ·····------------------------------------------------------------------------
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123 ·····------------------------------------------------------------------------123 ·····------------------------------------------------------------------------
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125 ·····------------------------------------------------------------------------125 ·····------------------------------------------------------------------------
126 ·····12,·13,·14,·18,·16,·19,·15,·17,·3},·{15,·9,·0,·7,·10,·8,·14,·12,·13,·1,126 ·····10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{3,·16,·14,·4,·5,·2,·7,·8,·1,
127 ·····------------------------------------------------------------------------127 ·····------------------------------------------------------------------------
128 ·····4,·2,·16,·17,·6,·11,·3,·5,·19,·20,·18},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,128 ·····11,·0,·12,·10,·15,·13,·6,·17,·9,·18,·19,·20},·{0,·1,·3,·19,·12,·11,·2,
129 ·····------------------------------------------------------------------------129 ·····------------------------------------------------------------------------
130 ·····10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{1,·6,·3,·19,·16,·2,·0,·8,130 ·····5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,·6},·{4,·16,·14,·3,·5,·2,
131 ·····------------------------------------------------------------------------131 ·····------------------------------------------------------------------------
132 ·····15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,·7},·{16,·1,·3,·19,·9,·2,·4,132 ·····7,·8,·1,·11,·0,·12,·10,·6,·13,·15,·9,·17,·18,·19,·20},·{16,·14,·17,·18,
133 ·····------------------------------------------------------------------------133 ·····------------------------------------------------------------------------
134 ·····8,·15,·6,·10,·12,·17,·0,·14,·20,·13,·18,·11,·5,·7},·{3,·6,·1,·19,·16,·2,134 ·····8,·7,·12,·11,·3,·2,·1,·5,·15,·0,·10,·19,·13,·20,·6,·9,·4},·{0,·1,·2,·3,
135 ·····------------------------------------------------------------------------135 ·····------------------------------------------------------------------------
136 ·····0,·8,·10,·13,·9,·12,·14,·15,·4,·20,·17,·18,·11,·5,·7},·{3,·1,·2,·19,·4,136 ·····4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,·19,·20},·{16,·7,
137 ·····------------------------------------------------------------------------137 ·····------------------------------------------------------------------------
138 ·····16,·6,·0,·8,·9,·10,·13,·12,·15,·14,·20,·17,·18,·11,·5,·7},·{1,·6,·3,·19,138 ·····12,·19,·9,·1,·4,·10,·2,·6,·11,·14,·8,·0,·5,·20,·13,·18,·3,·15,·17},·{2,
139 ·····------------------------------------------------------------------------139 ·····------------------------------------------------------------------------
140 ·····16,·2,·0,·8,·15,·13,·9,·12,·17,·10,·4,·20,·14,·18,·11,·5,·7},·{16,·6,·3,140 ·····0,·1,·3,·6,·5,·9,·7,·10,·4,·13,·11,·14,·8,·16,·15,·12,·17,·18,·19,·20},
141 ·····------------------------------------------------------------------------141 ·····------------------------------------------------------------------------
142 ·····19,·1,·2,·10,·8,·15,·14,·9,·12,·17,·0,·4,·20,·13,·18,·11,·5,·7},·{13,142 ·····{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,·16,·18,·9,·4,
143 ·····------------------------------------------------------------------------143 ·····------------------------------------------------------------------------
144 ·····10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,·0,·3,·19,·20,·18},144 ·····6},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,·16,·17,·18,
145 ·····------------------------------------------------------------------------145 ·····------------------------------------------------------------------------
146 ·····{19,·14,·4,·16,·2,·3,·8,·15,·6,·12,·1,·17,·9,·20,·10,·0,·18,·13,·11,·5,146 ·····19,·20},·{16,·14,·17,·18,·8,·7,·12,·11,·3,·2,·1,·5,·15,·0,·10,·19,·13,
147 ·····------------------------------------------------------------------------147 ·····------------------------------------------------------------------------
148 ·····7},·{13,·10,·8,·15,·6,·7,·9,·11,·12,·4,·14,·5,·2,·16,·1,·17,·0,·3,·19,148 ·····20,·6,·9,·4},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,·20,
149 ·····------------------------------------------------------------------------149 ·····------------------------------------------------------------------------
150 ·····20,·18},·{16,·14,·4,·19,·2,·3,·8,·15,·6,·12,·1,·17,·9,·0,·10,·20,·13,150 ·····16,·18,·9,·4,·6},·{0,·1,·3,·19,·12,·11,·2,·5,·15,·8,·10,·7,·17,·13,·14,
151 ·····------------------------------------------------------------------------151 ·····------------------------------------------------------------------------
152 ·····18,·11,·5,·7},·{0,·1,·2,·3,·4,·5,·6,·7,·8,·9,·10,·11,·12,·13,·14,·15,152 ·····20,·16,·18,·9,·4,·6},·{1,·7,·12,·19,·16,·4,·0,·6,·2,·13,·11,·9,·8,·10,
153 ·····------------------------------------------------------------------------153 ·····------------------------------------------------------------------------
154 ·····16,·17,·18,·19,·20}}154 ·····5,·20,·14,·18,·3,·15,·17}}
  
155 o9·:·List155 o9·:·List
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./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i2·:·I·=·ideal(x,·y,·z)9 i2·:·I·=·ideal(x,·y,·z)
  
10 o2·=·ideal·(x,·y,·z)10 o2·=·ideal·(x,·y,·z)
  
11 o2·:·Ideal·of·R11 o2·:·Ideal·of·R
  
12 i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)12 i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)
13 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3994339-0/0-in.ms·-o·/tmp/M2-3994339-0/0-out.ms13 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3018370-0/0-in.ms·-o·/tmp/M2-3018370-0/0-out.ms
  
14 ---------------·INPUT·DATA·---------------14 ---------------·INPUT·DATA·---------------
15 #variables·······················315 #variables·······················3
16 #equations·······················316 #equations·······················3
17 #invalid·equations···············017 #invalid·equations···············0
18 field·characteristic·············018 field·characteristic·············0
19 homogeneous·input?···············119 homogeneous·input?···············1
Offset 45, 24 lines modifiedOffset 45, 24 lines modified
45 time(rd)··time·of·the·current·f4·round·in·seconds·given45 time(rd)··time·of·the·current·f4·round·in·seconds·given
46 ··········for·real·and·cpu·time46 ··········for·real·and·cpu·time
47 --------------------------------------------------------47 --------------------------------------------------------
  
48 deg·····sel···pairs········mat··········density············new·data·········time(rd)·in·sec·(real|cpu)48 deg·····sel···pairs········mat··········density············new·data·········time(rd)·in·sec·(real|cpu)
49 ------------------------------------------------------------------------------------------------------49 ------------------------------------------------------------------------------------------------------
50 ------------------------------------------------------------------------------------------------------50 ------------------------------------------------------------------------------------------------------
51 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero·········0.00·|·0.00·········51 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero·········0.06·|·0.13·········
52 ------------------------------------------------------------------------------------------------------52 ------------------------------------------------------------------------------------------------------
  
53 ----------------·TIMINGS·----------------53 ----------------·TIMINGS·----------------
54 overall(elapsed)········0.03·sec54 overall(elapsed)········0.14·sec
55 overall(cpu)············0.08·sec55 overall(cpu)············0.26·sec
56 select··················0.00·sec···0.0%56 select··················0.00·sec···0.0%
57 symbolic·prep.··········0.00·sec···0.0%57 symbolic·prep.··········0.00·sec···0.0%
58 update··················0.03·sec··99.7%58 update··················0.07·sec··52.7%
59 convert·················0.00·sec···0.0%59 convert·················0.06·sec··47.2%
60 linear·algebra··········0.00·sec···0.0%60 linear·algebra··········0.00·sec···0.0%
61 reduce·gb···············0.00·sec···0.0%61 reduce·gb···············0.00·sec···0.0%
62 -----------------------------------------62 -----------------------------------------
  
63 ----------·COMPUTATIONAL·DATA·-----------63 ----------·COMPUTATIONAL·DATA·-----------
64 size·of·basis·····················364 size·of·basis·····················3
65 #terms·in·basis···················365 #terms·in·basis···················3
Offset 79, 40 lines modifiedOffset 79, 40 lines modified
  
79 ----------·COMPUTATIONAL·DATA·-----------79 ----------·COMPUTATIONAL·DATA·-----------
80 [3]80 [3]
81 #polynomials·to·lift··············381 #polynomials·to·lift··············3
82 -----------------------------------------82 -----------------------------------------
  
83 ----------------·TIMINGS·----------------83 ----------------·TIMINGS·----------------
84 multi-mod·overall(elapsed)······0.08·sec84 multi-mod·overall(elapsed)······0.12·sec
85 learning·phase··················0.00·Gops/sec85 learning·phase··················0.00·Gops/sec
86 application·phase···············0.00·Gops/sec86 application·phase···············0.00·Gops/sec
87 -----------------------------------------87 -----------------------------------------
  
88 multi-modular·steps88 multi-modular·steps
89 ------------------------------------------------------------------------------------------------------89 ------------------------------------------------------------------------------------------------------
90 {1}{2}<100.00%>·90 {1}{2}<100.00%>
  
 91 ------------------------------------------------------------------------------------
 92 msolve·overall·time···········0.59·sec·(elapsed)·/··1.06·sec·(cpu)
 93 ------------------------------------------------------------------------------------
 94 ·
91 ------------------------------------------------------------------------------------------------------95 ------------------------------------------------------------------------------------------------------
  
  
92 ----------·COMPUTATIONAL·DATA·-----------96 ----------·COMPUTATIONAL·DATA·-----------
93 Max·coeff.·bitsize················197 Max·coeff.·bitsize················1
94 #primes···························398 #primes···························3
95 #bad·primes·······················099 #bad·primes·······················0
96 -----------------------------------------100 -----------------------------------------
  
97 ----------------·TIMINGS·----------------101 ----------------·TIMINGS·----------------
98 CRT·····(elapsed)···············0.00·sec102 CRT·····(elapsed)···············0.00·sec
99 ratrecon(elapsed)···············0.00·sec103 ratrecon(elapsed)···············0.00·sec
100 -----------------------------------------104 -----------------------------------------
  
  
101 ------------------------------------------------------------------------------------ 
102 msolve·overall·time···········0.15·sec·(elapsed)·/··0.52·sec·(cpu) 
103 ------------------------------------------------------------------------------------ 
  
104 o3·=·|·z·y·x·|105 o3·=·|·z·y·x·|
  
105 ·············1······3106 ·············1······3
106 o3·:·Matrix·R··<--·R107 o3·:·Matrix·R··<--·R
  
107 i4·:·108 i4·:·
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./usr/share/doc/Macaulay2/Msolve/example-output/_msolve__Real__Solutions.out
    
Offset 11, 111 lines modifiedOffset 11, 103 lines modified
11 ·············2·······211 ·············2·······2
12 o2·=·ideal·(x··-·x,·y··-·5)12 o2·=·ideal·(x··-·x,·y··-·5)
  
13 o2·:·Ideal·of·R13 o2·:·Ideal·of·R
  
14 i3·:·rationalIntervalSols·=·msolveRealSolutions·I14 i3·:·rationalIntervalSols·=·msolveRealSolutions·I
  
15 ········8589934591··8589934593······9603838835····4801919417······· 
16 o3·=·{{{----------,·----------},·{-·----------,·-·----------}},·{{- 
17 ········8589934592··8589934592······4294967296····2147483648·······15 ······························7470786705····················
 16 o3·=·{{{-·-------------------------------------------------,
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20 ·····------------------------------------------------, 
21 ·····730750818665451459101842416358141509827966271488· 
22 ·····------------------------------------------------------------------------ 
23 ·························5187595997·························9603838835··· 
24 ·····-------------------------------------------------},·{-·----------,·-20 ·····--------------------------------------------------},·{----------,
25 ·····2923003274661805836407369665432566039311865085952······4294967296···21 ·····11692013098647223345629478661730264157247460343808····2147483648·
26 ·····------------------------------------------------------------------------22 ·····------------------------------------------------------------------------
27 ·····4801919417······8589934591··8589934593····4801919417··9603838835·······23 ·····9603838835······8589934591··8589934593····4801919417··9603838835·······
28 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-24 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-
29 ·····2147483648······8589934592··8589934592····2147483648··4294967296·······25 ·····4294967296······8589934592··8589934592····2147483648··4294967296·······
30 ·····------------------------------------------------------------------------26 ·····------------------------------------------------------------------------
31 ·························20429248941·····················27 ························6967040737····················
32 ·····---------------------------------------------------,28 ·····------------------------------------------------,
 29 ·····730750818665451459101842416358141509827966271488·
33 ·····187072209578355573530071658587684226515959365500928· 
34 ·····------------------------------------------------------------------------ 
35 ·························3386105301························4801919417· 
36 ·····--------------------------------------------------},·{----------, 
37 ·····93536104789177786765035829293842113257979682750464····2147483648· 
38 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
39 ·····9603838835 
40 ·····----------}}} 
41 ·····429496729631 ························4182019449·························9603838835···
 32 ·····------------------------------------------------},·{-·----------,·-
 33 ·····730750818665451459101842416358141509827966271488······4294967296···
 34 ·····------------------------------------------------------------------------
 35 ·····4801919417······8589934591··8589934593······9603838835····4801919417
 36 ·····----------}},·{{----------,·----------},·{-·----------,·-·----------}}}
 37 ·····2147483648······8589934592··8589934592······4294967296····2147483648
  
42 o3·:·List38 o3·:·List
  
43 i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)39 i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)
  
44 ············19207677669····· 
45 o4·=·{{1,·-·-----------},·{- 
46 ·············8589934592·····40 ···························290319759······················19207677669······
 41 o4·=·{{-------------------------------------------------,·-----------},·{1,
 42 ·······5846006549323611672814739330865132078623730171904···8589934592······
47 ·····------------------------------------------------------------------------43 ·····------------------------------------------------------------------------
48 ·························3381059991·······················19207677669······44 ·····19207677669·································24766267····················
49 ·····-------------------------------------------------,·-·-----------},·{1,45 ·····-----------},·{---------------------------------------------------------
50 ·····5846006549323611672814739330865132078623730171904·····8589934592······46 ······8589934592····421249166674228746791672110734681729275580381602196445017
51 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
 48 ··················19207677669·········19207677669
 49 ·····---------,·-·-----------},·{1,·-·-----------}}
 50 ·····243910144·····8589934592··········8589934592
52 ·····19207677669··························13657038339····················· 
53 ·····-----------},·{-·---------------------------------------------------, 
54 ······8589934592······374144419156711147060143317175368453031918731001856· 
55 ·····------------------------------------------------------------------------ 
56 ·····19207677669 
57 ·····-----------}} 
58 ······8589934592 
  
59 o4·:·List51 o4·:·List
  
60 i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)52 i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)
  
61 o5·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.93146e-39,1.77475e-39],53 o5·=·{{[-2.08337e-46,3.25491e-46],·[-2.23607,-2.23607]},·{[1,1],
62 ·····------------------------------------------------------------------------54 ·····------------------------------------------------------------------------
63 ·····[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},55 ·····[-2.23607,-2.23607]},·{[-6.75045e-58,4.57948e-58],·[2.23607,2.23607]},
64 ·····------------------------------------------------------------------------56 ·····------------------------------------------------------------------------
65 ·····{[-1.09205e-40,3.62011e-41],·[2.23607,2.23607]}}57 ·····{[1,1],·[2.23607,2.23607]}}
  
66 o5·:·List58 o5·:·List
  
67 i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)59 i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)
  
68 o6·=·{{[.999512,1.00049],·[-2.23633,-2.23535]},·{[-2.93156e-39,1.77501e-39],60 o6·=·{{[-1.69322e-39,3.45029e-39],·[-2.23633,-2.23535]},·{[.999512,1.00049],
69 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
70 ·····[-2.23633,-2.23535]},·{[.999512,1.00049],·[2.23535,2.23633]},62 ·····[-2.23633,-2.23535]},·{[-3.58732e-39,1.95581e-39],·[2.23535,2.23633]},
71 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
72 ·····{[-1.09212e-40,3.62096e-41],·[2.23535,2.23633]}}64 ·····{[.999512,1.00049],·[2.23535,2.23633]}}
  
73 o6·:·List65 o6·:·List
  
74 i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)66 i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)
  
75 o7·=·{{1,·2.23607},·{5.03732e-40,·2.23607},·{1,·-2.23607},·{-1.27767e-39,67 o7·=·{{7.14874e-42,·-2.23607},·{1,·-2.23607},·{-1.31296e-58,·2.23607},·{1,
76 ·····------------------------------------------------------------------------68 ·····------------------------------------------------------------------------
77 ·····-2.23607}}69 ·····2.23607}}
  
78 o7·:·List70 o7·:·List
  
79 i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)71 i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)
  
80 o8·=·{{1,·2.23584},·{5.0366e-40,·2.23584},·{1,·-2.23584},·{-1.2778e-39,72 o8·=·{{3.14339e-40,·-2.23584},·{1,·-2.23584},·{-3.58732e-41,·2.23584},·{1,
81 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
82 ·····-2.23584}}74 ·····2.23584}}
  
83 o8·:·List75 o8·:·List
  
84 i9·:·I·=·ideal·{(x-1)*x^3,·(y^2-5)^2}76 i9·:·I·=·ideal·{(x-1)*x^3,·(y^2-5)^2}
  
85 ·············4····3···4······277 ·············4····3···4······2
86 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)78 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)
  
87 o9·:·Ideal·of·R79 o9·:·Ideal·of·R
  
88 i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)80 i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)
  
89 o10·=·{{[-1.88775e-41,8.05504e-42],·[2.23607,2.23607]},·{[1,1],81 o10·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.82543e-39,4.39933e-39],
90 ······-----------------------------------------------------------------------82 ······-----------------------------------------------------------------------
91 ······[2.23607,2.23607]},·{[-8.32989e-58,9.7995e-58],·[-2.23607,-2.23607]},83 ······[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},
92 ······-----------------------------------------------------------------------84 ······-----------------------------------------------------------------------
93 ······{[1,1],·[-2.23607,-2.23607]}}85 ······{[-1.07007e-42,1.18482e-42],·[2.23607,2.23607]}}
  
94 o10·:·List86 o10·:·List
  
95 i11·:·87 i11·:·
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./usr/share/doc/Macaulay2/Msolve/html/_msolve__Real__Solutions.html
    
Offset 99, 102 lines modifiedOffset 99, 94 lines modified
99 o2·=·ideal·(x··-·x,·y··-·5)99 o2·=·ideal·(x··-·x,·y··-·5)
  
100 o2·:·Ideal·of·R</code></pre>100 o2·:·Ideal·of·R</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i3·:·rationalIntervalSols·=·msolveRealSolutions·I103 <td>··············<pre><code·class="language-macaulay2">i3·:·rationalIntervalSols·=·msolveRealSolutions·I
  
104 ········8589934591··8589934593······9603838835····4801919417······· 
105 o3·=·{{{----------,·----------},·{-·----------,·-·----------}},·{{- 
106 ········8589934592··8589934592······4294967296····2147483648·······104 ······························7470786705····················
 105 o3·=·{{{-·-------------------------------------------------,
 106 ··········5846006549323611672814739330865132078623730171904·
107 ·····------------------------------------------------------------------------107 ·····------------------------------------------------------------------------
108 ························2142163997····················108 ·························6533263071························4801919417·
109 ·····------------------------------------------------, 
110 ·····730750818665451459101842416358141509827966271488· 
111 ·····------------------------------------------------------------------------ 
112 ·························5187595997·························9603838835··· 
113 ·····-------------------------------------------------},·{-·----------,·-109 ·····--------------------------------------------------},·{----------,
114 ·····2923003274661805836407369665432566039311865085952······4294967296···110 ·····11692013098647223345629478661730264157247460343808····2147483648·
115 ·····------------------------------------------------------------------------111 ·····------------------------------------------------------------------------
116 ·····4801919417······8589934591··8589934593····4801919417··9603838835·······112 ·····9603838835······8589934591··8589934593····4801919417··9603838835·······
117 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-113 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-
118 ·····2147483648······8589934592··8589934592····2147483648··4294967296·······114 ·····4294967296······8589934592··8589934592····2147483648··4294967296·······
119 ·····------------------------------------------------------------------------115 ·····------------------------------------------------------------------------
120 ·························20429248941·····················116 ························6967040737····················
121 ·····---------------------------------------------------,117 ·····------------------------------------------------,
 118 ·····730750818665451459101842416358141509827966271488·
122 ·····187072209578355573530071658587684226515959365500928· 
123 ·····------------------------------------------------------------------------ 
124 ·························3386105301························4801919417· 
125 ·····--------------------------------------------------},·{----------, 
126 ·····93536104789177786765035829293842113257979682750464····2147483648· 
127 ·····------------------------------------------------------------------------119 ·····------------------------------------------------------------------------
128 ·····9603838835 
129 ·····----------}}} 
130 ·····4294967296120 ························4182019449·························9603838835···
 121 ·····------------------------------------------------},·{-·----------,·-
 122 ·····730750818665451459101842416358141509827966271488······4294967296···
 123 ·····------------------------------------------------------------------------
 124 ·····4801919417······8589934591··8589934593······9603838835····4801919417
 125 ·····----------}},·{{----------,·----------},·{-·----------,·-·----------}}}
 126 ·····2147483648······8589934592··8589934592······4294967296····2147483648
  
131 o3·:·List</code></pre>127 o3·:·List</code></pre>
132 </td>··········</tr>128 </td>··········</tr>
133 ··········<tr>129 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)130 <td>··············<pre><code·class="language-macaulay2">i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)
  
135 ············19207677669····· 
136 o4·=·{{1,·-·-----------},·{- 
137 ·············8589934592·····131 ···························290319759······················19207677669······
 132 o4·=·{{-------------------------------------------------,·-----------},·{1,
 133 ·······5846006549323611672814739330865132078623730171904···8589934592······
138 ·····------------------------------------------------------------------------134 ·····------------------------------------------------------------------------
139 ·························3381059991·······················19207677669······135 ·····19207677669·································24766267····················
140 ·····-------------------------------------------------,·-·-----------},·{1,136 ·····-----------},·{---------------------------------------------------------
141 ·····5846006549323611672814739330865132078623730171904·····8589934592······137 ······8589934592····421249166674228746791672110734681729275580381602196445017
142 ·····------------------------------------------------------------------------138 ·····------------------------------------------------------------------------
 139 ··················19207677669·········19207677669
 140 ·····---------,·-·-----------},·{1,·-·-----------}}
 141 ·····243910144·····8589934592··········8589934592
143 ·····19207677669··························13657038339····················· 
144 ·····-----------},·{-·---------------------------------------------------, 
145 ······8589934592······374144419156711147060143317175368453031918731001856· 
146 ·····------------------------------------------------------------------------ 
147 ·····19207677669 
148 ·····-----------}} 
149 ······8589934592 
  
150 o4·:·List</code></pre>142 o4·:·List</code></pre>
151 </td>··········</tr>143 </td>··········</tr>
152 ··········<tr>144 ··········<tr>
153 <td>··············<pre><code·class="language-macaulay2">i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)145 <td>··············<pre><code·class="language-macaulay2">i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)
  
154 o5·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.93146e-39,1.77475e-39],146 o5·=·{{[-2.08337e-46,3.25491e-46],·[-2.23607,-2.23607]},·{[1,1],
155 ·····------------------------------------------------------------------------147 ·····------------------------------------------------------------------------
156 ·····[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},148 ·····[-2.23607,-2.23607]},·{[-6.75045e-58,4.57948e-58],·[2.23607,2.23607]},
157 ·····------------------------------------------------------------------------149 ·····------------------------------------------------------------------------
158 ·····{[-1.09205e-40,3.62011e-41],·[2.23607,2.23607]}}150 ·····{[1,1],·[2.23607,2.23607]}}
  
159 o5·:·List</code></pre>151 o5·:·List</code></pre>
160 </td>··········</tr>152 </td>··········</tr>
161 ··········<tr>153 ··········<tr>
162 <td>··············<pre><code·class="language-macaulay2">i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)154 <td>··············<pre><code·class="language-macaulay2">i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)
  
163 o6·=·{{[.999512,1.00049],·[-2.23633,-2.23535]},·{[-2.93156e-39,1.77501e-39],155 o6·=·{{[-1.69322e-39,3.45029e-39],·[-2.23633,-2.23535]},·{[.999512,1.00049],
164 ·····------------------------------------------------------------------------156 ·····------------------------------------------------------------------------
165 ·····[-2.23633,-2.23535]},·{[.999512,1.00049],·[2.23535,2.23633]},157 ·····[-2.23633,-2.23535]},·{[-3.58732e-39,1.95581e-39],·[2.23535,2.23633]},
166 ·····------------------------------------------------------------------------158 ·····------------------------------------------------------------------------
167 ·····{[-1.09212e-40,3.62096e-41],·[2.23535,2.23633]}}159 ·····{[.999512,1.00049],·[2.23535,2.23633]}}
  
168 o6·:·List</code></pre>160 o6·:·List</code></pre>
169 </td>··········</tr>161 </td>··········</tr>
170 ··········<tr>162 ··········<tr>
171 <td>··············<pre><code·class="language-macaulay2">i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)163 <td>··············<pre><code·class="language-macaulay2">i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)
  
172 o7·=·{{1,·2.23607},·{5.03732e-40,·2.23607},·{1,·-2.23607},·{-1.27767e-39,164 o7·=·{{7.14874e-42,·-2.23607},·{1,·-2.23607},·{-1.31296e-58,·2.23607},·{1,
173 ·····------------------------------------------------------------------------165 ·····------------------------------------------------------------------------
174 ·····-2.23607}}166 ·····2.23607}}
  
175 o7·:·List</code></pre>167 o7·:·List</code></pre>
176 </td>··········</tr>168 </td>··········</tr>
177 ··········<tr>169 ··········<tr>
178 <td>··············<pre><code·class="language-macaulay2">i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)170 <td>··············<pre><code·class="language-macaulay2">i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)
  
179 o8·=·{{1,·2.23584},·{5.0366e-40,·2.23584},·{1,·-2.23584},·{-1.2778e-39,171 o8·=·{{3.14339e-40,·-2.23584},·{1,·-2.23584},·{-3.58732e-41,·2.23584},·{1,
180 ·····------------------------------------------------------------------------172 ·····------------------------------------------------------------------------
181 ·····-2.23584}}173 ·····2.23584}}
  
182 o8·:·List</code></pre>174 o8·:·List</code></pre>
183 </td>··········</tr>175 </td>··········</tr>
184 ········</table>176 ········</table>
185 ········<div>177 ········<div>
186 ··········<p>Note·in·cases·where·solutions·have·multiplicity·this·is·not·reflected·in·the·output.·While·the·solver·does·not·return·multiplicities,·it·reliably·outputs·the·verified·isolating·intervals·for·multiple·solutions.</p>178 ··········<p>Note·in·cases·where·solutions·have·multiplicity·this·is·not·reflected·in·the·output.·While·the·solver·does·not·return·multiplicities,·it·reliably·outputs·the·verified·isolating·intervals·for·multiple·solutions.</p>
187 ········</div>179 ········</div>
Offset 206, 19 lines modifiedOffset 198, 19 lines modified
206 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)198 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)
  
207 o9·:·Ideal·of·R</code></pre>199 o9·:·Ideal·of·R</code></pre>
208 </td>··········</tr>200 </td>··········</tr>
209 ··········<tr>201 ··········<tr>
210 <td>··············<pre><code·class="language-macaulay2">i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)202 <td>··············<pre><code·class="language-macaulay2">i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)
  
211 o10·=·{{[-1.88775e-41,8.05504e-42],·[2.23607,2.23607]},·{[1,1],203 o10·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.82543e-39,4.39933e-39],
212 ······-----------------------------------------------------------------------204 ······-----------------------------------------------------------------------
213 ······[2.23607,2.23607]},·{[-8.32989e-58,9.7995e-58],·[-2.23607,-2.23607]},205 ······[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},
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Offset 44, 110 lines modifiedOffset 44, 102 lines modified
  
44 ·············2·······244 ·············2·······2
45 o2·=·ideal·(x··-·x,·y··-·5)45 o2·=·ideal·(x··-·x,·y··-·5)
  
46 o2·:·Ideal·of·R46 o2·:·Ideal·of·R
47 i3·:·rationalIntervalSols·=·msolveRealSolutions·I47 i3·:·rationalIntervalSols·=·msolveRealSolutions·I
  
48 ········8589934591··8589934593······9603838835····4801919417 
49 o3·=·{{{----------,·----------},·{-·----------,·-·----------}},·{{- 
50 ········8589934592··8589934592······4294967296····214748364848 ······························7470786705
 49 o3·=·{{{-·-------------------------------------------------,
 50 ··········5846006549323611672814739330865132078623730171904
51 ·····------------------------------------------------------------------------51 ·····------------------------------------------------------------------------
 52 ·························6533263071························4801919417
52 ························2142163997 
53 ·····------------------------------------------------, 
54 ·····730750818665451459101842416358141509827966271488 
55 ·····------------------------------------------------------------------------ 
56 ·························5187595997·························9603838835 
57 ·····-------------------------------------------------},·{-·----------,·-53 ·····--------------------------------------------------},·{----------,
58 ·····2923003274661805836407369665432566039311865085952······429496729654 ·····11692013098647223345629478661730264157247460343808····2147483648
59 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
60 ·····4801919417······8589934591··8589934593····4801919417··960383883556 ·····9603838835······8589934591··8589934593····4801919417··9603838835
61 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-57 ·····----------}},·{{----------,·----------},·{----------,·----------}},·{{-
62 ·····2147483648······8589934592··8589934592····2147483648··429496729658 ·····4294967296······8589934592··8589934592····2147483648··4294967296
63 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
64 ·························2042924894160 ························6967040737
65 ·····---------------------------------------------------,61 ·····------------------------------------------------,
 62 ·····730750818665451459101842416358141509827966271488
66 ·····187072209578355573530071658587684226515959365500928 
67 ·····------------------------------------------------------------------------ 
68 ·························3386105301························4801919417 
69 ·····--------------------------------------------------},·{----------, 
70 ·····93536104789177786765035829293842113257979682750464····2147483648 
71 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
72 ·····9603838835 
73 ·····----------}}} 
74 ·····429496729664 ························4182019449·························9603838835
 65 ·····------------------------------------------------},·{-·----------,·-
 66 ·····730750818665451459101842416358141509827966271488······4294967296
 67 ·····------------------------------------------------------------------------
 68 ·····4801919417······8589934591··8589934593······9603838835····4801919417
 69 ·····----------}},·{{----------,·----------},·{-·----------,·-·----------}}}
 70 ·····2147483648······8589934592··8589934592······4294967296····2147483648
  
75 o3·:·List71 o3·:·List
76 i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)72 i4·:·rationalApproxSols·=·msolveRealSolutions(I,·QQ)
  
77 ············19207677669 
78 o4·=·{{1,·-·-----------},·{- 
79 ·············858993459273 ···························290319759······················19207677669
 74 o4·=·{{-------------------------------------------------,·-----------},·{1,
 75 ·······5846006549323611672814739330865132078623730171904···8589934592
80 ·····------------------------------------------------------------------------76 ·····------------------------------------------------------------------------
81 ·························3381059991·······················1920767766977 ·····19207677669·································24766267
82 ·····-------------------------------------------------,·-·-----------},·{1,78 ·····-----------},·{---------------------------------------------------------
83 ·····5846006549323611672814739330865132078623730171904·····858993459279 ······8589934592····421249166674228746791672110734681729275580381602196445017
84 ·····------------------------------------------------------------------------80 ·····------------------------------------------------------------------------
 81 ··················19207677669·········19207677669
 82 ·····---------,·-·-----------},·{1,·-·-----------}}
 83 ·····243910144·····8589934592··········8589934592
85 ·····19207677669··························13657038339 
86 ·····-----------},·{-·---------------------------------------------------, 
87 ······8589934592······374144419156711147060143317175368453031918731001856 
88 ·····------------------------------------------------------------------------ 
89 ·····19207677669 
90 ·····-----------}} 
91 ······8589934592 
  
92 o4·:·List84 o4·:·List
93 i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)85 i5·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi)
  
94 o5·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.93146e-39,1.77475e-39],86 o5·=·{{[-2.08337e-46,3.25491e-46],·[-2.23607,-2.23607]},·{[1,1],
95 ·····------------------------------------------------------------------------87 ·····------------------------------------------------------------------------
96 ·····[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},88 ·····[-2.23607,-2.23607]},·{[-6.75045e-58,4.57948e-58],·[2.23607,2.23607]},
97 ·····------------------------------------------------------------------------89 ·····------------------------------------------------------------------------
98 ·····{[-1.09205e-40,3.62011e-41],·[2.23607,2.23607]}}90 ·····{[1,1],·[2.23607,2.23607]}}
  
99 o5·:·List91 o5·:·List
100 i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)92 i6·:·floatIntervalSols·=·msolveRealSolutions(I,·RRi_10)
  
101 o6·=·{{[.999512,1.00049],·[-2.23633,-2.23535]},·{[-2.93156e-39,1.77501e-39],93 o6·=·{{[-1.69322e-39,3.45029e-39],·[-2.23633,-2.23535]},·{[.999512,1.00049],
102 ·····------------------------------------------------------------------------94 ·····------------------------------------------------------------------------
103 ·····[-2.23633,-2.23535]},·{[.999512,1.00049],·[2.23535,2.23633]},95 ·····[-2.23633,-2.23535]},·{[-3.58732e-39,1.95581e-39],·[2.23535,2.23633]},
104 ·····------------------------------------------------------------------------96 ·····------------------------------------------------------------------------
105 ·····{[-1.09212e-40,3.62096e-41],·[2.23535,2.23633]}}97 ·····{[.999512,1.00049],·[2.23535,2.23633]}}
  
106 o6·:·List98 o6·:·List
107 i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)99 i7·:·floatApproxSols·=·msolveRealSolutions(I,·RR)
  
108 o7·=·{{1,·2.23607},·{5.03732e-40,·2.23607},·{1,·-2.23607},·{-1.27767e-39,100 o7·=·{{7.14874e-42,·-2.23607},·{1,·-2.23607},·{-1.31296e-58,·2.23607},·{1,
109 ·····------------------------------------------------------------------------101 ·····------------------------------------------------------------------------
110 ·····-2.23607}}102 ·····2.23607}}
  
111 o7·:·List103 o7·:·List
112 i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)104 i8·:·floatApproxSols·=·msolveRealSolutions(I,·RR_10)
  
113 o8·=·{{1,·2.23584},·{5.0366e-40,·2.23584},·{1,·-2.23584},·{-1.2778e-39,105 o8·=·{{3.14339e-40,·-2.23584},·{1,·-2.23584},·{-3.58732e-41,·2.23584},·{1,
114 ·····------------------------------------------------------------------------106 ·····------------------------------------------------------------------------
115 ·····-2.23584}}107 ·····2.23584}}
  
116 o8·:·List108 o8·:·List
117 Note·in·cases·where·solutions·have·multiplicity·this·is·not·reflected·in·the109 Note·in·cases·where·solutions·have·multiplicity·this·is·not·reflected·in·the
118 output.·While·the·solver·does·not·return·multiplicities,·it·reliably·outputs110 output.·While·the·solver·does·not·return·multiplicities,·it·reliably·outputs
119 the·verified·isolating·intervals·for·multiple·solutions.111 the·verified·isolating·intervals·for·multiple·solutions.
120 i9·:·I·=·ideal·{(x-1)*x^3,·(y^2-5)^2}112 i9·:·I·=·ideal·{(x-1)*x^3,·(y^2-5)^2}
  
121 ·············4····3···4······2113 ·············4····3···4······2
122 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)114 o9·=·ideal·(x··-·x·,·y··-·10y··+·25)
  
123 o9·:·Ideal·of·R115 o9·:·Ideal·of·R
124 i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)116 i10·:·floatApproxSols·=·msolveRealSolutions(I,·RRi)
  
125 o10·=·{{[-1.88775e-41,8.05504e-42],·[2.23607,2.23607]},·{[1,1],117 o10·=·{{[1,1],·[-2.23607,-2.23607]},·{[-2.82543e-39,4.39933e-39],
126 ······-----------------------------------------------------------------------118 ······-----------------------------------------------------------------------
127 ······[2.23607,2.23607]},·{[-8.32989e-58,9.7995e-58],·[-2.23607,-2.23607]},119 ······[-2.23607,-2.23607]},·{[1,1],·[2.23607,2.23607]},
128 ······-----------------------------------------------------------------------120 ······-----------------------------------------------------------------------
129 ······{[1,1],·[-2.23607,-2.23607]}}121 ······{[-1.07007e-42,1.18482e-42],·[2.23607,2.23607]}}
  
130 o10·:·List122 o10·:·List
131 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8ms\x8so\x8ol\x8lv\x8ve\x8eR\x8Re\x8ea\x8al\x8lS\x8So\x8ol\x8lu\x8ut\x8ti\x8io\x8on\x8ns\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*123 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8ms\x8so\x8ol\x8lv\x8ve\x8eR\x8Re\x8ea\x8al\x8lS\x8So\x8ol\x8lu\x8ut\x8ti\x8io\x8on\x8ns\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
132 ····*·msolveRealSolutions(Ideal)124 ····*·msolveRealSolutions(Ideal)
133 ····*·msolveRealSolutions(Ideal,Ring)125 ····*·msolveRealSolutions(Ideal,Ring)
134 ····*·msolveRealSolutions(Ideal,RingFamily)126 ····*·msolveRealSolutions(Ideal,RingFamily)
135 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*127 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/Msolve/html/index.html
    
Offset 66, 15 lines modifiedOffset 66, 15 lines modified
  
66 o2·=·ideal·(x,·y,·z)66 o2·=·ideal·(x,·y,·z)
  
67 o2·:·Ideal·of·R</code></pre>67 o2·:·Ideal·of·R</code></pre>
68 </td>··········</tr>68 </td>··········</tr>
69 ··········<tr>69 ··········<tr>
70 <td>··············<pre><code·class="language-macaulay2">i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)70 <td>··············<pre><code·class="language-macaulay2">i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)
71 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3994339-0/0-in.ms·-o·/tmp/M2-3994339-0/0-out.ms71 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3018370-0/0-in.ms·-o·/tmp/M2-3018370-0/0-out.ms
  
72 ---------------·INPUT·DATA·---------------72 ---------------·INPUT·DATA·---------------
73 #variables·······················373 #variables·······················3
74 #equations·······················374 #equations·······················3
75 #invalid·equations···············075 #invalid·equations···············0
76 field·characteristic·············076 field·characteristic·············0
77 homogeneous·input?···············177 homogeneous·input?···············1
Offset 102, 24 lines modifiedOffset 102, 24 lines modified
102 time(rd)··time·of·the·current·f4·round·in·seconds·given102 time(rd)··time·of·the·current·f4·round·in·seconds·given
103 ··········for·real·and·cpu·time103 ··········for·real·and·cpu·time
104 --------------------------------------------------------104 --------------------------------------------------------
  
105 deg·····sel···pairs········mat··········density············new·data·········time(rd)·in·sec·(real|cpu)105 deg·····sel···pairs········mat··········density············new·data·········time(rd)·in·sec·(real|cpu)
106 ------------------------------------------------------------------------------------------------------106 ------------------------------------------------------------------------------------------------------
107 ------------------------------------------------------------------------------------------------------107 ------------------------------------------------------------------------------------------------------
108 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero·········0.00·|·0.00·········108 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero·········0.06·|·0.13·········
109 ------------------------------------------------------------------------------------------------------109 ------------------------------------------------------------------------------------------------------
  
110 ----------------·TIMINGS·----------------110 ----------------·TIMINGS·----------------
111 overall(elapsed)········0.03·sec111 overall(elapsed)········0.14·sec
112 overall(cpu)············0.08·sec112 overall(cpu)············0.26·sec
113 select··················0.00·sec···0.0%113 select··················0.00·sec···0.0%
114 symbolic·prep.··········0.00·sec···0.0%114 symbolic·prep.··········0.00·sec···0.0%
115 update··················0.03·sec··99.7%115 update··················0.07·sec··52.7%
116 convert·················0.00·sec···0.0%116 convert·················0.06·sec··47.2%
117 linear·algebra··········0.00·sec···0.0%117 linear·algebra··········0.00·sec···0.0%
118 reduce·gb···············0.00·sec···0.0%118 reduce·gb···············0.00·sec···0.0%
119 -----------------------------------------119 -----------------------------------------
  
120 ----------·COMPUTATIONAL·DATA·-----------120 ----------·COMPUTATIONAL·DATA·-----------
121 size·of·basis·····················3121 size·of·basis·····················3
122 #terms·in·basis···················3122 #terms·in·basis···················3
Offset 136, 41 lines modifiedOffset 136, 41 lines modified
  
136 ----------·COMPUTATIONAL·DATA·-----------136 ----------·COMPUTATIONAL·DATA·-----------
137 [3]137 [3]
138 #polynomials·to·lift··············3138 #polynomials·to·lift··············3
139 -----------------------------------------139 -----------------------------------------
  
140 ----------------·TIMINGS·----------------140 ----------------·TIMINGS·----------------
141 multi-mod·overall(elapsed)······0.08·sec141 multi-mod·overall(elapsed)······0.12·sec
142 learning·phase··················0.00·Gops/sec142 learning·phase··················0.00·Gops/sec
143 application·phase···············0.00·Gops/sec143 application·phase···············0.00·Gops/sec
144 -----------------------------------------144 -----------------------------------------
  
145 multi-modular·steps145 multi-modular·steps
146 ------------------------------------------------------------------------------------------------------146 ------------------------------------------------------------------------------------------------------
147 {1}{2}&lt;100.00%>·147 {1}{2}&lt;100.00%>
  
 148 ------------------------------------------------------------------------------------
 149 msolve·overall·time···········0.59·sec·(elapsed)·/··1.06·sec·(cpu)
 150 ------------------------------------------------------------------------------------
 151 ·
148 ------------------------------------------------------------------------------------------------------152 ------------------------------------------------------------------------------------------------------
  
  
149 ----------·COMPUTATIONAL·DATA·-----------153 ----------·COMPUTATIONAL·DATA·-----------
150 Max·coeff.·bitsize················1154 Max·coeff.·bitsize················1
151 #primes···························3155 #primes···························3
152 #bad·primes·······················0156 #bad·primes·······················0
153 -----------------------------------------157 -----------------------------------------
  
154 ----------------·TIMINGS·----------------158 ----------------·TIMINGS·----------------
155 CRT·····(elapsed)···············0.00·sec159 CRT·····(elapsed)···············0.00·sec
156 ratrecon(elapsed)···············0.00·sec160 ratrecon(elapsed)···············0.00·sec
157 -----------------------------------------161 -----------------------------------------
  
  
158 ------------------------------------------------------------------------------------ 
159 msolve·overall·time···········0.15·sec·(elapsed)·/··0.52·sec·(cpu) 
160 ------------------------------------------------------------------------------------ 
  
161 o3·=·|·z·y·x·|162 o3·=·|·z·y·x·|
  
162 ·············1······3163 ·············1······3
163 o3·:·Matrix·R··&lt;--·R</code></pre>164 o3·:·Matrix·R··&lt;--·R</code></pre>
164 </td>··········</tr>165 </td>··········</tr>
165 ········</table>166 ········</table>
166 ······</div>167 ······</div>
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Offset 31, 16 lines modifiedOffset 31, 16 lines modified
31 o1·:·PolynomialRing31 o1·:·PolynomialRing
32 i2·:·I·=·ideal(x,·y,·z)32 i2·:·I·=·ideal(x,·y,·z)
  
33 o2·=·ideal·(x,·y,·z)33 o2·=·ideal·(x,·y,·z)
  
34 o2·:·Ideal·of·R34 o2·:·Ideal·of·R
35 i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)35 i3·:·msolveGB(I,·Verbosity·=>·2,·Threads·=>·6)
36 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3994339-0/0-in.ms·-o·/36 ·--·running:·/usr/bin/msolve·-g·2·-t·6·-v·2·-f·/tmp/M2-3018370-0/0-in.ms·-o·/
37 tmp/M2-3994339-0/0-out.ms37 tmp/M2-3018370-0/0-out.ms
  
38 ---------------·INPUT·DATA·---------------38 ---------------·INPUT·DATA·---------------
39 #variables·······················339 #variables·······················3
40 #equations·······················340 #equations·······················3
41 #invalid·equations···············041 #invalid·equations···············0
42 field·characteristic·············042 field·characteristic·············0
43 homogeneous·input?···············143 homogeneous·input?···············1
Offset 72, 25 lines modifiedOffset 72, 25 lines modified
72 deg·····sel···pairs········mat··········density············new·data72 deg·····sel···pairs········mat··········density············new·data
73 time(rd)·in·sec·(real|cpu)73 time(rd)·in·sec·(real|cpu)
74 -------------------------------------------------------------------------------74 -------------------------------------------------------------------------------
75 -----------------------75 -----------------------
76 -------------------------------------------------------------------------------76 -------------------------------------------------------------------------------
77 -----------------------77 -----------------------
78 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero78 reduce·final·basis········3·x·3··········33.33%········3·new·······0·zero
79 0.00·|·0.0079 0.06·|·0.13
80 -------------------------------------------------------------------------------80 -------------------------------------------------------------------------------
81 -----------------------81 -----------------------
  
82 ----------------·TIMINGS·----------------82 ----------------·TIMINGS·----------------
83 overall(elapsed)········0.03·sec83 overall(elapsed)········0.14·sec
84 overall(cpu)············0.08·sec84 overall(cpu)············0.26·sec
85 select··················0.00·sec···0.0%85 select··················0.00·sec···0.0%
86 symbolic·prep.··········0.00·sec···0.0%86 symbolic·prep.··········0.00·sec···0.0%
87 update··················0.03·sec··99.7%87 update··················0.07·sec··52.7%
88 convert·················0.00·sec···0.0%88 convert·················0.06·sec··47.2%
89 linear·algebra··········0.00·sec···0.0%89 linear·algebra··········0.00·sec···0.0%
90 reduce·gb···············0.00·sec···0.0%90 reduce·gb···············0.00·sec···0.0%
91 -----------------------------------------91 -----------------------------------------
  
92 ----------·COMPUTATIONAL·DATA·-----------92 ----------·COMPUTATIONAL·DATA·-----------
93 size·of·basis·····················393 size·of·basis·····················3
94 #terms·in·basis···················394 #terms·in·basis···················3
Offset 107, 23 lines modifiedOffset 107, 30 lines modified
  
107 ----------·COMPUTATIONAL·DATA·-----------107 ----------·COMPUTATIONAL·DATA·-----------
108 [3]108 [3]
109 #polynomials·to·lift··············3109 #polynomials·to·lift··············3
110 -----------------------------------------110 -----------------------------------------
  
111 ----------------·TIMINGS·----------------111 ----------------·TIMINGS·----------------
112 multi-mod·overall(elapsed)······0.08·sec112 multi-mod·overall(elapsed)······0.12·sec
113 learning·phase··················0.00·Gops/sec113 learning·phase··················0.00·Gops/sec
114 application·phase···············0.00·Gops/sec114 application·phase···············0.00·Gops/sec
115 -----------------------------------------115 -----------------------------------------
  
116 multi-modular·steps116 multi-modular·steps
117 -------------------------------------------------------------------------------117 -------------------------------------------------------------------------------
118 -----------------------118 -----------------------
119 {1}{2}<100.00%>119 {1}{2}<100.00%>
  
 120 -------------------------------------------------------------------------------
 121 -----
 122 msolve·overall·time···········0.59·sec·(elapsed)·/··1.06·sec·(cpu)
 123 -------------------------------------------------------------------------------
 124 -----
  
120 -------------------------------------------------------------------------------125 -------------------------------------------------------------------------------
121 -----------------------126 -----------------------
  
  
122 ----------·COMPUTATIONAL·DATA·-----------127 ----------·COMPUTATIONAL·DATA·-----------
123 Max·coeff.·bitsize················1128 Max·coeff.·bitsize················1
124 #primes···························3129 #primes···························3
Offset 131, 21 lines modifiedOffset 138, 14 lines modified
131 -----------------------------------------138 -----------------------------------------
  
132 ----------------·TIMINGS·----------------139 ----------------·TIMINGS·----------------
133 CRT·····(elapsed)···············0.00·sec140 CRT·····(elapsed)···············0.00·sec
134 ratrecon(elapsed)···············0.00·sec141 ratrecon(elapsed)···············0.00·sec
135 -----------------------------------------142 -----------------------------------------
  
  
136 ------------------------------------------------------------------------------- 
137 ----- 
138 msolve·overall·time···········0.15·sec·(elapsed)·/··0.52·sec·(cpu) 
139 ------------------------------------------------------------------------------- 
140 ----- 
  
141 o3·=·|·z·y·x·|143 o3·=·|·z·y·x·|
  
142 ·············1······3144 ·············1······3
143 o3·:·Matrix·R··<--·R145 o3·:·Matrix·R··<--·R
144 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*146 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
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9 i2·:·I·=·ideal"xy,yz,zx"9 i2·:·I·=·ideal"xy,yz,zx"
  
10 o2·=·ideal·(x*y,·y*z,·x*z)10 o2·=·ideal·(x*y,·y*z,·x*z)
  
11 o2·:·Ideal·of·R11 o2·:·Ideal·of·R
  
12 i3·:·elapsedTime·jMult·I12 i3·:·elapsedTime·jMult·I
13 ·--·.0376545s·elapsed13 ·--·.115819s·elapsed
  
14 o3·=·214 o3·=·2
  
15 i4·:·elapsedTime·monjMult·I15 i4·:·elapsedTime·monjMult·I
16 ·--·.233236s·elapsed16 ·--·.501802s·elapsed
  
17 o4·=·217 o4·=·2
  
18 i5·:·elapsedTime·multiplicitySequence·I18 i5·:·elapsedTime·multiplicitySequence·I
19 ·--·1.09887s·elapsed19 ·--·1.18457s·elapsed
  
20 o5·=·HashTable{2·=>·3}20 o5·=·HashTable{2·=>·3}
21 ···············3·=>·221 ···············3·=>·2
  
22 o5·:·HashTable22 o5·:·HashTable
  
23 i6·:·23 i6·:·
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./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_mon__Analytic__Spread.out
    
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12 o2·:·Ideal·of·R12 o2·:·Ideal·of·R
  
13 i3·:·elapsedTime·monAnalyticSpread·I13 i3·:·elapsedTime·monAnalyticSpread·I
14 ·--·.245606s·elapsed14 ·--·.247574s·elapsed
  
15 o3·=·215 o3·=·2
  
16 i4·:·16 i4·:·
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./usr/share/doc/Macaulay2/MultiplicitySequence/example-output/_monj__Mult.out
    
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13 ·····------------------------------------------------------------------------13 ·····------------------------------------------------------------------------
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16 o2·:·Ideal·of·R16 o2·:·Ideal·of·R
  
17 i3·:·elapsedTime·monjMult·I17 i3·:·elapsedTime·monjMult·I
18 ·--·.304239s·elapsed18 ·--·.586319s·elapsed
  
19 o3·=·19219 o3·=·192
  
20 i4·:·elapsedTime·jMult·I20 i4·:·elapsedTime·jMult·I
21 ·--·1.43869s·elapsed21 ·--·2.31904s·elapsed
  
22 o4·=·19222 o4·=·192
  
23 i5·:·23 i5·:·
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./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
    
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83 o2·=·ideal·(x*y,·y*z,·x*z)83 o2·=·ideal·(x*y,·y*z,·x*z)
  
84 o2·:·Ideal·of·R</code></pre>84 o2·:·Ideal·of·R</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·jMult·I87 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·jMult·I
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89 o3·=·2</code></pre>89 o3·=·2</code></pre>
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94 o4·=·2</code></pre>94 o4·=·2</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
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99 o5·=·HashTable{2·=>·3}99 o5·=·HashTable{2·=>·3}
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21 o1·:·PolynomialRing21 o1·:·PolynomialRing
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23 o2·=·ideal·(x*y,·y*z,·x*z)23 o2·=·ideal·(x*y,·y*z,·x*z)
  
24 o2·:·Ideal·of·R24 o2·:·Ideal·of·R
25 i3·:·elapsedTime·jMult·I25 i3·:·elapsedTime·jMult·I
26 ·--·.0376545s·elapsed26 ·--·.115819s·elapsed
  
27 o3·=·227 o3·=·2
28 i4·:·elapsedTime·monjMult·I28 i4·:·elapsedTime·monjMult·I
29 ·--·.233236s·elapsed29 ·--·.501802s·elapsed
  
30 o4·=·230 o4·=·2
31 i5·:·elapsedTime·multiplicitySequence·I31 i5·:·elapsedTime·multiplicitySequence·I
32 ·--·1.09887s·elapsed32 ·--·1.18457s·elapsed
  
33 o5·=·HashTable{2·=>·3}33 o5·=·HashTable{2·=>·3}
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35 o5·:·HashTable35 o5·:·HashTable
36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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84 ·············2········384 ·············2········3
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94 ······</div>94 ······</div>
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29 o3·=·229 o3·=·2
30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
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98 ·--·1.43869s·elapsed98 ·--·2.31904s·elapsed
  
99 o4·=·192</code></pre>99 o4·=·192</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
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26 ·····------------------------------------------------------------------------26 ·····------------------------------------------------------------------------
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34 ·--·1.43869s·elapsed34 ·--·2.31904s·elapsed
  
35 o4·=·19235 o4·=·192
36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
37 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8p_\x8l_\x8i_\x8c_\x8i_\x8t_\x8y_\x8S_\x8e_\x8q_\x8u_\x8e_\x8n_\x8c_\x8e·--·the·multiplicity·sequence·of·an·ideal37 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8p_\x8l_\x8i_\x8c_\x8i_\x8t_\x8y_\x8S_\x8e_\x8q_\x8u_\x8e_\x8n_\x8c_\x8e·--·the·multiplicity·sequence·of·an·ideal
38 ····*·_\x8j_\x8M_\x8u_\x8l_\x8t·--·the·j-multiplicity·of·an·ideal38 ····*·_\x8j_\x8M_\x8u_\x8l_\x8t·--·the·j-multiplicity·of·an·ideal
39 ····*·_\x8m_\x8o_\x8n_\x8R_\x8e_\x8d_\x8u_\x8c_\x8t_\x8i_\x8o_\x8n·--·the·minimal·monomial·reduction·of·a·monomial·ideal39 ····*·_\x8m_\x8o_\x8n_\x8R_\x8e_\x8d_\x8u_\x8c_\x8t_\x8i_\x8o_\x8n·--·the·minimal·monomial·reduction·of·a·monomial·ideal
40 ····*·_\x8N_\x8P·--·the·Newton·polyhedron·of·a·monomial·ideal40 ····*·_\x8N_\x8P·--·the·Newton·polyhedron·of·a·monomial·ideal
⊟
773 B
./usr/share/doc/Macaulay2/MultiplierIdeals/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=417 #:len=41
8 bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk=8 bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk=
9 #:len=3619 #:len=361
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA2OSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA2OSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB
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737 B
./usr/share/doc/Macaulay2/MultiplierIdealsDim2/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=97 #:len=9
8 TXVsdElkZWFs8 TXVsdElkZWFs
9 #:len=13599 #:len=1359
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg
11 aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT11 aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT
⊟
749 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=127 #:len=12
8 dGV4TWF0aChSQVQp8 dGV4TWF0aChSQVQp
9 #:len=2099 #:len=209
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OSwgInVuZG9jdW1lbnRlZCIgPT4g10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OSwgInVuZG9jdW1lbnRlZCIgPT4g
11 dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF011 dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF0
⊟
1.56 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.out
    
Offset 13, 15 lines modifiedOffset 13, 15 lines modified
13 o4·:·ProjectiveVariety,·curve·in·PP^813 o4·:·ProjectiveVariety,·curve·in·PP^8
  
14 i5·:·?·X14 i5·:·?·X
  
15 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^15·15 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^15·
  
16 i6·:·time·f·=·X·===>·Y;16 i6·:·time·f·=·X·===>·Y;
17 ·--·used·3.54716s·(cpu);·1.54735s·(thread);·0s·(gc)17 ·--·used·3.70499s·(cpu);·1.75293s·(thread);·0s·(gc)
  
18 o6·:·MultirationalMap·(automorphism·of·PP^8)18 o6·:·MultirationalMap·(automorphism·of·PP^8)
  
19 i7·:·f·X19 i7·:·f·X
  
20 o7·=·Y20 o7·=·Y
  
Offset 38, 15 lines modifiedOffset 38, 15 lines modified
38 o9·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^838 o9·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8
  
39 i10·:·W·=·random({{2},{1}},Y);39 i10·:·W·=·random({{2},{1}},Y);
  
40 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^840 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8
  
41 i11·:·time·g·=·V·===>·W;41 i11·:·time·g·=·V·===>·W;
42 ·--·used·5.91992s·(cpu);·1.97699s·(thread);·0s·(gc)42 ·--·used·6.04597s·(cpu);·2.00781s·(thread);·0s·(gc)
  
43 o11·:·MultirationalMap·(automorphism·of·PP^8)43 o11·:·MultirationalMap·(automorphism·of·PP^8)
  
44 i12·:·g||W44 i12·:·g||W
  
45 o12·=·multi-rational·map·consisting·of·one·single·rational·map45 o12·=·multi-rational·map·consisting·of·one·single·rational·map
46 ······source·variety:·6-dimensional·subvariety·of·PP^8·cut·out·by·2·hypersurfaces·of·degrees·1^1·2^1·46 ······source·variety:·6-dimensional·subvariety·of·PP^8·cut·out·by·2·hypersurfaces·of·degrees·1^1·2^1·
Offset 129, 15 lines modifiedOffset 129, 15 lines modified
129 o15·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^9129 o15·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^9
  
130 i16·:·?·Z130 i16·:·?·Z
  
131 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2131 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2
  
132 i17·:·time·h·=·Z·===>·GG_K(1,4)132 i17·:·time·h·=·Z·===>·GG_K(1,4)
133 ·--·used·6.20544s·(cpu);·3.75724s·(thread);·0s·(gc)133 ·--·used·6.47215s·(cpu);·3.81843s·(thread);·0s·(gc)
  
134 o17·=·h134 o17·=·h
  
135 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)135 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)
  
136 i18·:·h·||·GG_K(1,4)136 i18·:·h·||·GG_K(1,4)
  
⊟
1.04 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.out
    
Offset 7, 15 lines modifiedOffset 7, 15 lines modified
7 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)7 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)
  
8 i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random({1,1},ring·target·Phi));8 i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random({1,1},ring·target·Phi));
  
9 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^49 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^4
  
10 i4·:·time·X·=·Phi^*·Y;10 i4·:·time·X·=·Phi^*·Y;
11 ·--·used·2.99629s·(cpu);·2.45374s·(thread);·0s·(gc)11 ·--·used·3.77536s·(cpu);·3.0669s·(thread);·0s·(gc)
  
12 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension·2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)12 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension·2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)
  
13 i5·:·dim·X,·degree·X,·degrees·X13 i5·:·dim·X,·degree·X,·degrees·X
  
14 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),14 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),
15 ·····------------------------------------------------------------------------15 ·····------------------------------------------------------------------------
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1.12 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multirational__Map_sp__Multiprojective__Variety.out
    
Offset 11, 26 lines modifiedOffset 11, 26 lines modified
11 o3·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^7·x·PP^7)11 o3·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^7·x·PP^7)
  
12 i4·:·Z·=·source·Phi;12 i4·:·Z·=·source·Phi;
  
13 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^713 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^7
  
14 i5·:·time·Phi·Z;14 i5·:·time·Phi·Z;
15 ·--·used·0.353551s·(cpu);·0.111677s·(thread);·0s·(gc)15 ·--·used·0.307538s·(cpu);·0.119111s·(thread);·0s·(gc)
  
16 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^716 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^7
  
17 i6·:·dim·oo,·degree·oo,·degrees·oo17 i6·:·dim·oo,·degree·oo,·degrees·oo
  
18 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})18 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})
  
19 o6·:·Sequence19 o6·:·Sequence
  
20 i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)20 i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)
21 ·--·used·1.66337s·(cpu);·1.0477s·(thread);·0s·(gc)21 ·--·used·1.96298s·(cpu);·1.10521s·(thread);·0s·(gc)
  
22 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of·multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^3·22 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of·multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^3·
  
23 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^723 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^7
  
24 i8·:·dim·oo,·degree·oo,·degrees·oo24 i8·:·dim·oo,·degree·oo,·degrees·oo
  
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1.07 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_cm__Option_rp.out
    
Offset 11, 22 lines modifiedOffset 11, 22 lines modified
11 o3·=·multi-rational·map·consisting·of·one·single·rational·map11 o3·=·multi-rational·map·consisting·of·one·single·rational·map
12 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of12 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of
13 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·213 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·2
14 ·····------------------------------------------------------------------------14 ·····------------------------------------------------------------------------
15 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^115 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^1
  
16 i4·:·time·degree(Phi,Strategy=>"random·point")16 i4·:·time·degree(Phi,Strategy=>"random·point")
17 ·--·used·3.64249s·(cpu);·1.93164s·(thread);·0s·(gc)17 ·--·used·4.80968s·(cpu);·2.62178s·(thread);·0s·(gc)
  
18 o4·=·218 o4·=·2
  
19 i5·:·time·degree(Phi,Strategy=>"0-th·projective·degree")19 i5·:·time·degree(Phi,Strategy=>"0-th·projective·degree")
20 ·--·used·0.257288s·(cpu);·0.198792s·(thread);·0s·(gc)20 ·--·used·0.381043s·(cpu);·0.288489s·(thread);·0s·(gc)
  
21 o5·=·221 o5·=·2
  
22 i6·:·time·degree·Phi22 i6·:·time·degree·Phi
23 ·--·used·0.177037s·(cpu);·0.179623s·(thread);·0s·(gc)23 ·--·used·0.338079s·(cpu);·0.261095s·(thread);·0s·(gc)
  
24 o6·=·224 o6·=·2
  
25 i7·:·25 i7·:·
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672 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_degree_lp__Multirational__Map_rp.out
    
Offset 3, 12 lines modifiedOffset 3, 12 lines modified
3 i1·:·ZZ/300007[x_0..x_3],·f·=·rationalMap·{x_2^2-x_1*x_3,·x_1*x_2-x_0*x_3,·x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,·x_3^2};3 i1·:·ZZ/300007[x_0..x_3],·f·=·rationalMap·{x_2^2-x_1*x_3,·x_1*x_2-x_0*x_3,·x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,·x_3^2};
  
4 i2·:·Phi·=·last·graph·rationalMap·{f,g};4 i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
5 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)5 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)
  
6 i3·:·time·degree·Phi6 i3·:·time·degree·Phi
7 ·--·used·0.650485s·(cpu);·0.309242s·(thread);·0s·(gc)7 ·--·used·0.774467s·(cpu);·0.406148s·(thread);·0s·(gc)
  
8 o3·=·18 o3·=·1
  
9 i4·:·9 i4·:·
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2.66 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_describe_lp__Multirational__Map_rp.out
    
Offset 1, 52 lines modifiedOffset 1, 52 lines modified
1 --·-*-·M2-comint·-*-·hash:·115337213248520721611 --·-*-·M2-comint·-*-·hash:·11533721324852072161
  
2 i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);2 i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);
  
3 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4·x·PP^5)3 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4·x·PP^5)
  
4 i2·:·time·?·Phi4 i2·:·time·?·Phi
5 ·--·used·0.00342738s·(cpu);·0.000130076s·(thread);·0s·(gc)5 ·--·used·0.00206852s·(cpu);·0.000144972s·(thread);·0s·(gc)
  
6 o2·=·multi-rational·map·consisting·of·2·rational·maps6 o2·=·multi-rational·map·consisting·of·2·rational·maps
7 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·97 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
8 ·····target·variety:·PP^4·x·PP^58 ·····target·variety:·PP^4·x·PP^5
9 ·····------------------------------------------------------------------------9 ·····------------------------------------------------------------------------
10 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^810 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8
  
11 i3·:·image·Phi;11 i3·:·image·Phi;
  
12 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^512 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^5
  
13 i4·:·time·?·Phi13 i4·:·time·?·Phi
14 ·--·used·0.00218014s·(cpu);·0.000190787s·(thread);·0s·(gc)14 ·--·used·0.00045313s·(cpu);·0.000386114s·(thread);·0s·(gc)
  
15 o4·=·multi-rational·map·consisting·of·2·rational·maps15 o4·=·multi-rational·map·consisting·of·2·rational·maps
16 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·16 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
17 ·····target·variety:·PP^4·x·PP^517 ·····target·variety:·PP^4·x·PP^5
18 ·····dominance:·false18 ·····dominance:·false
19 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·19 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
  
20 i5·:·time·describe·Phi20 i5·:·time·describe·Phi
21 ·--·used·0.922696s·(cpu);·0.800107s·(thread);·0s·(gc)21 ·--·used·0.985958s·(cpu);·0.821223s·(thread);·0s·(gc)
  
22 o5·=·multi-rational·map·consisting·of·2·rational·maps22 o5·=·multi-rational·map·consisting·of·2·rational·maps
23 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·23 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
24 ·····target·variety:·PP^4·x·PP^524 ·····target·variety:·PP^4·x·PP^5
25 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^525 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
26 ·····dominance:·false26 ·····dominance:·false
27 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·27 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
28 ·····multidegree:·{51,·51,·51,·51,·51}28 ·····multidegree:·{51,·51,·51,·51,·51}
29 ·····degree:·129 ·····degree:·1
30 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]30 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]
31 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]31 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]
32 ·····coefficient·ring:·ZZ/6552132 ·····coefficient·ring:·ZZ/65521
  
33 i6·:·time·?·Phi33 i6·:·time·?·Phi
34 ·--·used·0.000120181s·(cpu);·0.000327323s·(thread);·0s·(gc)34 ·--·used·0.000116318s·(cpu);·0.000358022s·(thread);·0s·(gc)
  
35 o6·=·multi-rational·map·consisting·of·2·rational·maps35 o6·=·multi-rational·map·consisting·of·2·rational·maps
36 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·36 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
37 ·····target·variety:·PP^4·x·PP^537 ·····target·variety:·PP^4·x·PP^5
38 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^538 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
39 ·····dominance:·false39 ·····dominance:·false
40 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·40 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
⊟
1.47 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_graph_lp__Multirational__Map_rp.out
    
Offset 3, 45 lines modifiedOffset 3, 45 lines modified
3 i1·:·Phi·=·rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)3 i1·:·Phi·=·rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
  
4 o1·=·Phi4 o1·=·Phi
  
5 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)5 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
  
6 i2·:·time·(Phi1,Phi2)·=·graph·Phi6 i2·:·time·(Phi1,Phi2)·=·graph·Phi
7 ·--·used·0.213793s·(cpu);·0.0353456s·(thread);·0s·(gc)7 ·--·used·0.241683s·(cpu);·0.0513733s·(thread);·0s·(gc)
  
8 o2·=·(Phi1,·Phi2)8 o2·=·(Phi1,·Phi2)
  
9 o2·:·Sequence9 o2·:·Sequence
  
10 i3·:·Phi1;10 i3·:·Phi1;
  
11 o3·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4)11 o3·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4)
  
12 i4·:·Phi2;12 i4·:·Phi2;
  
13 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)13 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)
  
14 i5·:·time·(Phi21,Phi22)·=·graph·Phi214 i5·:·time·(Phi21,Phi22)·=·graph·Phi2
15 ·--·used·0.261904s·(cpu);·0.0484107s·(thread);·0s·(gc)15 ·--·used·0.21007s·(cpu);·0.0572863s·(thread);·0s·(gc)
  
16 o5·=·(Phi21,·Phi22)16 o5·=·(Phi21,·Phi22)
  
17 o5·:·Sequence17 o5·:·Sequence
  
18 i6·:·Phi21;18 i6·:·Phi21;
  
19 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)19 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)
  
20 i7·:·Phi22;20 i7·:·Phi22;
  
21 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)21 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)
  
22 i8·:·time·(Phi211,Phi212)·=·graph·Phi2122 i8·:·time·(Phi211,Phi212)·=·graph·Phi21
23 ·--·used·0.338021s·(cpu);·0.134171s·(thread);·0s·(gc)23 ·--·used·0.497146s·(cpu);·0.172417s·(thread);·0s·(gc)
  
24 o8·=·(Phi211,·Phi212)24 o8·=·(Phi211,·Phi212)
  
25 o8·:·Sequence25 o8·:·Sequence
  
26 i9·:·Phi211;26 i9·:·Phi211;
  
⊟
998 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_image_lp__Multirational__Map_rp.out
    
Offset 11, 25 lines modifiedOffset 11, 25 lines modified
11 o3·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)11 o3·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)
  
12 i4·:·Phi·=·rationalMap·{f,g};12 i4·:·Phi·=·rationalMap·{f,g};
  
13 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)13 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)
  
14 i5·:·time·Z·=·image·Phi;14 i5·:·time·Z·=·image·Phi;
15 ·--·used·0.39011s·(cpu);·0.13779s·(thread);·0s·(gc)15 ·--·used·0.326121s·(cpu);·0.129263s·(thread);·0s·(gc)
  
16 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^416 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4
  
17 i6·:·dim·Z,·degree·Z,·degrees·Z17 i6·:·dim·Z,·degree·Z,·degrees·Z
  
18 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})18 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})
  
19 o6·:·Sequence19 o6·:·Sequence
  
20 i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,"F4");20 i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,"F4");
21 ·--·used·7.74487s·(cpu);·2.28877s·(thread);·0s·(gc)21 ·--·used·8.40908s·(cpu);·2.37569s·(thread);·0s·(gc)
  
22 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^422 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4
  
23 i8·:·assert(Z·==·Z')23 i8·:·assert(Z·==·Z')
  
24 i9·:·24 i9·:·
⊟
1.02 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse2.out
    
Offset 4, 25 lines modifiedOffset 4, 25 lines modified
  
4 i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational·normal·curve·of·degree·64 i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational·normal·curve·of·degree·6
5 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal·PP_K([6],2));5 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal·PP_K([6],2));
  
6 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))6 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))
  
7 i3·:·time·Psi·=·inverse2·Phi;7 i3·:·time·Psi·=·inverse2·Phi;
8 ·--·used·0.491906s·(cpu);·0.261695s·(thread);·0s·(gc)8 ·--·used·0.580318s·(cpu);·0.301422s·(thread);·0s·(gc)
  
9 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)9 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)
  
10 i4·:·assert(Phi·*·Psi·==·1)10 i4·:·assert(Phi·*·Psi·==·1)
  
11 i5·:·Phi'·=·Phi·||·Phi;11 i5·:·Phi'·=·Phi·||·Phi;
  
12 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))12 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))
  
13 i6·:·time·Psi'·=·inverse2·Phi';13 i6·:·time·Psi'·=·inverse2·Phi';
14 ·--·used·1.26599s·(cpu);·0.922462s·(thread);·0s·(gc)14 ·--·used·1.32228s·(cpu);·0.896273s·(thread);·0s·(gc)
  
15 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)15 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)
  
16 i7·:·assert(Phi'·*·Psi'·==·1)16 i7·:·assert(Phi'·*·Psi'·==·1)
  
17 i8·:·17 i8·:·
⊟
1.51 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_inverse_lp__Multirational__Map_rp.out
    
Offset 7, 33 lines modifiedOffset 7, 33 lines modified
  
7 i2·:·--·we·see·Phi·as·a·dominant·map7 i2·:·--·we·see·Phi·as·a·dominant·map
8 ·····Phi·=·rationalMap(Phi,image·Phi);8 ·····Phi·=·rationalMap(Phi,image·Phi);
  
9 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)9 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
  
10 i3·:·time·inverse·Phi;10 i3·:·time·inverse·Phi;
11 ·--·used·0.285898s·(cpu);·0.0614978s·(thread);·0s·(gc)11 ·--·used·0.295307s·(cpu);·0.0646031s·(thread);·0s·(gc)
  
12 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)12 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)
  
13 i4·:·Psi·=·last·graph·Phi;13 i4·:·Psi·=·last·graph·Phi;
  
14 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)14 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)
  
15 i5·:·time·inverse·Psi;15 i5·:·time·inverse·Psi;
16 ·--·used·0.556178s·(cpu);·0.118223s·(thread);·0s·(gc)16 ·--·used·0.573546s·(cpu);·0.13748s·(thread);·0s·(gc)
  
17 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)17 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)
  
18 i6·:·Eta·=·first·graph·Psi;18 i6·:·Eta·=·first·graph·Psi;
  
19 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)19 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)
  
20 i7·:·time·inverse·Eta;20 i7·:·time·inverse·Eta;
21 ·--·used·0.856787s·(cpu);·0.296816s·(thread);·0s·(gc)21 ·--·used·0.835829s·(cpu);·0.335785s·(thread);·0s·(gc)
  
22 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)22 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)
  
23 i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)23 i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)
  
24 i9·:·assert(Psi·*·Psi^-1·==·1·and·Psi^-1·*·Psi·==·1)24 i9·:·assert(Psi·*·Psi^-1·==·1·and·Psi^-1·*·Psi·==·1)
  
⊟
1.17 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Isomorphism_lp__Multirational__Map_rp.out
    
Offset 6, 32 lines modifiedOffset 6, 32 lines modified
6 o2·:·RationalMap·(quadratic·rational·map·from·PP^3·to·PP^2)6 o2·:·RationalMap·(quadratic·rational·map·from·PP^3·to·PP^2)
  
7 i3·:·Phi·=·rationalMap·{f,f};7 i3·:·Phi·=·rationalMap·{f,f};
  
8 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)8 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)
  
9 i4·:·time·isIsomorphism·Phi9 i4·:·time·isIsomorphism·Phi
10 ·--·used·0.00236091s·(cpu);·5.147e-06s·(thread);·0s·(gc)10 ·--·used·0.00148525s·(cpu);·9.017e-06s·(thread);·0s·(gc)
  
11 o4·=·false11 o4·=·false
  
12 i5·:·Psi·=·first·graph·Phi;12 i5·:·Psi·=·first·graph·Phi;
  
13 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to·PP^3)13 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to·PP^3)
  
14 i6·:·time·isIsomorphism·Psi14 i6·:·time·isIsomorphism·Psi
15 ·--·used·0.696023s·(cpu);·0.196833s·(thread);·0s·(gc)15 ·--·used·0.671814s·(cpu);·0.170447s·(thread);·0s·(gc)
  
16 o6·=·false16 o6·=·false
  
17 i7·:·Eta·=·first·graph·Psi;17 i7·:·Eta·=·first·graph·Psi;
  
18 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x·PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)18 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x·PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)
  
19 i8·:·time·isIsomorphism·Eta19 i8·:·time·isIsomorphism·Eta
20 ·--·used·1.54989s·(cpu);·0.688024s·(thread);·0s·(gc)20 ·--·used·1.67481s·(cpu);·0.719784s·(thread);·0s·(gc)
  
21 o8·=·true21 o8·=·true
  
22 i9·:·assert(o8·and·(not·o6)·and·(not·o4))22 i9·:·assert(o8·and·(not·o6)·and·(not·o4))
  
23 i10·:·23 i10·:·
⊟
1.1 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_is__Morphism_lp__Multirational__Map_rp.out
    
Offset 3, 24 lines modifiedOffset 3, 24 lines modified
3 i1·:·ZZ/300007[a..e],·f·=·first·graph·rationalMap·ideal(c^2-b*d,b*c-a*d,b^2-a*c,e),·g·=·rationalMap·submatrix(matrix·f,{0..2});3 i1·:·ZZ/300007[a..e],·f·=·first·graph·rationalMap·ideal(c^2-b*d,b*c-a*d,b^2-a*c,e),·g·=·rationalMap·submatrix(matrix·f,{0..2});
  
4 i2·:·Phi·=·rationalMap·{f,g};4 i2·:·Phi·=·rationalMap·{f,g};
  
5 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^4·x·PP^2)5 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^4·x·PP^2)
  
6 i3·:·time·isMorphism·Phi6 i3·:·time·isMorphism·Phi
7 ·--·used·0.177148s·(cpu);·0.139134s·(thread);·0s·(gc)7 ·--·used·0.212445s·(cpu);·0.154119s·(thread);·0s·(gc)
  
8 o3·=·false8 o3·=·false
  
9 i4·:·time·Psi·=·first·graph·Phi;9 i4·:·time·Psi·=·first·graph·Phi;
10 ·--·used·0.362469s·(cpu);·0.0960295s·(thread);·0s·(gc)10 ·--·used·0.327153s·(cpu);·0.0990364s·(thread);·0s·(gc)
  
11 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)11 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)
  
12 i5·:·time·isMorphism·Psi12 i5·:·time·isMorphism·Psi
13 ·--·used·2.83708s·(cpu);·2.48281s·(thread);·0s·(gc)13 ·--·used·2.99672s·(cpu);·2.47521s·(thread);·0s·(gc)
  
14 o5·=·true14 o5·=·true
  
15 i6·:·assert((not·o3)·and·o5)15 i6·:·assert((not·o3)·and·o5)
  
16 i7·:·16 i7·:·
⊟
968 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_linearly__Normal__Embedding.out
    
Offset 3, 24 lines modifiedOffset 3, 24 lines modified
3 i1·:·K·=·ZZ/333331;3 i1·:·K·=·ZZ/333331;
  
4 i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·74 i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·7
  
5 o2·:·ProjectiveVariety,·curve·in·PP^75 o2·:·ProjectiveVariety,·curve·in·PP^7
  
6 i3·:·time·f·=·linearlyNormalEmbedding·X;6 i3·:·time·f·=·linearlyNormalEmbedding·X;
7 ·--·used·0.00500658s·(cpu);·0.00768908s·(thread);·0s·(gc)7 ·--·used·0.00802597s·(cpu);·0.008841s·(thread);·0s·(gc)
  
8 o3·:·MultirationalMap·(automorphism·of·X)8 o3·:·MultirationalMap·(automorphism·of·X)
  
9 i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an·isomorphic·projection·of·X·in·PP^39 i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an·isomorphic·projection·of·X·in·PP^3
  
10 o4·:·ProjectiveVariety,·curve·in·PP^310 o4·:·ProjectiveVariety,·curve·in·PP^3
  
11 i5·:·time·g·=·linearlyNormalEmbedding·Y;11 i5·:·time·g·=·linearlyNormalEmbedding·Y;
12 ·--·used·0.383948s·(cpu);·0.345654s·(thread);·0s·(gc)12 ·--·used·0.410114s·(cpu);·0.35555s·(thread);·0s·(gc)
  
13 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)13 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)
  
14 i6·:·assert(isIsomorphism·g)14 i6·:·assert(isIsomorphism·g)
  
15 i7·:·describe·g15 i7·:·describe·g
  
⊟
752 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Multirational__Map_rp.out
    
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3 i1·:·ZZ/300007[x_0..x_3],·f·=·rationalMap·{x_2^2-x_1*x_3,·x_1*x_2-x_0*x_3,·x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,·x_3^2};3 i1·:·ZZ/300007[x_0..x_3],·f·=·rationalMap·{x_2^2-x_1*x_3,·x_1*x_2-x_0*x_3,·x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,·x_3^2};
  
4 i2·:·Phi·=·last·graph·rationalMap·{f,g};4 i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
5 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)5 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)
  
6 i3·:·time·multidegree·Phi6 i3·:·time·multidegree·Phi
7 ·--·used·0.64645s·(cpu);·0.319383s·(thread);·0s·(gc)7 ·--·used·0.627121s·(cpu);·0.326963s·(thread);·0s·(gc)
  
8 o3·=·{66,·46,·31,·20}8 o3·=·{66,·46,·31,·20}
  
9 o3·:·List9 o3·:·List
  
10 i4·:·(degree·source·Phi,degree·image·Phi)10 i4·:·(degree·source·Phi,degree·image·Phi)
  
⊟
1.25 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.out
    
Offset 1, 21 lines modifiedOffset 1, 21 lines modified
1 --·-*-·M2-comint·-*-·hash:·161997332192100812141 --·-*-·M2-comint·-*-·hash:·16199733219210081214
  
2 i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);2 i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);
  
3 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^5)3 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^5)
  
4 i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)4 i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)
 5 ·--·used·0.00178884s·(cpu);·0.00116877s·(thread);·0s·(gc)
 6 ·--·used·0.166747s·(cpu);·0.116878s·(thread);·0s·(gc)
5 ·--·used·0.00397065s·(cpu);·0.00103293s·(thread);·0s·(gc)7 ·--·used·0.209552s·(cpu);·0.137466s·(thread);·0s·(gc)
6 ·--·used·0.140164s·(cpu);·0.0956171s·(thread);·0s·(gc)8 ·--·used·0.160699s·(cpu);·0.0976585s·(thread);·0s·(gc)
7 ·--·used·0.168172s·(cpu);·0.126466s·(thread);·0s·(gc) 
8 ·--·used·0.144435s·(cpu);·0.0993572s·(thread);·0s·(gc) 
9 ·--·used·0.115045s·(cpu);·0.0734805s·(thread);·0s·(gc)9 ·--·used·0.14151s·(cpu);·0.0880439s·(thread);·0s·(gc)
  
10 o2·=·{51,·28,·14,·6,·2}10 o2·=·{51,·28,·14,·6,·2}
  
11 o2·:·List11 o2·:·List
  
12 i3·:·time·assert(oo·==·multidegree·Phi)12 i3·:·time·assert(oo·==·multidegree·Phi)
13 ·--·used·0.142749s·(cpu);·0.0455335s·(thread);·0s·(gc)13 ·--·used·0.270609s·(cpu);·0.0620393s·(thread);·0s·(gc)
  
14 i4·:·14 i4·:·
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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_point_lp__Multiprojective__Variety_rp.out
    
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3 i1·:·K·=·ZZ/1000003;3 i1·:·K·=·ZZ/1000003;
  
4 i2·:·X·=·PP_K^({1,1,2},{3,2,3});4 i2·:·X·=·PP_K^({1,1,2},{3,2,3});
  
5 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^95 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^9
  
6 i3·:·time·p·:=·point·X6 i3·:·time·p·:=·point·X
7 ·--·used·0.241285s·(cpu);·0.0300762s·(thread);·0s·(gc)7 ·--·used·0.207929s·(cpu);·0.0323835s·(thread);·0s·(gc)
  
8 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],[3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])8 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],[3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])
  
9 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^99 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9
  
10 i4·:·Y·=·random({2,1,2},X);10 i4·:·Y·=·random({2,1,2},X);
  
11 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^911 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^9
  
12 i5·:·time·q·=·point·Y12 i5·:·time·q·=·point·Y
13 ·--·used·1.25059s·(cpu);·0.746661s·(thread);·0s·(gc)13 ·--·used·1.39438s·(cpu);·0.750503s·(thread);·0s·(gc)
  
14 o5·=·q14 o5·=·q
  
15 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^915 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9
  
16 i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))16 i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))
  
⊟
699 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_segre_lp__Multirational__Map_rp.out
    
Offset 15, 15 lines modifiedOffset 15, 15 lines modified
15 o4·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)15 o4·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)
  
16 i5·:·Phi·=·rationalMap·{f,g,h};16 i5·:·Phi·=·rationalMap·{f,g,h};
  
17 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x·PP^4)17 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x·PP^4)
  
18 i6·:·time·segre·Phi;18 i6·:·time·segre·Phi;
19 ·--·used·4.27397s·(cpu);·0.868122s·(thread);·0s·(gc)19 ·--·used·3.80843s·(cpu);·0.731163s·(thread);·0s·(gc)
  
20 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)20 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)
  
21 i7·:·describe·segre·Phi21 i7·:·describe·segre·Phi
  
22 o7·=·rational·map·defined·by·forms·of·degree·622 o7·=·rational·map·defined·by·forms·of·degree·6
23 ·····source·variety:·PP^423 ·····source·variety:·PP^4
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938 B
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/_show_lp__Multirational__Map_rp.out
    
Offset 3, 15 lines modifiedOffset 3, 15 lines modified
3 i1·:·Phi·=·inverse·first·graph·last·graph·rationalMap·PP_(ZZ/33331)^(1,3)3 i1·:·Phi·=·inverse·first·graph·last·graph·rationalMap·PP_(ZZ/33331)^(1,3)
  
4 o1·=·Phi4 o1·=·Phi
  
5 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to·threefold·in·PP^3·x·PP^2·x·PP^2)5 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to·threefold·in·PP^3·x·PP^2·x·PP^2)
  
6 i2·:·time·describe·Phi6 i2·:·time·describe·Phi
7 ·--·used·0.157974s·(cpu);·0.117367s·(thread);·0s·(gc)7 ·--·used·0.18467s·(cpu);·0.123938s·(thread);·0s·(gc)
  
8 o2·=·multi-rational·map·consisting·of·3·rational·maps8 o2·=·multi-rational·map·consisting·of·3·rational·maps
9 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of·multi-degree·(1,1)9 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of·multi-degree·(1,1)
10 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces·of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^2·10 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces·of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^2·
11 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^211 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^2
12 ·····dominance:·true12 ·····dominance:·true
13 ·····multidegree:·{10,·14,·19,·25}13 ·····multidegree:·{10,·14,·19,·25}
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3.58 KB
./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html
    
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94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i5·:·?·X95 <td>··············<pre><code·class="language-macaulay2">i5·:·?·X
  
96 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^15·</code></pre>96 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^15·</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i6·:·time·f·=·X·===>·Y;99 <td>··············<pre><code·class="language-macaulay2">i6·:·time·f·=·X·===>·Y;
100 ·--·used·3.54716s·(cpu);·1.54735s·(thread);·0s·(gc)100 ·--·used·3.70499s·(cpu);·1.75293s·(thread);·0s·(gc)
  
101 o6·:·MultirationalMap·(automorphism·of·PP^8)</code></pre>101 o6·:·MultirationalMap·(automorphism·of·PP^8)</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i7·:·f·X104 <td>··············<pre><code·class="language-macaulay2">i7·:·f·X
  
105 o7·=·Y105 o7·=·Y
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124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i10·:·W·=·random({{2},{1}},Y);125 <td>··············<pre><code·class="language-macaulay2">i10·:·W·=·random({{2},{1}},Y);
  
126 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8</code></pre>126 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8</code></pre>
127 </td>··········</tr>127 </td>··········</tr>
128 ··········<tr>128 ··········<tr>
129 <td>··············<pre><code·class="language-macaulay2">i11·:·time·g·=·V·===>·W;129 <td>··············<pre><code·class="language-macaulay2">i11·:·time·g·=·V·===>·W;
130 ·--·used·5.91992s·(cpu);·1.97699s·(thread);·0s·(gc)130 ·--·used·6.04597s·(cpu);·2.00781s·(thread);·0s·(gc)
  
131 o11·:·MultirationalMap·(automorphism·of·PP^8)</code></pre>131 o11·:·MultirationalMap·(automorphism·of·PP^8)</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i12·:·g||W134 <td>··············<pre><code·class="language-macaulay2">i12·:·g||W
  
135 o12·=·multi-rational·map·consisting·of·one·single·rational·map135 o12·=·multi-rational·map·consisting·of·one·single·rational·map
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223 ··········<tr>223 ··········<tr>
224 <td>··············<pre><code·class="language-macaulay2">i16·:·?·Z224 <td>··············<pre><code·class="language-macaulay2">i16·:·?·Z
  
225 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2</code></pre>225 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2</code></pre>
226 </td>··········</tr>226 </td>··········</tr>
227 ··········<tr>227 ··········<tr>
228 <td>··············<pre><code·class="language-macaulay2">i17·:·time·h·=·Z·===>·GG_K(1,4)228 <td>··············<pre><code·class="language-macaulay2">i17·:·time·h·=·Z·===>·GG_K(1,4)
229 ·--·used·6.20544s·(cpu);·3.75724s·(thread);·0s·(gc)229 ·--·used·6.47215s·(cpu);·3.81843s·(thread);·0s·(gc)
  
230 o17·=·h230 o17·=·h
  
231 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)</code></pre>231 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)</code></pre>
232 </td>··········</tr>232 </td>··········</tr>
233 ··········<tr>233 ··········<tr>
234 <td>··············<pre><code·class="language-macaulay2">i18·:·h·||·GG_K(1,4)234 <td>··············<pre><code·class="language-macaulay2">i18·:·h·||·GG_K(1,4)
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34 take(N,-2));34 take(N,-2));
  
35 o4·:·ProjectiveVariety,·curve·in·PP^835 o4·:·ProjectiveVariety,·curve·in·PP^8
36 i5·:·?·X36 i5·:·?·X
  
37 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^1537 o5·=·curve·in·PP^8·cut·out·by·17·hypersurfaces·of·degrees·1^2·2^15
38 i6·:·time·f·=·X·===>·Y;38 i6·:·time·f·=·X·===>·Y;
39 ·--·used·3.54716s·(cpu);·1.54735s·(thread);·0s·(gc)39 ·--·used·3.70499s·(cpu);·1.75293s·(thread);·0s·(gc)
  
40 o6·:·MultirationalMap·(automorphism·of·PP^8)40 o6·:·MultirationalMap·(automorphism·of·PP^8)
41 i7·:·f·X41 i7·:·f·X
  
42 o7·=·Y42 o7·=·Y
  
43 o7·:·ProjectiveVariety,·curve·in·PP^843 o7·:·ProjectiveVariety,·curve·in·PP^8
Offset 54, 15 lines modifiedOffset 54, 15 lines modified
54 i9·:·V·=·random({{2},{1}},X);54 i9·:·V·=·random({{2},{1}},X);
  
55 o9·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^855 o9·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8
56 i10·:·W·=·random({{2},{1}},Y);56 i10·:·W·=·random({{2},{1}},Y);
  
57 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^857 o10·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^8
58 i11·:·time·g·=·V·===>·W;58 i11·:·time·g·=·V·===>·W;
59 ·--·used·5.91992s·(cpu);·1.97699s·(thread);·0s·(gc)59 ·--·used·6.04597s·(cpu);·2.00781s·(thread);·0s·(gc)
  
60 o11·:·MultirationalMap·(automorphism·of·PP^8)60 o11·:·MultirationalMap·(automorphism·of·PP^8)
61 i12·:·g||W61 i12·:·g||W
  
62 o12·=·multi-rational·map·consisting·of·one·single·rational·map62 o12·=·multi-rational·map·consisting·of·one·single·rational·map
63 ······source·variety:·6-dimensional·subvariety·of·PP^8·cut·out·by·263 ······source·variety:·6-dimensional·subvariety·of·PP^8·cut·out·by·2
64 hypersurfaces·of·degrees·1^1·2^164 hypersurfaces·of·degrees·1^1·2^1
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145 i15·:·Z·=·projectiveVariety·pfaffians(4,A);145 i15·:·Z·=·projectiveVariety·pfaffians(4,A);
  
146 o15·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^9146 o15·:·ProjectiveVariety,·6-dimensional·subvariety·of·PP^9
147 i16·:·?·Z147 i16·:·?·Z
  
148 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2148 o16·=·6-dimensional·subvariety·of·PP^9·cut·out·by·5·hypersurfaces·of·degree·2
149 i17·:·time·h·=·Z·===>·GG_K(1,4)149 i17·:·time·h·=·Z·===>·GG_K(1,4)
150 ·--·used·6.20544s·(cpu);·3.75724s·(thread);·0s·(gc)150 ·--·used·6.47215s·(cpu);·3.81843s·(thread);·0s·(gc)
  
151 o17·=·h151 o17·=·h
  
152 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)152 o17·:·MultirationalMap·(isomorphism·from·PP^9·to·PP^9)
153 i18·:·h·||·GG_K(1,4)153 i18·:·h·||·GG_K(1,4)
  
154 o18·=·multi-rational·map·consisting·of·one·single·rational·map154 o18·=·multi-rational·map·consisting·of·one·single·rational·map
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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html
    
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84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random({1,1},ring·target·Phi));85 <td>··············<pre><code·class="language-macaulay2">i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random({1,1},ring·target·Phi));
  
86 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^4</code></pre>86 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^4</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i4·:·time·X·=·Phi^*·Y;89 <td>··············<pre><code·class="language-macaulay2">i4·:·time·X·=·Phi^*·Y;
90 ·--·used·2.99629s·(cpu);·2.45374s·(thread);·0s·(gc)90 ·--·used·3.77536s·(cpu);·3.0669s·(thread);·0s·(gc)
  
91 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension·2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)</code></pre>91 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension·2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i5·:·dim·X,·degree·X,·degrees·X94 <td>··············<pre><code·class="language-macaulay2">i5·:·dim·X,·degree·X,·degrees·X
  
95 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),95 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),
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27 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to27 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to
28 PP^2·x·PP^4)28 PP^2·x·PP^4)
29 i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random(29 i3·:·Y·=·projectiveVariety·ideal(random({1,1},ring·target·Phi),·random(
30 {1,1},ring·target·Phi));30 {1,1},ring·target·Phi));
  
31 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^431 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^2·x·PP^4
32 i4·:·time·X·=·Phi^*·Y;32 i4·:·time·X·=·Phi^*·Y;
33 ·--·used·2.99629s·(cpu);·2.45374s·(thread);·0s·(gc)33 ·--·used·3.77536s·(cpu);·3.0669s·(thread);·0s·(gc)
  
34 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension34 o4·:·ProjectiveVariety,·curve·in·PP^3·x·PP^2·x·PP^4·(subvariety·of·codimension
35 2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-35 2·in·threefold·in·PP^3·x·PP^2·x·PP^4·cut·out·by·12·hypersurfaces·of·multi-
36 degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)36 degrees·(0,0,2)^1·(0,1,1)^2·(1,0,1)^7·(1,1,0)^2·)
37 i5·:·dim·X,·degree·X,·degrees·X37 i5·:·dim·X,·degree·X,·degrees·X
  
38 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),38 o5·=·(1,·31,·{({0,·0,·2},·1),·({0,·0,·3},·4),·({0,·1,·1},·4),·({0,·4,·1},·1),
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88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i4·:·Z·=·source·Phi;89 <td>··············<pre><code·class="language-macaulay2">i4·:·Z·=·source·Phi;
  
90 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^7</code></pre>90 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^7</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Phi·Z;93 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Phi·Z;
94 ·--·used·0.353551s·(cpu);·0.111677s·(thread);·0s·(gc)94 ·--·used·0.307538s·(cpu);·0.119111s·(thread);·0s·(gc)
  
95 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^7</code></pre>95 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^7</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i6·:·dim·oo,·degree·oo,·degrees·oo98 <td>··············<pre><code·class="language-macaulay2">i6·:·dim·oo,·degree·oo,·degrees·oo
  
99 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})99 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})
  
100 o6·:·Sequence</code></pre>100 o6·:·Sequence</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)103 <td>··············<pre><code·class="language-macaulay2">i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)
104 ·--·used·1.66337s·(cpu);·1.0477s·(thread);·0s·(gc)104 ·--·used·1.96298s·(cpu);·1.10521s·(thread);·0s·(gc)
  
105 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of·multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^3·105 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of·multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^3·
  
106 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^7</code></pre>106 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^7</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i8·:·dim·oo,·degree·oo,·degrees·oo109 <td>··············<pre><code·class="language-macaulay2">i8·:·dim·oo,·degree·oo,·degrees·oo
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26 o3·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x26 o3·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x
27 PP^7·to·PP^7·x·PP^7)27 PP^7·to·PP^7·x·PP^7)
28 i4·:·Z·=·source·Phi;28 i4·:·Z·=·source·Phi;
  
29 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^729 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^7
30 i5·:·time·Phi·Z;30 i5·:·time·Phi·Z;
31 ·--·used·0.353551s·(cpu);·0.111677s·(thread);·0s·(gc)31 ·--·used·0.307538s·(cpu);·0.119111s·(thread);·0s·(gc)
  
32 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^732 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^7
33 i6·:·dim·oo,·degree·oo,·degrees·oo33 i6·:·dim·oo,·degree·oo,·degrees·oo
  
34 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})34 o6·=·(4,·80,·{({0,·2},·5),·({1,·1},·33),·({2,·0},·5)})
  
35 o6·:·Sequence35 o6·:·Sequence
36 i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)36 i7·:·time·Phi·(point·Z·+·point·Z·+·point·Z)
37 ·--·used·1.66337s·(cpu);·1.0477s·(thread);·0s·(gc)37 ·--·used·1.96298s·(cpu);·1.10521s·(thread);·0s·(gc)
  
38 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of38 o7·=·0-dimensional·subvariety·of·PP^7·x·PP^7·cut·out·by·22·hypersurfaces·of
39 multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^339 multi-degrees·(0,1)^5·(0,2)^3·(1,0)^5·(1,1)^6·(2,0)^3
  
40 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^740 o7·:·ProjectiveVariety,·0-dimensional·subvariety·of·PP^7·x·PP^7
41 i8·:·dim·oo,·degree·oo,·degrees·oo41 i8·:·dim·oo,·degree·oo,·degrees·oo
  
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88 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of88 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of
89 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·289 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·2
90 ·····------------------------------------------------------------------------90 ·····------------------------------------------------------------------------
91 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^1</code></pre>91 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^1</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degree(Phi,Strategy=>&quot;random·point&quot;)94 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degree(Phi,Strategy=>&quot;random·point&quot;)
95 ·--·used·3.64249s·(cpu);·1.93164s·(thread);·0s·(gc)95 ·--·used·4.80968s·(cpu);·2.62178s·(thread);·0s·(gc)
  
96 o4·=·2</code></pre>96 o4·=·2</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·time·degree(Phi,Strategy=>&quot;0-th·projective·degree&quot;)99 <td>··············<pre><code·class="language-macaulay2">i5·:·time·degree(Phi,Strategy=>&quot;0-th·projective·degree&quot;)
100 ·--·used·0.257288s·(cpu);·0.198792s·(thread);·0s·(gc)100 ·--·used·0.381043s·(cpu);·0.288489s·(thread);·0s·(gc)
  
101 o5·=·2</code></pre>101 o5·=·2</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i6·:·time·degree·Phi104 <td>··············<pre><code·class="language-macaulay2">i6·:·time·degree·Phi
105 ·--·used·0.177037s·(cpu);·0.179623s·(thread);·0s·(gc)105 ·--·used·0.338079s·(cpu);·0.261095s·(thread);·0s·(gc)
  
106 o6·=·2</code></pre>106 o6·=·2</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ········</table>108 ········</table>
109 ········<p>Note,·as·in·the·example·above,·that·calculation·times·may·vary·depending·on·the·strategy·used.</p>109 ········<p>Note,·as·in·the·example·above,·that·calculation·times·may·vary·depending·on·the·strategy·used.</p>
110 ······</div>110 ······</div>
111 ······<div>111 ······<div>
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28 o3·=·multi-rational·map·consisting·of·one·single·rational·map28 o3·=·multi-rational·map·consisting·of·one·single·rational·map
29 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of29 ·····source·variety:·threefold·in·PP^4·x·PP^4·cut·out·by·13·hypersurfaces·of
30 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·230 ·····target·variety:·hypersurface·in·PP^4·defined·by·a·form·of·degree·2
31 ·····------------------------------------------------------------------------31 ·····------------------------------------------------------------------------
32 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^132 ·····multi-degrees·(0,2)^1·(1,1)^3·(2,1)^8·(4,0)^1
33 i4·:·time·degree(Phi,Strategy=>"random·point")33 i4·:·time·degree(Phi,Strategy=>"random·point")
34 ·--·used·3.64249s·(cpu);·1.93164s·(thread);·0s·(gc)34 ·--·used·4.80968s·(cpu);·2.62178s·(thread);·0s·(gc)
  
35 o4·=·235 o4·=·2
36 i5·:·time·degree(Phi,Strategy=>"0-th·projective·degree")36 i5·:·time·degree(Phi,Strategy=>"0-th·projective·degree")
37 ·--·used·0.257288s·(cpu);·0.198792s·(thread);·0s·(gc)37 ·--·used·0.381043s·(cpu);·0.288489s·(thread);·0s·(gc)
  
38 o5·=·238 o5·=·2
39 i6·:·time·degree·Phi39 i6·:·time·degree·Phi
40 ·--·used·0.177037s·(cpu);·0.179623s·(thread);·0s·(gc)40 ·--·used·0.338079s·(cpu);·0.261095s·(thread);·0s·(gc)
  
41 o6·=·241 o6·=·2
42 Note,·as·in·the·example·above,·that·calculation·times·may·vary·depending·on·the42 Note,·as·in·the·example·above,·that·calculation·times·may·vary·depending·on·the
43 strategy·used.43 strategy·used.
44 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*44 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
45 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·multi-rational·map45 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·multi-rational·map
46 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8M_\x8a_\x8p_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map46 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8M_\x8a_\x8p_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map
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77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·last·graph·rationalMap·{f,g};78 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
79 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)</code></pre>79 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·time·degree·Phi82 <td>··············<pre><code·class="language-macaulay2">i3·:·time·degree·Phi
83 ·--·used·0.650485s·(cpu);·0.309242s·(thread);·0s·(gc)83 ·--·used·0.774467s·(cpu);·0.406148s·(thread);·0s·(gc)
  
84 o3·=·1</code></pre>84 o3·=·1</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ········</table>86 ········</table>
87 ······</div>87 ······</div>
88 ······<div>88 ······<div>
89 ········<h2>See·also</h2>89 ········<h2>See·also</h2>
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19 x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,19 x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,
20 x_3^2};20 x_3^2};
21 i2·:·Phi·=·last·graph·rationalMap·{f,g};21 i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
22 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to22 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to
23 PP^2·x·PP^4)23 PP^2·x·PP^4)
24 i3·:·time·degree·Phi24 i3·:·time·degree·Phi
25 ·--·used·0.650485s·(cpu);·0.309242s·(thread);·0s·(gc)25 ·--·used·0.774467s·(cpu);·0.406148s·(thread);·0s·(gc)
  
26 o3·=·126 o3·=·1
27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
28 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8,_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8)·--·degree·of·a·multi-rational·map·using·a28 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8,_\x8O_\x8p_\x8t_\x8i_\x8o_\x8n_\x8)·--·degree·of·a·multi-rational·map·using·a
29 ······probabilistic·approach29 ······probabilistic·approach
30 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map30 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·degree·of·a·rational·map
31 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·multi-rational31 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·multi-rational
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75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);76 <td>··············<pre><code·class="language-macaulay2">i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);
  
77 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4·x·PP^5)</code></pre>77 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^4·x·PP^5)</code></pre>
78 </td>··········</tr>78 </td>··········</tr>
79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i2·:·time·?·Phi80 <td>··············<pre><code·class="language-macaulay2">i2·:·time·?·Phi
81 ·--·used·0.00342738s·(cpu);·0.000130076s·(thread);·0s·(gc)81 ·--·used·0.00206852s·(cpu);·0.000144972s·(thread);·0s·(gc)
  
82 o2·=·multi-rational·map·consisting·of·2·rational·maps82 o2·=·multi-rational·map·consisting·of·2·rational·maps
83 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·983 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
84 ·····target·variety:·PP^4·x·PP^584 ·····target·variety:·PP^4·x·PP^5
85 ·····------------------------------------------------------------------------85 ·····------------------------------------------------------------------------
86 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8</code></pre>86 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·image·Phi;89 <td>··············<pre><code·class="language-macaulay2">i3·:·image·Phi;
  
90 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^5</code></pre>90 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^5</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·?·Phi93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·?·Phi
94 ·--·used·0.00218014s·(cpu);·0.000190787s·(thread);·0s·(gc)94 ·--·used·0.00045313s·(cpu);·0.000386114s·(thread);·0s·(gc)
  
95 o4·=·multi-rational·map·consisting·of·2·rational·maps95 o4·=·multi-rational·map·consisting·of·2·rational·maps
96 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·96 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
97 ·····target·variety:·PP^4·x·PP^597 ·····target·variety:·PP^4·x·PP^5
98 ·····dominance:·false98 ·····dominance:·false
99 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·</code></pre>99 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i5·:·time·describe·Phi102 <td>··············<pre><code·class="language-macaulay2">i5·:·time·describe·Phi
103 ·--·used·0.922696s·(cpu);·0.800107s·(thread);·0s·(gc)103 ·--·used·0.985958s·(cpu);·0.821223s·(thread);·0s·(gc)
  
104 o5·=·multi-rational·map·consisting·of·2·rational·maps104 o5·=·multi-rational·map·consisting·of·2·rational·maps
105 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·105 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
106 ·····target·variety:·PP^4·x·PP^5106 ·····target·variety:·PP^4·x·PP^5
107 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5107 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
108 ·····dominance:·false108 ·····dominance:·false
109 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·109 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
Offset 116, 15 lines modifiedOffset 116, 15 lines modified
116 ·····degree:·1116 ·····degree:·1
117 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]117 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]
118 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]118 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]
119 ·····coefficient·ring:·ZZ/65521</code></pre>119 ·····coefficient·ring:·ZZ/65521</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i6·:·time·?·Phi122 <td>··············<pre><code·class="language-macaulay2">i6·:·time·?·Phi
123 ·--·used·0.000120181s·(cpu);·0.000327323s·(thread);·0s·(gc)123 ·--·used·0.000116318s·(cpu);·0.000358022s·(thread);·0s·(gc)
  
124 o6·=·multi-rational·map·consisting·of·2·rational·maps124 o6·=·multi-rational·map·consisting·of·2·rational·maps
125 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·125 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
126 ·····target·variety:·PP^4·x·PP^5126 ·····target·variety:·PP^4·x·PP^5
127 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5127 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
128 ·····dominance:·false128 ·····dominance:·false
129 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·129 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8·
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17 ?·Phi·is·a·lite·version·of·describe·Phi.·The·latter·has·a·different·behavior17 ?·Phi·is·a·lite·version·of·describe·Phi.·The·latter·has·a·different·behavior
18 than·_\x8d_\x8e_\x8s_\x8c_\x8r_\x8i_\x8b_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8),·since·it·performs·computations.18 than·_\x8d_\x8e_\x8s_\x8c_\x8r_\x8i_\x8b_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8),·since·it·performs·computations.
19 i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);19 i1·:·Phi·=·multirationalMap·graph·rationalMap·PP_(ZZ/65521)^(1,4);
  
20 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x20 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x
21 PP^5·to·PP^4·x·PP^5)21 PP^5·to·PP^4·x·PP^5)
22 i2·:·time·?·Phi22 i2·:·time·?·Phi
23 ·--·used·0.00342738s·(cpu);·0.000130076s·(thread);·0s·(gc)23 ·--·used·0.00206852s·(cpu);·0.000144972s·(thread);·0s·(gc)
  
24 o2·=·multi-rational·map·consisting·of·2·rational·maps24 o2·=·multi-rational·map·consisting·of·2·rational·maps
25 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·925 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
26 ·····target·variety:·PP^4·x·PP^526 ·····target·variety:·PP^4·x·PP^5
27 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
28 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^828 ·····hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8
29 i3·:·image·Phi;29 i3·:·image·Phi;
  
30 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^530 o3·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^4·x·PP^5
31 i4·:·time·?·Phi31 i4·:·time·?·Phi
32 ·--·used·0.00218014s·(cpu);·0.000190787s·(thread);·0s·(gc)32 ·--·used·0.00045313s·(cpu);·0.000386114s·(thread);·0s·(gc)
  
33 o4·=·multi-rational·map·consisting·of·2·rational·maps33 o4·=·multi-rational·map·consisting·of·2·rational·maps
34 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·934 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
35 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^835 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8
36 ·····target·variety:·PP^4·x·PP^536 ·····target·variety:·PP^4·x·PP^5
37 ·····dominance:·false37 ·····dominance:·false
38 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces38 ·····image:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9·hypersurfaces
39 of·multi-degrees·(0,2)^1·(1,1)^839 of·multi-degrees·(0,2)^1·(1,1)^8
40 i5·:·time·describe·Phi40 i5·:·time·describe·Phi
41 ·--·used·0.922696s·(cpu);·0.800107s·(thread);·0s·(gc)41 ·--·used·0.985958s·(cpu);·0.821223s·(thread);·0s·(gc)
  
42 o5·=·multi-rational·map·consisting·of·2·rational·maps42 o5·=·multi-rational·map·consisting·of·2·rational·maps
43 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·943 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
44 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^844 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8
45 ·····target·variety:·PP^4·x·PP^545 ·····target·variety:·PP^4·x·PP^5
46 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^546 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
47 ·····dominance:·false47 ·····dominance:·false
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54 of·multi-degrees·(0,2)^1·(1,1)^854 of·multi-degrees·(0,2)^1·(1,1)^8
55 ·····multidegree:·{51,·51,·51,·51,·51}55 ·····multidegree:·{51,·51,·51,·51,·51}
56 ·····degree:·156 ·····degree:·1
57 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]57 ·····degree·sequence·(map·1/2):·[(1,0),·(0,2)]
58 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]58 ·····degree·sequence·(map·2/2):·[(0,1),·(2,0)]
59 ·····coefficient·ring:·ZZ/6552159 ·····coefficient·ring:·ZZ/65521
60 i6·:·time·?·Phi60 i6·:·time·?·Phi
61 ·--·used·0.000120181s·(cpu);·0.000327323s·(thread);·0s·(gc)61 ·--·used·0.000116318s·(cpu);·0.000358022s·(thread);·0s·(gc)
  
62 o6·=·multi-rational·map·consisting·of·2·rational·maps62 o6·=·multi-rational·map·consisting·of·2·rational·maps
63 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·963 ·····source·variety:·4-dimensional·subvariety·of·PP^4·x·PP^5·cut·out·by·9
64 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^864 hypersurfaces·of·multi-degrees·(0,2)^1·(1,1)^8
65 ·····target·variety:·PP^4·x·PP^565 ·····target·variety:·PP^4·x·PP^5
66 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^566 ·····base·locus:·empty·subscheme·of·PP^4·x·PP^5
67 ·····dominance:·false67 ·····dominance:·false
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84 o1·=·Phi84 o1·=·Phi
  
85 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>85 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·time·(Phi1,Phi2)·=·graph·Phi88 <td>··············<pre><code·class="language-macaulay2">i2·:·time·(Phi1,Phi2)·=·graph·Phi
89 ·--·used·0.213793s·(cpu);·0.0353456s·(thread);·0s·(gc)89 ·--·used·0.241683s·(cpu);·0.0513733s·(thread);·0s·(gc)
  
90 o2·=·(Phi1,·Phi2)90 o2·=·(Phi1,·Phi2)
  
91 o2·:·Sequence</code></pre>91 o2·:·Sequence</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi1;94 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi1;
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102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i4·:·Phi2;103 <td>··············<pre><code·class="language-macaulay2">i4·:·Phi2;
  
104 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)</code></pre>104 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i5·:·time·(Phi21,Phi22)·=·graph·Phi2107 <td>··············<pre><code·class="language-macaulay2">i5·:·time·(Phi21,Phi22)·=·graph·Phi2
108 ·--·used·0.261904s·(cpu);·0.0484107s·(thread);·0s·(gc)108 ·--·used·0.21007s·(cpu);·0.0572863s·(thread);·0s·(gc)
  
109 o5·=·(Phi21,·Phi22)109 o5·=·(Phi21,·Phi22)
  
110 o5·:·Sequence</code></pre>110 o5·:·Sequence</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i6·:·Phi21;113 <td>··············<pre><code·class="language-macaulay2">i6·:·Phi21;
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120 ··········<tr>120 ··········<tr>
121 <td>··············<pre><code·class="language-macaulay2">i7·:·Phi22;121 <td>··············<pre><code·class="language-macaulay2">i7·:·Phi22;
  
122 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)</code></pre>122 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i8·:·time·(Phi211,Phi212)·=·graph·Phi21125 <td>··············<pre><code·class="language-macaulay2">i8·:·time·(Phi211,Phi212)·=·graph·Phi21
126 ·--·used·0.338021s·(cpu);·0.134171s·(thread);·0s·(gc)126 ·--·used·0.497146s·(cpu);·0.172417s·(thread);·0s·(gc)
  
127 o8·=·(Phi211,·Phi212)127 o8·=·(Phi211,·Phi212)
  
128 o8·:·Sequence</code></pre>128 o8·:·Sequence</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i9·:·Phi211;131 <td>··············<pre><code·class="language-macaulay2">i9·:·Phi211;
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20 Phi)^-1·*·(last·graph·Phi)·==·Phi·are·always·satisfied.20 Phi)^-1·*·(last·graph·Phi)·==·Phi·are·always·satisfied.
21 i1·:·Phi·=·rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)21 i1·:·Phi·=·rationalMap(PP_(ZZ/333331)^(1,4),Dominant=>true)
  
22 o1·=·Phi22 o1·=·Phi
  
23 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)23 o1·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
24 i2·:·time·(Phi1,Phi2)·=·graph·Phi24 i2·:·time·(Phi1,Phi2)·=·graph·Phi
25 ·--·used·0.213793s·(cpu);·0.0353456s·(thread);·0s·(gc)25 ·--·used·0.241683s·(cpu);·0.0513733s·(thread);·0s·(gc)
  
26 o2·=·(Phi1,·Phi2)26 o2·=·(Phi1,·Phi2)
  
27 o2·:·Sequence27 o2·:·Sequence
28 i3·:·Phi1;28 i3·:·Phi1;
  
29 o3·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x29 o3·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
30 PP^5·to·PP^4)30 PP^5·to·PP^4)
31 i4·:·Phi2;31 i4·:·Phi2;
  
32 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of32 o4·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of
33 PP^4·x·PP^5·to·hypersurface·in·PP^5)33 PP^4·x·PP^5·to·hypersurface·in·PP^5)
34 i5·:·time·(Phi21,Phi22)·=·graph·Phi234 i5·:·time·(Phi21,Phi22)·=·graph·Phi2
35 ·--·used·0.261904s·(cpu);·0.0484107s·(thread);·0s·(gc)35 ·--·used·0.21007s·(cpu);·0.0572863s·(thread);·0s·(gc)
  
36 o5·=·(Phi21,·Phi22)36 o5·=·(Phi21,·Phi22)
  
37 o5·:·Sequence37 o5·:·Sequence
38 i6·:·Phi21;38 i6·:·Phi21;
  
39 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x39 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
40 PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)40 PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)
41 i7·:·Phi22;41 i7·:·Phi22;
  
42 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of42 o7·:·MultirationalMap·(dominant·rational·map·from·4-dimensional·subvariety·of
43 PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)43 PP^4·x·PP^5·x·PP^5·to·hypersurface·in·PP^5)
44 i8·:·time·(Phi211,Phi212)·=·graph·Phi2144 i8·:·time·(Phi211,Phi212)·=·graph·Phi21
45 ·--·used·0.338021s·(cpu);·0.134171s·(thread);·0s·(gc)45 ·--·used·0.497146s·(cpu);·0.172417s·(thread);·0s·(gc)
  
46 o8·=·(Phi211,·Phi212)46 o8·=·(Phi211,·Phi212)
  
47 o8·:·Sequence47 o8·:·Sequence
48 i9·:·Phi211;48 i9·:·Phi211;
  
49 o9·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x49 o9·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
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87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·Phi·=·rationalMap·{f,g};88 <td>··············<pre><code·class="language-macaulay2">i4·:·Phi·=·rationalMap·{f,g};
  
89 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)</code></pre>89 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Z·=·image·Phi;92 <td>··············<pre><code·class="language-macaulay2">i5·:·time·Z·=·image·Phi;
93 ·--·used·0.39011s·(cpu);·0.13779s·(thread);·0s·(gc)93 ·--·used·0.326121s·(cpu);·0.129263s·(thread);·0s·(gc)
  
94 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4</code></pre>94 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i6·:·dim·Z,·degree·Z,·degrees·Z97 <td>··············<pre><code·class="language-macaulay2">i6·:·dim·Z,·degree·Z,·degrees·Z
  
98 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})98 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})
Offset 103, 15 lines modifiedOffset 103, 15 lines modified
103 o6·:·Sequence</code></pre>103 o6·:·Sequence</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ········</table>105 ········</table>
106 ········<p>Alternatively,·the·calculation·can·be·performed·using·the·Segre·embedding·as·follows:</p>106 ········<p>Alternatively,·the·calculation·can·be·performed·using·the·Segre·embedding·as·follows:</p>
107 ········<table·class="examples">107 ········<table·class="examples">
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,&quot;F4&quot;);109 <td>··············<pre><code·class="language-macaulay2">i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,&quot;F4&quot;);
110 ·--·used·7.74487s·(cpu);·2.28877s·(thread);·0s·(gc)110 ·--·used·8.40908s·(cpu);·2.37569s·(thread);·0s·(gc)
  
111 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4</code></pre>111 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(Z·==·Z')</code></pre>114 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(Z·==·Z')</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ········</table>116 ········</table>
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24 3*x_2^2+2*x_1*x_3+x_0*x_4,·2*x_1*x_2-2*x_0*x_3,·-x_1^2+x_0*x_2};24 3*x_2^2+2*x_1*x_3+x_0*x_4,·2*x_1*x_2-2*x_0*x_3,·-x_1^2+x_0*x_2};
  
25 o3·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)25 o3·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)
26 i4·:·Phi·=·rationalMap·{f,g};26 i4·:·Phi·=·rationalMap·{f,g};
  
27 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)27 o4·:·MultirationalMap·(rational·map·from·PP^4·to·PP^7·x·PP^4)
28 i5·:·time·Z·=·image·Phi;28 i5·:·time·Z·=·image·Phi;
29 ·--·used·0.39011s·(cpu);·0.13779s·(thread);·0s·(gc)29 ·--·used·0.326121s·(cpu);·0.129263s·(thread);·0s·(gc)
  
30 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^430 o5·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4
31 i6·:·dim·Z,·degree·Z,·degrees·Z31 i6·:·dim·Z,·degree·Z,·degrees·Z
  
32 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})32 o6·=·(4,·151,·{({1,·1},·4),·({1,·2},·3),·({2,·0},·5),·({2,·1},·13)})
  
33 o6·:·Sequence33 o6·:·Sequence
34 Alternatively,·the·calculation·can·be·performed·using·the·Segre·embedding·as34 Alternatively,·the·calculation·can·be·performed·using·the·Segre·embedding·as
35 follows:35 follows:
36 i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,"F4");36 i7·:·time·Z'·=·projectiveVariety·(map·segre·target·Phi)·image(segre·Phi,"F4");
37 ·--·used·7.74487s·(cpu);·2.28877s·(thread);·0s·(gc)37 ·--·used·8.40908s·(cpu);·2.37569s·(thread);·0s·(gc)
  
38 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^438 o7·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^7·x·PP^4
39 i8·:·assert(Z·==·Z')39 i8·:·assert(Z·==·Z')
40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
41 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·direct·image·via·a·multi-41 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·direct·image·via·a·multi-
42 ······rational·map42 ······rational·map
43 ····*·_\x8i_\x8m_\x8a_\x8g_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·closure·of·the·image·of·a·rational·map43 ····*·_\x8i_\x8m_\x8a_\x8g_\x8e_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·closure·of·the·image·of·a·rational·map
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78 <td>··············<pre><code·class="language-macaulay2">i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational·normal·curve·of·degree·678 <td>··············<pre><code·class="language-macaulay2">i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational·normal·curve·of·degree·6
79 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal·PP_K([6],2));79 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal·PP_K([6],2));
  
80 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))</code></pre>80 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ··········<tr>82 ··········<tr>
83 <td>··············<pre><code·class="language-macaulay2">i3·:·time·Psi·=·inverse2·Phi;83 <td>··············<pre><code·class="language-macaulay2">i3·:·time·Psi·=·inverse2·Phi;
84 ·--·used·0.491906s·(cpu);·0.261695s·(thread);·0s·(gc)84 ·--·used·0.580318s·(cpu);·0.301422s·(thread);·0s·(gc)
  
85 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)</code></pre>85 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·assert(Phi·*·Psi·==·1)</code></pre>88 <td>··············<pre><code·class="language-macaulay2">i4·:·assert(Phi·*·Psi·==·1)</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi'·=·Phi·||·Phi;91 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi'·=·Phi·||·Phi;
  
92 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))</code></pre>92 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Psi'·=·inverse2·Phi';95 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Psi'·=·inverse2·Phi';
96 ·--·used·1.26599s·(cpu);·0.922462s·(thread);·0s·(gc)96 ·--·used·1.32228s·(cpu);·0.896273s·(thread);·0s·(gc)
  
97 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)</code></pre>97 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i7·:·assert(Phi'·*·Psi'·==·1)</code></pre>100 <td>··············<pre><code·class="language-macaulay2">i7·:·assert(Phi'·*·Psi'·==·1)</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ········</table>102 ········</table>
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25 i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational25 i2·:·--·map·defined·by·the·cubics·through·the·secant·variety·to·the·rational
26 normal·curve·of·degree·626 normal·curve·of·degree·6
27 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal27 ·····Phi·=·multirationalMap·rationalMap(ring·PP_K^6,ring·GG_K(2,4),gens·ideal
28 PP_K([6],2));28 PP_K([6],2));
  
29 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))29 o2·:·MultirationalMap·(rational·map·from·PP^6·to·GG(2,4))
30 i3·:·time·Psi·=·inverse2·Phi;30 i3·:·time·Psi·=·inverse2·Phi;
31 ·--·used·0.491906s·(cpu);·0.261695s·(thread);·0s·(gc)31 ·--·used·0.580318s·(cpu);·0.301422s·(thread);·0s·(gc)
  
32 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)32 o3·:·MultirationalMap·(birational·map·from·GG(2,4)·to·PP^6)
33 i4·:·assert(Phi·*·Psi·==·1)33 i4·:·assert(Phi·*·Psi·==·1)
34 i5·:·Phi'·=·Phi·||·Phi;34 i5·:·Phi'·=·Phi·||·Phi;
  
35 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))35 o5·:·MultirationalMap·(rational·map·from·PP^6·x·PP^6·to·GG(2,4)·x·GG(2,4))
36 i6·:·time·Psi'·=·inverse2·Phi';36 i6·:·time·Psi'·=·inverse2·Phi';
37 ·--·used·1.26599s·(cpu);·0.922462s·(thread);·0s·(gc)37 ·--·used·1.32228s·(cpu);·0.896273s·(thread);·0s·(gc)
  
38 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)38 o6·:·MultirationalMap·(birational·map·from·GG(2,4)·x·GG(2,4)·to·PP^6·x·PP^6)
39 i7·:·assert(Phi'·*·Psi'·==·1)39 i7·:·assert(Phi'·*·Psi'·==·1)
40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
41 ····*·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·inverse·of·a·birational·map41 ····*·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·inverse·of·a·birational·map
42 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8·_\x8<_\x8=_\x8=_\x8>_\x8·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·--·equality·of·multi-rational·maps42 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8·_\x8<_\x8=_\x8=_\x8>_\x8·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p·--·equality·of·multi-rational·maps
43 ······with·checks·on·internal·data43 ······with·checks·on·internal·data
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84 <td>··············<pre><code·class="language-macaulay2">i2·:·--·we·see·Phi·as·a·dominant·map84 <td>··············<pre><code·class="language-macaulay2">i2·:·--·we·see·Phi·as·a·dominant·map
85 ·····Phi·=·rationalMap(Phi,image·Phi);85 ·····Phi·=·rationalMap(Phi,image·Phi);
  
86 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>86 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·time·inverse·Phi;89 <td>··············<pre><code·class="language-macaulay2">i3·:·time·inverse·Phi;
90 ·--·used·0.285898s·(cpu);·0.0614978s·(thread);·0s·(gc)90 ·--·used·0.295307s·(cpu);·0.0646031s·(thread);·0s·(gc)
  
91 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)</code></pre>91 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·Psi·=·last·graph·Phi;94 <td>··············<pre><code·class="language-macaulay2">i4·:·Psi·=·last·graph·Phi;
  
95 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)</code></pre>95 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·hypersurface·in·PP^5)</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i5·:·time·inverse·Psi;98 <td>··············<pre><code·class="language-macaulay2">i5·:·time·inverse·Psi;
99 ·--·used·0.556178s·(cpu);·0.118223s·(thread);·0s·(gc)99 ·--·used·0.573546s·(cpu);·0.13748s·(thread);·0s·(gc)
  
100 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)</code></pre>100 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i6·:·Eta·=·first·graph·Psi;103 <td>··············<pre><code·class="language-macaulay2">i6·:·Eta·=·first·graph·Psi;
  
104 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)</code></pre>104 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i7·:·time·inverse·Eta;107 <td>··············<pre><code·class="language-macaulay2">i7·:·time·inverse·Eta;
108 ·--·used·0.856787s·(cpu);·0.296816s·(thread);·0s·(gc)108 ·--·used·0.835829s·(cpu);·0.335785s·(thread);·0s·(gc)
  
109 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)</code></pre>109 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)</code></pre>112 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
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25 o1·:·MultirationalMap·(rational·map·from·PP^4·to·PP^5)25 o1·:·MultirationalMap·(rational·map·from·PP^4·to·PP^5)
26 i2·:·--·we·see·Phi·as·a·dominant·map26 i2·:·--·we·see·Phi·as·a·dominant·map
27 ·····Phi·=·rationalMap(Phi,image·Phi);27 ·····Phi·=·rationalMap(Phi,image·Phi);
  
28 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)28 o2·:·MultirationalMap·(dominant·rational·map·from·PP^4·to·hypersurface·in·PP^5)
29 i3·:·time·inverse·Phi;29 i3·:·time·inverse·Phi;
30 ·--·used·0.285898s·(cpu);·0.0614978s·(thread);·0s·(gc)30 ·--·used·0.295307s·(cpu);·0.0646031s·(thread);·0s·(gc)
  
31 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)31 o3·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·PP^4)
32 i4·:·Psi·=·last·graph·Phi;32 i4·:·Psi·=·last·graph·Phi;
  
33 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x33 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
34 PP^5·to·hypersurface·in·PP^5)34 PP^5·to·hypersurface·in·PP^5)
35 i5·:·time·inverse·Psi;35 i5·:·time·inverse·Psi;
36 ·--·used·0.556178s·(cpu);·0.118223s·(thread);·0s·(gc)36 ·--·used·0.573546s·(cpu);·0.13748s·(thread);·0s·(gc)
  
37 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-37 o5·:·MultirationalMap·(birational·map·from·hypersurface·in·PP^5·to·4-
38 dimensional·subvariety·of·PP^4·x·PP^5)38 dimensional·subvariety·of·PP^4·x·PP^5)
39 i6·:·Eta·=·first·graph·Psi;39 i6·:·Eta·=·first·graph·Psi;
  
40 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x40 o6·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
41 PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)41 PP^5·x·PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5)
42 i7·:·time·inverse·Eta;42 i7·:·time·inverse·Eta;
43 ·--·used·0.856787s·(cpu);·0.296816s·(thread);·0s·(gc)43 ·--·used·0.835829s·(cpu);·0.335785s·(thread);·0s·(gc)
  
44 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x44 o7·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
45 PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)45 PP^5·to·4-dimensional·subvariety·of·PP^4·x·PP^5·x·PP^5)
46 i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)46 i8·:·assert(Phi·*·Phi^-1·==·1·and·Phi^-1·*·Phi·==·1)
47 i9·:·assert(Psi·*·Psi^-1·==·1·and·Psi^-1·*·Psi·==·1)47 i9·:·assert(Psi·*·Psi^-1·==·1·and·Psi^-1·*·Psi·==·1)
48 i10·:·assert(Eta·*·Eta^-1·==·1·and·Eta^-1·*·Eta·==·1)48 i10·:·assert(Eta·*·Eta^-1·==·1·and·Eta^-1·*·Eta·==·1)
49 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*49 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
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79 ··········<tr>79 ··········<tr>
80 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi·=·rationalMap·{f,f};80 <td>··············<pre><code·class="language-macaulay2">i3·:·Phi·=·rationalMap·{f,f};
  
81 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)</code></pre>81 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isIsomorphism·Phi84 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isIsomorphism·Phi
85 ·--·used·0.00236091s·(cpu);·5.147e-06s·(thread);·0s·(gc)85 ·--·used·0.00148525s·(cpu);·9.017e-06s·(thread);·0s·(gc)
  
86 o4·=·false</code></pre>86 o4·=·false</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·Psi·=·first·graph·Phi;89 <td>··············<pre><code·class="language-macaulay2">i5·:·Psi·=·first·graph·Phi;
  
90 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to·PP^3)</code></pre>90 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to·PP^3)</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i6·:·time·isIsomorphism·Psi93 <td>··············<pre><code·class="language-macaulay2">i6·:·time·isIsomorphism·Psi
94 ·--·used·0.696023s·(cpu);·0.196833s·(thread);·0s·(gc)94 ·--·used·0.671814s·(cpu);·0.170447s·(thread);·0s·(gc)
  
95 o6·=·false</code></pre>95 o6·=·false</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i7·:·Eta·=·first·graph·Psi;98 <td>··············<pre><code·class="language-macaulay2">i7·:·Eta·=·first·graph·Psi;
  
99 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x·PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)</code></pre>99 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x·PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i8·:·time·isIsomorphism·Eta102 <td>··············<pre><code·class="language-macaulay2">i8·:·time·isIsomorphism·Eta
103 ·--·used·1.54989s·(cpu);·0.688024s·(thread);·0s·(gc)103 ·--·used·1.67481s·(cpu);·0.719784s·(thread);·0s·(gc)
  
104 o8·=·true</code></pre>104 o8·=·true</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i9·:·assert(o8·and·(not·o6)·and·(not·o4))</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i9·:·assert(o8·and·(not·o6)·and·(not·o4))</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ········</table>109 ········</table>
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18 ·····ZZ/33331[a..d];·f·=·rationalMap·{c^2-b*d,b*c-a*d,b^2-a*c};18 ·····ZZ/33331[a..d];·f·=·rationalMap·{c^2-b*d,b*c-a*d,b^2-a*c};
  
19 o2·:·RationalMap·(quadratic·rational·map·from·PP^3·to·PP^2)19 o2·:·RationalMap·(quadratic·rational·map·from·PP^3·to·PP^2)
20 i3·:·Phi·=·rationalMap·{f,f};20 i3·:·Phi·=·rationalMap·{f,f};
  
21 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)21 o3·:·MultirationalMap·(rational·map·from·PP^3·to·PP^2·x·PP^2)
22 i4·:·time·isIsomorphism·Phi22 i4·:·time·isIsomorphism·Phi
23 ·--·used·0.00236091s·(cpu);·5.147e-06s·(thread);·0s·(gc)23 ·--·used·0.00148525s·(cpu);·9.017e-06s·(thread);·0s·(gc)
  
24 o4·=·false24 o4·=·false
25 i5·:·Psi·=·first·graph·Phi;25 i5·:·Psi·=·first·graph·Phi;
  
26 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to26 o5·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·to
27 PP^3)27 PP^3)
28 i6·:·time·isIsomorphism·Psi28 i6·:·time·isIsomorphism·Psi
29 ·--·used·0.696023s·(cpu);·0.196833s·(thread);·0s·(gc)29 ·--·used·0.671814s·(cpu);·0.170447s·(thread);·0s·(gc)
  
30 o6·=·false30 o6·=·false
31 i7·:·Eta·=·first·graph·Psi;31 i7·:·Eta·=·first·graph·Psi;
  
32 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x32 o7·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·x·PP^2·x
33 PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)33 PP^3·to·threefold·in·PP^3·x·PP^2·x·PP^2)
34 i8·:·time·isIsomorphism·Eta34 i8·:·time·isIsomorphism·Eta
35 ·--·used·1.54989s·(cpu);·0.688024s·(thread);·0s·(gc)35 ·--·used·1.67481s·(cpu);·0.719784s·(thread);·0s·(gc)
  
36 o8·=·true36 o8·=·true
37 i9·:·assert(o8·and·(not·o6)·and·(not·o4))37 i9·:·assert(o8·and·(not·o6)·and·(not·o4))
38 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*38 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
39 ····*·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·inverse·of·a·birational·map39 ····*·_\x8i_\x8n_\x8v_\x8e_\x8r_\x8s_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·inverse·of·a·birational·map
40 ····*·_\x8i_\x8s_\x8M_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·multi-rational·map·is·a40 ····*·_\x8i_\x8s_\x8M_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·multi-rational·map·is·a
41 ······morphism41 ······morphism
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·rationalMap·{f,g};77 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·rationalMap·{f,g};
  
78 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^4·x·PP^2)</code></pre>78 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·to·PP^4·x·PP^2)</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isMorphism·Phi81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·isMorphism·Phi
82 ·--·used·0.177148s·(cpu);·0.139134s·(thread);·0s·(gc)82 ·--·used·0.212445s·(cpu);·0.154119s·(thread);·0s·(gc)
  
83 o3·=·false</code></pre>83 o3·=·false</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Psi·=·first·graph·Phi;86 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Psi·=·first·graph·Phi;
87 ·--·used·0.362469s·(cpu);·0.0960295s·(thread);·0s·(gc)87 ·--·used·0.327153s·(cpu);·0.0990364s·(thread);·0s·(gc)
  
88 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)</code></pre>88 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isMorphism·Psi91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isMorphism·Psi
92 ·--·used·2.83708s·(cpu);·2.48281s·(thread);·0s·(gc)92 ·--·used·2.99672s·(cpu);·2.47521s·(thread);·0s·(gc)
  
93 o5·=·true</code></pre>93 o5·=·true</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i6·:·assert((not·o3)·and·o5)</code></pre>96 <td>··············<pre><code·class="language-macaulay2">i6·:·assert((not·o3)·and·o5)</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ········</table>98 ········</table>
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18 i1·:·ZZ/300007[a..e],·f·=·first·graph·rationalMap·ideal(c^2-b*d,b*c-a*d,b^2-18 i1·:·ZZ/300007[a..e],·f·=·first·graph·rationalMap·ideal(c^2-b*d,b*c-a*d,b^2-
19 a*c,e),·g·=·rationalMap·submatrix(matrix·f,{0..2});19 a*c,e),·g·=·rationalMap·submatrix(matrix·f,{0..2});
20 i2·:·Phi·=·rationalMap·{f,g};20 i2·:·Phi·=·rationalMap·{f,g};
  
21 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x21 o2·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x
22 PP^7·to·PP^4·x·PP^2)22 PP^7·to·PP^4·x·PP^2)
23 i3·:·time·isMorphism·Phi23 i3·:·time·isMorphism·Phi
24 ·--·used·0.177148s·(cpu);·0.139134s·(thread);·0s·(gc)24 ·--·used·0.212445s·(cpu);·0.154119s·(thread);·0s·(gc)
  
25 o3·=·false25 o3·=·false
26 i4·:·time·Psi·=·first·graph·Phi;26 i4·:·time·Psi·=·first·graph·Phi;
27 ·--·used·0.362469s·(cpu);·0.0960295s·(thread);·0s·(gc)27 ·--·used·0.327153s·(cpu);·0.0990364s·(thread);·0s·(gc)
  
28 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x28 o4·:·MultirationalMap·(birational·map·from·4-dimensional·subvariety·of·PP^4·x
29 PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)29 PP^7·x·PP^4·x·PP^2·to·4-dimensional·subvariety·of·PP^4·x·PP^7)
30 i5·:·time·isMorphism·Psi30 i5·:·time·isMorphism·Psi
31 ·--·used·2.83708s·(cpu);·2.48281s·(thread);·0s·(gc)31 ·--·used·2.99672s·(cpu);·2.47521s·(thread);·0s·(gc)
  
32 o5·=·true32 o5·=·true
33 i6·:·assert((not·o3)·and·o5)33 i6·:·assert((not·o3)·and·o5)
34 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*34 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
35 ····*·_\x8i_\x8s_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·birational·map·is·an35 ····*·_\x8i_\x8s_\x8I_\x8s_\x8o_\x8m_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·birational·map·is·an
36 ······isomorphism36 ······isomorphism
37 ····*·_\x8i_\x8s_\x8M_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·rational·map·is·a·morphism37 ····*·_\x8i_\x8s_\x8M_\x8o_\x8r_\x8p_\x8h_\x8i_\x8s_\x8m_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·whether·a·rational·map·is·a·morphism
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74 ··········<tr>74 ··········<tr>
75 <td>··············<pre><code·class="language-macaulay2">i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·775 <td>··············<pre><code·class="language-macaulay2">i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·7
  
76 o2·:·ProjectiveVariety,·curve·in·PP^7</code></pre>76 o2·:·ProjectiveVariety,·curve·in·PP^7</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·linearlyNormalEmbedding·X;79 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·linearlyNormalEmbedding·X;
80 ·--·used·0.00500658s·(cpu);·0.00768908s·(thread);·0s·(gc)80 ·--·used·0.00802597s·(cpu);·0.008841s·(thread);·0s·(gc)
  
81 o3·:·MultirationalMap·(automorphism·of·X)</code></pre>81 o3·:·MultirationalMap·(automorphism·of·X)</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an·isomorphic·projection·of·X·in·PP^384 <td>··············<pre><code·class="language-macaulay2">i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an·isomorphic·projection·of·X·in·PP^3
  
85 o4·:·ProjectiveVariety,·curve·in·PP^3</code></pre>85 o4·:·ProjectiveVariety,·curve·in·PP^3</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i5·:·time·g·=·linearlyNormalEmbedding·Y;88 <td>··············<pre><code·class="language-macaulay2">i5·:·time·g·=·linearlyNormalEmbedding·Y;
89 ·--·used·0.383948s·(cpu);·0.345654s·(thread);·0s·(gc)89 ·--·used·0.410114s·(cpu);·0.35555s·(thread);·0s·(gc)
  
90 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)</code></pre>90 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(isIsomorphism·g)</code></pre>93 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(isIsomorphism·g)</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
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14 ············is·a·linear·projection14 ············is·a·linear·projection
15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 i1·:·K·=·ZZ/333331;16 i1·:·K·=·ZZ/333331;
17 i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·717 i2·:·X·=·PP_K^(1,7);·--·rational·normal·curve·of·degree·7
  
18 o2·:·ProjectiveVariety,·curve·in·PP^718 o2·:·ProjectiveVariety,·curve·in·PP^7
19 i3·:·time·f·=·linearlyNormalEmbedding·X;19 i3·:·time·f·=·linearlyNormalEmbedding·X;
20 ·--·used·0.00500658s·(cpu);·0.00768908s·(thread);·0s·(gc)20 ·--·used·0.00802597s·(cpu);·0.008841s·(thread);·0s·(gc)
  
21 o3·:·MultirationalMap·(automorphism·of·X)21 o3·:·MultirationalMap·(automorphism·of·X)
22 i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an22 i4·:·Y·=·(rationalMap·{for·i·to·3·list·random(1,ring·ambient·X)})·X;·--·an
23 isomorphic·projection·of·X·in·PP^323 isomorphic·projection·of·X·in·PP^3
  
24 o4·:·ProjectiveVariety,·curve·in·PP^324 o4·:·ProjectiveVariety,·curve·in·PP^3
25 i5·:·time·g·=·linearlyNormalEmbedding·Y;25 i5·:·time·g·=·linearlyNormalEmbedding·Y;
26 ·--·used·0.383948s·(cpu);·0.345654s·(thread);·0s·(gc)26 ·--·used·0.410114s·(cpu);·0.35555s·(thread);·0s·(gc)
  
27 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)27 o5·:·MultirationalMap·(birational·map·from·Y·to·curve·in·PP^7)
28 i6·:·assert(isIsomorphism·g)28 i6·:·assert(isIsomorphism·g)
29 i7·:·describe·g29 i7·:·describe·g
  
30 o7·=·multi-rational·map·consisting·of·one·single·rational·map30 o7·=·multi-rational·map·consisting·of·one·single·rational·map
31 ·····source·variety:·curve·in·PP^3·cut·out·by·6·hypersurfaces·of·degree·431 ·····source·variety:·curve·in·PP^3·cut·out·by·6·hypersurfaces·of·degree·4
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77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·last·graph·rationalMap·{f,g};78 <td>··············<pre><code·class="language-macaulay2">i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
79 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)</code></pre>79 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to·PP^2·x·PP^4)</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·time·multidegree·Phi82 <td>··············<pre><code·class="language-macaulay2">i3·:·time·multidegree·Phi
83 ·--·used·0.64645s·(cpu);·0.319383s·(thread);·0s·(gc)83 ·--·used·0.627121s·(cpu);·0.326963s·(thread);·0s·(gc)
  
84 o3·=·{66,·46,·31,·20}84 o3·=·{66,·46,·31,·20}
  
85 o3·:·List</code></pre>85 o3·:·List</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·(degree·source·Phi,degree·image·Phi)88 <td>··············<pre><code·class="language-macaulay2">i4·:·(degree·source·Phi,degree·image·Phi)
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20 x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,20 x_1^2-x_0*x_2},·g·=·rationalMap·{x_1^2-x_0*x_2,·x_0*x_3,·x_1*x_3,·x_2*x_3,
21 x_3^2};21 x_3^2};
22 i2·:·Phi·=·last·graph·rationalMap·{f,g};22 i2·:·Phi·=·last·graph·rationalMap·{f,g};
  
23 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to23 o2·:·MultirationalMap·(rational·map·from·threefold·in·PP^3·x·PP^2·x·PP^4·to
24 PP^2·x·PP^4)24 PP^2·x·PP^4)
25 i3·:·time·multidegree·Phi25 i3·:·time·multidegree·Phi
26 ·--·used·0.64645s·(cpu);·0.319383s·(thread);·0s·(gc)26 ·--·used·0.627121s·(cpu);·0.326963s·(thread);·0s·(gc)
  
27 o3·=·{66,·46,·31,·20}27 o3·=·{66,·46,·31,·20}
  
28 o3·:·List28 o3·:·List
29 i4·:·(degree·source·Phi,degree·image·Phi)29 i4·:·(degree·source·Phi,degree·image·Phi)
  
30 o4·=·(66,·20)30 o4·=·(66,·20)
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);77 <td>··············<pre><code·class="language-macaulay2">i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);
  
78 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^5)</code></pre>78 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x·PP^5·to·PP^5)</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)81 <td>··············<pre><code·class="language-macaulay2">i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)
 82 ·--·used·0.00178884s·(cpu);·0.00116877s·(thread);·0s·(gc)
 83 ·--·used·0.166747s·(cpu);·0.116878s·(thread);·0s·(gc)
82 ·--·used·0.00397065s·(cpu);·0.00103293s·(thread);·0s·(gc)84 ·--·used·0.209552s·(cpu);·0.137466s·(thread);·0s·(gc)
83 ·--·used·0.140164s·(cpu);·0.0956171s·(thread);·0s·(gc)85 ·--·used·0.160699s·(cpu);·0.0976585s·(thread);·0s·(gc)
84 ·--·used·0.168172s·(cpu);·0.126466s·(thread);·0s·(gc) 
85 ·--·used·0.144435s·(cpu);·0.0993572s·(thread);·0s·(gc) 
86 ·--·used·0.115045s·(cpu);·0.0734805s·(thread);·0s·(gc)86 ·--·used·0.14151s·(cpu);·0.0880439s·(thread);·0s·(gc)
  
87 o2·=·{51,·28,·14,·6,·2}87 o2·=·{51,·28,·14,·6,·2}
  
88 o2·:·List</code></pre>88 o2·:·List</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·time·assert(oo·==·multidegree·Phi)91 <td>··············<pre><code·class="language-macaulay2">i3·:·time·assert(oo·==·multidegree·Phi)
92 ·--·used·0.142749s·(cpu);·0.0455335s·(thread);·0s·(gc)</code></pre>92 ·--·used·0.270609s·(cpu);·0.0620393s·(thread);·0s·(gc)</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ········</table>94 ········</table>
95 ······</div>95 ······</div>
96 ······<div>96 ······<div>
97 ········<h2>References</h2>97 ········<h2>References</h2>
98 ArXiv·preprint:·<a·href="https://arxiv.org/abs/2101.04503">Computations·with·rational·maps·between·multi-projective·varieties</a>.······</div>98 ArXiv·preprint:·<a·href="https://arxiv.org/abs/2101.04503">Computations·with·rational·maps·between·multi-projective·varieties</a>.······</div>
99 ······<div>99 ······<div>
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18 This·is·calculated·by·means·of·the·inverse·image·of·an·appropriate·random18 This·is·calculated·by·means·of·the·inverse·image·of·an·appropriate·random
19 subvariety·of·the·target.19 subvariety·of·the·target.
20 i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);20 i1·:·Phi·=·last·graph·rationalMap·PP_(ZZ/300007)^(1,4);
  
21 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x21 o1·:·MultirationalMap·(rational·map·from·4-dimensional·subvariety·of·PP^4·x
22 PP^5·to·PP^5)22 PP^5·to·PP^5)
23 i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)23 i2·:·for·i·in·{4,3,2,1,0}·list·time·multidegree(i,Phi)
 24 ·--·used·0.00178884s·(cpu);·0.00116877s·(thread);·0s·(gc)
 25 ·--·used·0.166747s·(cpu);·0.116878s·(thread);·0s·(gc)
24 ·--·used·0.00397065s·(cpu);·0.00103293s·(thread);·0s·(gc)26 ·--·used·0.209552s·(cpu);·0.137466s·(thread);·0s·(gc)
25 ·--·used·0.140164s·(cpu);·0.0956171s·(thread);·0s·(gc)27 ·--·used·0.160699s·(cpu);·0.0976585s·(thread);·0s·(gc)
26 ·--·used·0.168172s·(cpu);·0.126466s·(thread);·0s·(gc) 
27 ·--·used·0.144435s·(cpu);·0.0993572s·(thread);·0s·(gc) 
28 ·--·used·0.115045s·(cpu);·0.0734805s·(thread);·0s·(gc)28 ·--·used·0.14151s·(cpu);·0.0880439s·(thread);·0s·(gc)
  
29 o2·=·{51,·28,·14,·6,·2}29 o2·=·{51,·28,·14,·6,·2}
  
30 o2·:·List30 o2·:·List
31 i3·:·time·assert(oo·==·multidegree·Phi)31 i3·:·time·assert(oo·==·multidegree·Phi)
32 ·--·used·0.142749s·(cpu);·0.0455335s·(thread);·0s·(gc)32 ·--·used·0.270609s·(cpu);·0.0620393s·(thread);·0s·(gc)
33 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*33 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
34 ArXiv·preprint:·_\x8C_\x8o_\x8m_\x8p_\x8u_\x8t_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p_\x8s_\x8·_\x8b_\x8e_\x8t_\x8w_\x8e_\x8e_\x8n_\x8·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8-_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e34 ArXiv·preprint:·_\x8C_\x8o_\x8m_\x8p_\x8u_\x8t_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8·_\x8m_\x8a_\x8p_\x8s_\x8·_\x8b_\x8e_\x8t_\x8w_\x8e_\x8e_\x8n_\x8·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8-_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e
35 _\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s.35 _\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s.
36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
37 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·multi-rational37 ····*·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8r_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·multi-rational
38 ······map38 ······map
39 ····*·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8s_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·rational·map39 ····*·_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8D_\x8e_\x8g_\x8r_\x8e_\x8e_\x8s_\x8(_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8p_\x8)·--·projective·degrees·of·a·rational·map
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76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i2·:·X·=·PP_K^({1,1,2},{3,2,3});77 <td>··············<pre><code·class="language-macaulay2">i2·:·X·=·PP_K^({1,1,2},{3,2,3});
  
78 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^9</code></pre>78 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^9</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·p·:=·point·X81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·p·:=·point·X
82 ·--·used·0.241285s·(cpu);·0.0300762s·(thread);·0s·(gc)82 ·--·used·0.207929s·(cpu);·0.0323835s·(thread);·0s·(gc)
  
83 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],[3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])83 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],[3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])
  
84 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9</code></pre>84 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·Y·=·random({2,1,2},X);87 <td>··············<pre><code·class="language-macaulay2">i4·:·Y·=·random({2,1,2},X);
  
88 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^9</code></pre>88 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^9</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·q·=·point·Y91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·q·=·point·Y
92 ·--·used·1.25059s·(cpu);·0.746661s·(thread);·0s·(gc)92 ·--·used·1.39438s·(cpu);·0.750503s·(thread);·0s·(gc)
  
93 o5·=·q93 o5·=·q
  
94 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9</code></pre>94 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))</code></pre>97 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))</code></pre>
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15 ··········o·a·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8-_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y,·a·random·rational·point·on·$X$15 ··········o·a·_\x8m_\x8u_\x8l_\x8t_\x8i_\x8-_\x8p_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y,·a·random·rational·point·on·$X$
16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
17 i1·:·K·=·ZZ/1000003;17 i1·:·K·=·ZZ/1000003;
18 i2·:·X·=·PP_K^({1,1,2},{3,2,3});18 i2·:·X·=·PP_K^({1,1,2},{3,2,3});
  
19 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^919 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^3·x·PP^2·x·PP^9
20 i3·:·time·p·:=·point·X20 i3·:·time·p·:=·point·X
21 ·--·used·0.241285s·(cpu);·0.0300762s·(thread);·0s·(gc)21 ·--·used·0.207929s·(cpu);·0.0323835s·(thread);·0s·(gc)
  
22 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],22 o3·=·point·of·coordinates·([421369,·39917,·-212481,·1],[-128795,·-176966,·1],
23 [3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])23 [3870,·-390108,·-496127,·-308581,·46649,·164926,·-446111,·48038,·415309,·1])
  
24 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^924 o3·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9
25 i4·:·Y·=·random({2,1,2},X);25 i4·:·Y·=·random({2,1,2},X);
  
26 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^926 o4·:·ProjectiveVariety,·hypersurface·in·PP^3·x·PP^2·x·PP^9
27 i5·:·time·q·=·point·Y27 i5·:·time·q·=·point·Y
28 ·--·used·1.25059s·(cpu);·0.746661s·(thread);·0s·(gc)28 ·--·used·1.39438s·(cpu);·0.750503s·(thread);·0s·(gc)
  
29 o5·=·q29 o5·=·q
  
30 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^930 o5·:·ProjectiveVariety,·a·point·in·PP^3·x·PP^2·x·PP^9
31 i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))31 i6·:·assert(isSubset(p,X)·and·isSubset(q,Y))
32 The·list·of·homogeneous·coordinates·can·be·obtained·with·the·operator·|-.32 The·list·of·homogeneous·coordinates·can·be·obtained·with·the·operator·|-.
33 i7·:·|-·p33 i7·:·|-·p
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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html
    
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91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi·=·rationalMap·{f,g,h};92 <td>··············<pre><code·class="language-macaulay2">i5·:·Phi·=·rationalMap·{f,g,h};
  
93 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x·PP^4)</code></pre>93 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x·PP^4)</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i6·:·time·segre·Phi;96 <td>··············<pre><code·class="language-macaulay2">i6·:·time·segre·Phi;
97 ·--·used·4.27397s·(cpu);·0.868122s·(thread);·0s·(gc)97 ·--·used·3.80843s·(cpu);·0.731163s·(thread);·0s·(gc)
  
98 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)</code></pre>98 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i7·:·describe·segre·Phi101 <td>··············<pre><code·class="language-macaulay2">i7·:·describe·segre·Phi
  
102 o7·=·rational·map·defined·by·forms·of·degree·6102 o7·=·rational·map·defined·by·forms·of·degree·6
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30 o4·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)30 o4·:·RationalMap·(quadratic·rational·map·from·PP^4·to·PP^4)
31 i5·:·Phi·=·rationalMap·{f,g,h};31 i5·:·Phi·=·rationalMap·{f,g,h};
  
32 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x32 o5·:·MultirationalMap·(rational·map·from·PP^4·to·hypersurface·in·PP^5·x·PP^4·x
33 PP^4)33 PP^4)
34 i6·:·time·segre·Phi;34 i6·:·time·segre·Phi;
35 ·--·used·4.27397s·(cpu);·0.868122s·(thread);·0s·(gc)35 ·--·used·3.80843s·(cpu);·0.731163s·(thread);·0s·(gc)
  
36 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)36 o6·:·RationalMap·(rational·map·from·PP^4·to·PP^149)
37 i7·:·describe·segre·Phi37 i7·:·describe·segre·Phi
  
38 o7·=·rational·map·defined·by·forms·of·degree·638 o7·=·rational·map·defined·by·forms·of·degree·6
39 ·····source·variety:·PP^439 ·····source·variety:·PP^4
40 ·····target·variety:·PP^14940 ·····target·variety:·PP^149
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./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
    
Offset 75, 15 lines modifiedOffset 75, 15 lines modified
  
75 o1·=·Phi75 o1·=·Phi
  
76 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to·threefold·in·PP^3·x·PP^2·x·PP^2)</code></pre>76 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to·threefold·in·PP^3·x·PP^2·x·PP^2)</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·Phi79 <td>··············<pre><code·class="language-macaulay2">i2·:·time·describe·Phi
80 ·--·used·0.157974s·(cpu);·0.117367s·(thread);·0s·(gc)80 ·--·used·0.18467s·(cpu);·0.123938s·(thread);·0s·(gc)
  
81 o2·=·multi-rational·map·consisting·of·3·rational·maps81 o2·=·multi-rational·map·consisting·of·3·rational·maps
82 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of·multi-degree·(1,1)82 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of·multi-degree·(1,1)
83 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces·of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^2·83 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces·of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^2·
84 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^284 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^2
85 ·····dominance:·true85 ·····dominance:·true
86 ·····multidegree:·{10,·14,·19,·25}86 ·····multidegree:·{10,·14,·19,·25}
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16 i1·:·Phi·=·inverse·first·graph·last·graph·rationalMap·PP_(ZZ/33331)^(1,3)16 i1·:·Phi·=·inverse·first·graph·last·graph·rationalMap·PP_(ZZ/33331)^(1,3)
  
17 o1·=·Phi17 o1·=·Phi
  
18 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to18 o1·:·MultirationalMap·(birational·map·from·threefold·in·PP^3·x·PP^2·to
19 threefold·in·PP^3·x·PP^2·x·PP^2)19 threefold·in·PP^3·x·PP^2·x·PP^2)
20 i2·:·time·describe·Phi20 i2·:·time·describe·Phi
21 ·--·used·0.157974s·(cpu);·0.117367s·(thread);·0s·(gc)21 ·--·used·0.18467s·(cpu);·0.123938s·(thread);·0s·(gc)
  
22 o2·=·multi-rational·map·consisting·of·3·rational·maps22 o2·=·multi-rational·map·consisting·of·3·rational·maps
23 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of23 ·····source·variety:·threefold·in·PP^3·x·PP^2·cut·out·by·2·hypersurfaces·of
24 multi-degree·(1,1)24 multi-degree·(1,1)
25 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces25 ·····target·variety:·threefold·in·PP^3·x·PP^2·x·PP^2·cut·out·by·7·hypersurfaces
26 of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^226 of·multi-degrees·(0,1,1)^3·(1,0,1)^2·(1,1,0)^2
27 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^227 ·····base·locus:·empty·subscheme·of·PP^3·x·PP^2
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712 B
./usr/share/doc/Macaulay2/NAGtypes/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=97 #:len=9
8 UG9seVNwYWNl8 UG9seVNwYWNl
9 #:len=8279 #:len=827
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz
11 cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv11 cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv
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731 B
./usr/share/doc/Macaulay2/NCAlgebra/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=207 #:len=20
8 UVEgJSBOQ0dyb2VibmVyQmFzaXM=8 UVEgJSBOQ0dyb2VibmVyQmFzaXM=
9 #:len=3179 #:len=317
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi
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./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=307 #:len=30
8 Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop8 Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop
9 #:len=2759 #:len=275
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMiwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMiwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop
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./usr/share/doc/Macaulay2/Nauty/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
    
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26 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>·true};26 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>·true};
  
27 i8·:·prob·=·n·->·log(n)/n;27 i8·:·prob·=·n·->·log(n)/n;
  
28 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))28 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))
  
29 o9·=·(71,·88,·91,·93,·93,·94,·92,·99,·97,·96,·96,·98,·96,·98,·98,·97,·98,29 o9·=·(68,·84,·92,·94,·99,·95,·98,·98,·95,·96,·100,·97,·97,·99,·96,·98,·99,
30 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
31 ·····100,·98,·96,·98,·98,·99,·99,·98,·97,·100,·98,·99)31 ·····98,·98,·97,·98,·98,·97,·99,·100,·100,·98,·97,·99)
  
32 o9·:·Sequence32 o9·:·Sequence
  
33 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))33 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))
  
34 o10·=·(13,·10,·6,·4,·3,·4,·5,·1,·1,·1,·0,·1,·0,·0,·1,·0,·1,·0,·0,·1,·1,·1,·0,34 o10·=·(20,·3,·6,·10,·5,·1,·1,·1,·2,·2,·0,·1,·1,·1,·0,·1,·0,·0,·0,·0,·0,·0,·0,
35 ······-----------------------------------------------------------------------35 ······-----------------------------------------------------------------------
36 ······1,·0,·0,·0,·0,·0)36 ······1,·1,·0,·0,·1,·0)
  
37 o10·:·Sequence37 o10·:·Sequence
  
38 i11·:·38 i11·:·
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424 B
./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Graphs.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}4 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
5 o1·:·List5 o1·:·List
  
6 i2·:·generateRandomGraphs(5,·5)6 i2·:·generateRandomGraphs(5,·5)
  
7 o2·=·{D@c,·DJs,·Doo,·Dwk,·DJG}7 o2·=·{Dz[,·DAC,·Di?,·Dr{,·D_[}
  
8 o2·:·List8 o2·:·List
  
9 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)9 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
10 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}10 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
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./usr/share/doc/Macaulay2/Nauty/example-output/_generate__Random__Regular__Graphs.out
    
Offset 1, 21 lines modifiedOffset 1, 21 lines modified
1 --·-*-·M2-comint·-*-·hash:·17298311710600676751 --·-*-·M2-comint·-*-·hash:·1729831171060067675
  
2 i1·:·R·=·QQ[a..e];2 i1·:·R·=·QQ[a..e];
  
3 i2·:·generateRandomRegularGraphs(R,·3,·2)3 i2·:·generateRandomRegularGraphs(R,·3,·2)
  
4 o2·=·{Graph{"edges"·=>·{{a,·c},·{b,·c},·{b,·d},·{a,·e},·{d,·e}}},4 o2·=·{Graph{"edges"·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
5 ············"ring"·=>·R··········································5 ············"ring"·=>·R··········································
6 ············"vertices"·=>·{a,·b,·c,·d,·e}························6 ············"vertices"·=>·{a,·b,·c,·d,·e}························
7 ·····------------------------------------------------------------------------7 ·····------------------------------------------------------------------------
8 ·····Graph{"edges"·=>·{{b,·c},·{a,·d},·{c,·d},·{a,·e},·{b,·e}}},8 ·····Graph{"edges"·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
9 ···········"ring"·=>·R··········································9 ···········"ring"·=>·R··········································
10 ···········"vertices"·=>·{a,·b,·c,·d,·e}························10 ···········"vertices"·=>·{a,·b,·c,·d,·e}························
11 ·····------------------------------------------------------------------------11 ·····------------------------------------------------------------------------
12 ·····Graph{"edges"·=>·{{a,·b},·{a,·d},·{c,·d},·{b,·e},·{c,·e}}}}12 ·····Graph{"edges"·=>·{{a,·c},·{b,·d},·{c,·d},·{a,·e},·{b,·e}}}}
13 ···········"ring"·=>·R13 ···········"ring"·=>·R
14 ···········"vertices"·=>·{a,·b,·c,·d,·e}14 ···········"vertices"·=>·{a,·b,·c,·d,·e}
  
15 o2·:·List15 o2·:·List
  
16 i3·:·16 i3·:·
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550 B
./usr/share/doc/Macaulay2/Nauty/example-output/_graph__Complement.out
    
Offset 13, 13 lines modifiedOffset 13, 13 lines modified
13 i3·:·graphComplement·"Dhc"13 i3·:·graphComplement·"Dhc"
  
14 o3·=·DUW14 o3·=·DUW
  
15 i4·:·G·=·generateBipartiteGraphs·7;15 i4·:·G·=·generateBipartiteGraphs·7;
  
16 i5·:·time·graphComplement·G;16 i5·:·time·graphComplement·G;
17 ·--·used·0.000576146s·(cpu);·0.000526572s·(thread);·0s·(gc)17 ·--·used·0.000652915s·(cpu);·0.000525586s·(thread);·0s·(gc)
  
18 i6·:·time·(graphComplement·\·G);18 i6·:·time·(graphComplement·\·G);
19 ·--·used·0.0532041s·(cpu);·0.0518868s·(thread);·0s·(gc)19 ·--·used·0.0775188s·(cpu);·0.0744009s·(thread);·0s·(gc)
  
20 i7·:·20 i7·:·
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93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i8·:·prob·=·n·->·log(n)/n;</code></pre>95 <td>··············<pre><code·class="language-macaulay2">i8·:·prob·=·n·->·log(n)/n;</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))98 <td>··············<pre><code·class="language-macaulay2">i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))
  
99 o9·=·(71,·88,·91,·93,·93,·94,·92,·99,·97,·96,·96,·98,·96,·98,·98,·97,·98,99 o9·=·(68,·84,·92,·94,·99,·95,·98,·98,·95,·96,·100,·97,·97,·99,·96,·98,·99,
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
101 ·····100,·98,·96,·98,·98,·99,·99,·98,·97,·100,·98,·99)101 ·····98,·98,·97,·98,·98,·97,·99,·100,·100,·98,·97,·99)
  
102 o9·:·Sequence</code></pre>102 o9·:·Sequence</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))105 <td>··············<pre><code·class="language-macaulay2">i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))
  
106 o10·=·(13,·10,·6,·4,·3,·4,·5,·1,·1,·1,·0,·1,·0,·0,·1,·0,·1,·0,·0,·1,·1,·1,·0,106 o10·=·(20,·3,·6,·10,·5,·1,·1,·1,·2,·2,·0,·1,·1,·1,·0,·1,·0,·0,·0,·0,·0,·0,·0,
107 ······-----------------------------------------------------------------------107 ······-----------------------------------------------------------------------
108 ······1,·0,·0,·0,·0,·0)108 ······1,·1,·0,·0,·1,·0)
  
109 o10·:·Sequence</code></pre>109 o10·:·Sequence</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ········</table>111 ········</table>
112 ······</div>112 ······</div>
113 ······<div>113 ······<div>
114 ········<h2>See·also</h2>114 ········<h2>See·also</h2>
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38 connected,·at·least·as·$n$·tends·to·infinity.38 connected,·at·least·as·$n$·tends·to·infinity.
39 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>39 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>
40 true};40 true};
41 i8·:·prob·=·n·->·log(n)/n;41 i8·:·prob·=·n·->·log(n)/n;
42 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),42 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),
43 connected))43 connected))
  
44 o9·=·(71,·88,·91,·93,·93,·94,·92,·99,·97,·96,·96,·98,·96,·98,·98,·97,·98,44 o9·=·(68,·84,·92,·94,·99,·95,·98,·98,·95,·96,·100,·97,·97,·99,·96,·98,·99,
45 ·····------------------------------------------------------------------------45 ·····------------------------------------------------------------------------
46 ·····100,·98,·96,·98,·98,·99,·99,·98,·97,·100,·98,·99)46 ·····98,·98,·97,·98,·98,·97,·99,·100,·100,·98,·97,·99)
  
47 o9·:·Sequence47 o9·:·Sequence
48 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),48 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),
49 connected))49 connected))
  
50 o10·=·(13,·10,·6,·4,·3,·4,·5,·1,·1,·1,·0,·1,·0,·0,·1,·0,·1,·0,·0,·1,·1,·1,·0,50 o10·=·(20,·3,·6,·10,·5,·1,·1,·1,·2,·2,·0,·1,·1,·1,·0,·1,·0,·0,·0,·0,·0,·0,·0,
51 ······-----------------------------------------------------------------------51 ······-----------------------------------------------------------------------
52 ······1,·0,·0,·0,·0,·0)52 ······1,·1,·0,·0,·1,·0)
  
53 o10·:·Sequence53 o10·:·Sequence
54 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*54 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
55 ····*·_\x8b_\x8u_\x8i_\x8l_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8F_\x8i_\x8l_\x8t_\x8e_\x8r·--·creates·the·appropriate·filter·string·for·use·with55 ····*·_\x8b_\x8u_\x8i_\x8l_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8F_\x8i_\x8l_\x8t_\x8e_\x8r·--·creates·the·appropriate·filter·string·for·use·with
56 ······filterGraphs·and·countGraphs56 ······filterGraphs·and·countGraphs
57 ····*·_\x8f_\x8i_\x8l_\x8t_\x8e_\x8r_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·filters·(i.e.,·selects)·graphs·in·a·list·for·given57 ····*·_\x8f_\x8i_\x8l_\x8t_\x8e_\x8r_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·filters·(i.e.,·selects)·graphs·in·a·list·for·given
58 ······properties58 ······properties
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104 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}104 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
105 o1·:·List</code></pre>105 o1·:·List</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomGraphs(5,·5)108 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomGraphs(5,·5)
  
109 o2·=·{D@c,·DJs,·Doo,·Dwk,·DJG}109 o2·=·{Dz[,·DAC,·Di?,·Dr{,·D_[}
  
110 o2·:·List</code></pre>110 o2·:·List</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)113 <td>··············<pre><code·class="language-macaulay2">i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
114 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}114 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
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38 i1·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)38 i1·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
39 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}39 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
40 o1·:·List40 o1·:·List
41 i2·:·generateRandomGraphs(5,·5)41 i2·:·generateRandomGraphs(5,·5)
  
42 o2·=·{D@c,·DJs,·Doo,·Dwk,·DJG}42 o2·=·{Dz[,·DAC,·Di?,·Dr{,·D_[}
  
43 o2·:·List43 o2·:·List
44 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)44 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
45 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}45 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
46 o3·:·List46 o3·:·List
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89 ········<table·class="examples">89 ········<table·class="examples">
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[a..e];</code></pre>91 <td>··············<pre><code·class="language-macaulay2">i1·:·R·=·QQ[a..e];</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomRegularGraphs(R,·3,·2)94 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomRegularGraphs(R,·3,·2)
  
95 o2·=·{Graph{&quot;edges&quot;·=>·{{a,·c},·{b,·c},·{b,·d},·{a,·e},·{d,·e}}},95 o2·=·{Graph{&quot;edges&quot;·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
96 ············&quot;ring&quot;·=>·R··········································96 ············&quot;ring&quot;·=>·R··········································
97 ············&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}························97 ············&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}························
98 ·····------------------------------------------------------------------------98 ·····------------------------------------------------------------------------
99 ·····Graph{&quot;edges&quot;·=>·{{b,·c},·{a,·d},·{c,·d},·{a,·e},·{b,·e}}},99 ·····Graph{&quot;edges&quot;·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
100 ···········&quot;ring&quot;·=>·R··········································100 ···········&quot;ring&quot;·=>·R··········································
101 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}························101 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}························
102 ·····------------------------------------------------------------------------102 ·····------------------------------------------------------------------------
103 ·····Graph{&quot;edges&quot;·=>·{{a,·b},·{a,·d},·{c,·d},·{b,·e},·{c,·e}}}}103 ·····Graph{&quot;edges&quot;·=>·{{a,·c},·{b,·d},·{c,·d},·{a,·e},·{b,·e}}}}
104 ···········&quot;ring&quot;·=>·R104 ···········&quot;ring&quot;·=>·R
105 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}105 ···········&quot;vertices&quot;·=>·{a,·b,·c,·d,·e}
  
106 o2·:·List</code></pre>106 o2·:·List</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ········</table>108 ········</table>
109 ······</div>109 ······</div>
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25 vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.25 vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.
26 If·a·_\x8P_\x8o_\x8l_\x8y_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8R_\x8i_\x8n_\x8g·$R$·is·supplied·instead,·then·the·number·of·vertices·is·the26 If·a·_\x8P_\x8o_\x8l_\x8y_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8R_\x8i_\x8n_\x8g·$R$·is·supplied·instead,·then·the·number·of·vertices·is·the
27 number·of·generators.·Moreover,·the·nauty-based·strings·are·automatically27 number·of·generators.·Moreover,·the·nauty-based·strings·are·automatically
28 converted·to·instances·of·the·class·_\x8G_\x8r_\x8a_\x8p_\x8h·in·$R$.28 converted·to·instances·of·the·class·_\x8G_\x8r_\x8a_\x8p_\x8h·in·$R$.
29 i1·:·R·=·QQ[a..e];29 i1·:·R·=·QQ[a..e];
30 i2·:·generateRandomRegularGraphs(R,·3,·2)30 i2·:·generateRandomRegularGraphs(R,·3,·2)
  
31 o2·=·{Graph{"edges"·=>·{{a,·c},·{b,·c},·{b,·d},·{a,·e},·{d,·e}}},31 o2·=·{Graph{"edges"·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
32 ············"ring"·=>·R32 ············"ring"·=>·R
33 ············"vertices"·=>·{a,·b,·c,·d,·e}33 ············"vertices"·=>·{a,·b,·c,·d,·e}
34 ·····------------------------------------------------------------------------34 ·····------------------------------------------------------------------------
35 ·····Graph{"edges"·=>·{{b,·c},·{a,·d},·{c,·d},·{a,·e},·{b,·e}}},35 ·····Graph{"edges"·=>·{{b,·c},·{a,·d},·{b,·d},·{a,·e},·{c,·e}}},
36 ···········"ring"·=>·R36 ···········"ring"·=>·R
37 ···········"vertices"·=>·{a,·b,·c,·d,·e}37 ···········"vertices"·=>·{a,·b,·c,·d,·e}
38 ·····------------------------------------------------------------------------38 ·····------------------------------------------------------------------------
39 ·····Graph{"edges"·=>·{{a,·b},·{a,·d},·{c,·d},·{b,·e},·{c,·e}}}}39 ·····Graph{"edges"·=>·{{a,·c},·{b,·d},·{c,·d},·{a,·e},·{b,·e}}}}
40 ···········"ring"·=>·R40 ···········"ring"·=>·R
41 ···········"vertices"·=>·{a,·b,·c,·d,·e}41 ···········"vertices"·=>·{a,·b,·c,·d,·e}
  
42 o2·:·List42 o2·:·List
43 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*43 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
44 The·number·of·vertices·$n$·must·be·positive·as·nauty·cannot·handle·graphs·with44 The·number·of·vertices·$n$·must·be·positive·as·nauty·cannot·handle·graphs·with
45 zero·vertices.45 zero·vertices.
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113 ········</div>113 ········</div>
114 ········<table·class="examples">114 ········<table·class="examples">
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i4·:·G·=·generateBipartiteGraphs·7;</code></pre>116 <td>··············<pre><code·class="language-macaulay2">i4·:·G·=·generateBipartiteGraphs·7;</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·graphComplement·G;119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·graphComplement·G;
120 ·--·used·0.000576146s·(cpu);·0.000526572s·(thread);·0s·(gc)</code></pre>120 ·--·used·0.000652915s·(cpu);·0.000525586s·(thread);·0s·(gc)</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i6·:·time·(graphComplement·\·G);123 <td>··············<pre><code·class="language-macaulay2">i6·:·time·(graphComplement·\·G);
124 ·--·used·0.0532041s·(cpu);·0.0518868s·(thread);·0s·(gc)</code></pre>124 ·--·used·0.0775188s·(cpu);·0.0744009s·(thread);·0s·(gc)</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ········</table>126 ········</table>
127 ······</div>127 ······</div>
128 ······<div>128 ······<div>
129 ········<h2>See·also</h2>129 ········<h2>See·also</h2>
130 ········<ul>130 ········<ul>
131 ··········<li>131 ··········<li>
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42 o3·=·DUW42 o3·=·DUW
43 Batch·calls·can·be·performed·considerably·faster·when·using·the·List·input43 Batch·calls·can·be·performed·considerably·faster·when·using·the·List·input
44 format.·However,·care·should·be·taken·as·the·returned·list·is·entirely·in44 format.·However,·care·should·be·taken·as·the·returned·list·is·entirely·in
45 Graph6·or·Sparse6·format.45 Graph6·or·Sparse6·format.
46 i4·:·G·=·generateBipartiteGraphs·7;46 i4·:·G·=·generateBipartiteGraphs·7;
47 i5·:·time·graphComplement·G;47 i5·:·time·graphComplement·G;
48 ·--·used·0.000576146s·(cpu);·0.000526572s·(thread);·0s·(gc)48 ·--·used·0.000652915s·(cpu);·0.000525586s·(thread);·0s·(gc)
49 i6·:·time·(graphComplement·\·G);49 i6·:·time·(graphComplement·\·G);
50 ·--·used·0.0532041s·(cpu);·0.0518868s·(thread);·0s·(gc)50 ·--·used·0.0775188s·(cpu);·0.0744009s·(thread);·0s·(gc)
51 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*51 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
52 ····*·_\x8c_\x8o_\x8m_\x8p_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8G_\x8r_\x8a_\x8p_\x8h·--·returns·the·complement·of·a·graph·or·hypergraph52 ····*·_\x8c_\x8o_\x8m_\x8p_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8G_\x8r_\x8a_\x8p_\x8h·--·returns·the·complement·of·a·graph·or·hypergraph
53 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8gr\x8ra\x8ap\x8ph\x8hC\x8Co\x8om\x8mp\x8pl\x8le\x8em\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*53 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8gr\x8ra\x8ap\x8ph\x8hC\x8Co\x8om\x8mp\x8pl\x8le\x8em\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
54 ····*·graphComplement(Graph)54 ····*·graphComplement(Graph)
55 ····*·graphComplement(List)55 ····*·graphComplement(List)
56 ····*·graphComplement(String)56 ····*·graphComplement(String)
57 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*57 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
⊟
751 B
./usr/share/doc/Macaulay2/NautyGraphs/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=317 #:len=31
8 YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ==8 YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ==
9 #:len=2619 #:len=261
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl
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./usr/share/doc/Macaulay2/NautyGraphs/example-output/___Example_co_sp__Generating_spand_spfiltering_spgraphs.out
    
Offset 26, 22 lines modifiedOffset 26, 22 lines modified
  
26 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>·true};26 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>·true};
  
27 i8·:·prob·=·n·->·log(n)/n;27 i8·:·prob·=·n·->·log(n)/n;
  
28 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))28 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))
  
29 o9·=·(74,·85,·92,·88,·93,·95,·100,·98,·95,·94,·96,·99,·96,·99,·99,·99,·98,29 o9·=·(80,·84,·91,·88,·91,·96,·98,·97,·96,·96,·95,·99,·99,·97,·98,·99,·98,·99,
30 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
31 ·····98,·93,·96,·99,·97,·98,·96,·99,·99,·98,·96,·99)31 ·····99,·100,·100,·95,·99,·98,·98,·97,·99,·100,·99)
  
32 o9·:·Sequence32 o9·:·Sequence
  
33 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))33 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))
  
34 o10·=·(15,·5,·2,·4,·1,·1,·7,·1,·0,·0,·1,·0,·1,·1,·0,·0,·0,·0,·0,·0,·0,·0,·1,34 o10·=·(21,·9,·11,·5,·3,·2,·4,·1,·2,·2,·1,·0,·0,·0,·1,·0,·1,·0,·2,·1,·1,·0,·1,
35 ······-----------------------------------------------------------------------35 ······-----------------------------------------------------------------------
36 ······0,·3,·0,·0,·0,·0)36 ······0,·0,·0,·0,·1,·0)
  
37 o10·:·Sequence37 o10·:·Sequence
  
38 i11·:·38 i11·:·
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./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Graphs.out
    
Offset 4, 15 lines modifiedOffset 4, 15 lines modified
  
4 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}4 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
5 o1·:·List5 o1·:·List
  
6 i2·:·generateRandomGraphs(5,·5)6 i2·:·generateRandomGraphs(5,·5)
  
7 o2·=·{DyG,·DlW,·DaG,·Dlw,·DBg}7 o2·=·{Dnw,·Di?,·Dfg,·D^K,·Dm[}
  
8 o2·:·List8 o2·:·List
  
9 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)9 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
10 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}10 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
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353 B
./usr/share/doc/Macaulay2/NautyGraphs/example-output/_generate__Random__Regular__Graphs.out
    
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1 --·-*-·M2-comint·-*-·hash:·13312873922681 --·-*-·M2-comint·-*-·hash:·1331287392268
  
2 i1·:·generateRandomRegularGraphs(5,·3,·2)2 i1·:·generateRandomRegularGraphs(5,·3,·2)
  
3 o1·=·{DYc,·Dbg,·D[S}3 o1·=·{Dhc,·DYc,·DMg}
  
4 o1·:·List4 o1·:·List
  
5 i2·:·5 i2·:·
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./usr/share/doc/Macaulay2/NautyGraphs/example-output/_graph__Complement.out
    
Offset 13, 13 lines modifiedOffset 13, 13 lines modified
13 ···········4·=>·{2,·1}13 ···········4·=>·{2,·1}
  
14 o2·:·Graph14 o2·:·Graph
  
15 i3·:·G·=·generateBipartiteGraphs·7;15 i3·:·G·=·generateBipartiteGraphs·7;
  
16 i4·:·time·graphComplement·G;16 i4·:·time·graphComplement·G;
17 ·--·used·0.000666844s·(cpu);·0.000554565s·(thread);·0s·(gc)17 ·--·used·0.000988884s·(cpu);·0.000620312s·(thread);·0s·(gc)
  
18 i5·:·time·(graphComplement·\·G);18 i5·:·time·(graphComplement·\·G);
19 ·--·used·0.110816s·(cpu);·0.0666706s·(thread);·0s·(gc)19 ·--·used·0.158517s·(cpu);·0.0842207s·(thread);·0s·(gc)
  
20 i6·:·20 i6·:·
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./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
    
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93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i8·:·prob·=·n·->·log(n)/n;</code></pre>95 <td>··············<pre><code·class="language-macaulay2">i8·:·prob·=·n·->·log(n)/n;</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))98 <td>··············<pre><code·class="language-macaulay2">i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),·connected))
  
99 o9·=·(74,·85,·92,·88,·93,·95,·100,·98,·95,·94,·96,·99,·96,·99,·99,·99,·98,99 o9·=·(80,·84,·91,·88,·91,·96,·98,·97,·96,·96,·95,·99,·99,·97,·98,·99,·98,·99,
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
101 ·····98,·93,·96,·99,·97,·98,·96,·99,·99,·98,·96,·99)101 ·····99,·100,·100,·95,·99,·98,·98,·97,·99,·100,·99)
  
102 o9·:·Sequence</code></pre>102 o9·:·Sequence</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))105 <td>··············<pre><code·class="language-macaulay2">i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),·connected))
  
106 o10·=·(15,·5,·2,·4,·1,·1,·7,·1,·0,·0,·1,·0,·1,·1,·0,·0,·0,·0,·0,·0,·0,·0,·1,106 o10·=·(21,·9,·11,·5,·3,·2,·4,·1,·2,·2,·1,·0,·0,·0,·1,·0,·1,·0,·2,·1,·1,·0,·1,
107 ······-----------------------------------------------------------------------107 ······-----------------------------------------------------------------------
108 ······0,·3,·0,·0,·0,·0)108 ······0,·0,·0,·0,·1,·0)
  
109 o10·:·Sequence</code></pre>109 o10·:·Sequence</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ········</table>111 ········</table>
112 ······</div>112 ······</div>
113 ······<div>113 ······<div>
114 ········<h2>See·also</h2>114 ········<h2>See·also</h2>
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Offset 38, 25 lines modifiedOffset 38, 25 lines modified
38 connected,·at·least·as·$n$·tends·to·infinity.38 connected,·at·least·as·$n$·tends·to·infinity.
39 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>39 i7·:·connected·=·buildGraphFilter·{"Connectivity"·=>·0,·"NegateConnectivity"·=>
40 true};40 true};
41 i8·:·prob·=·n·->·log(n)/n;41 i8·:·prob·=·n·->·log(n)/n;
42 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),42 i9·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·2*(prob·n)),
43 connected))43 connected))
  
44 o9·=·(74,·85,·92,·88,·93,·95,·100,·98,·95,·94,·96,·99,·96,·99,·99,·99,·98,44 o9·=·(80,·84,·91,·88,·91,·96,·98,·97,·96,·96,·95,·99,·99,·97,·98,·99,·98,·99,
45 ·····------------------------------------------------------------------------45 ·····------------------------------------------------------------------------
46 ·····98,·93,·96,·99,·97,·98,·96,·99,·99,·98,·96,·99)46 ·····99,·100,·100,·95,·99,·98,·98,·97,·99,·100,·99)
  
47 o9·:·Sequence47 o9·:·Sequence
48 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),48 i10·:·apply(2..30,·n·->·#filterGraphs(generateRandomGraphs(n,·100,·(prob·n)/2),
49 connected))49 connected))
  
50 o10·=·(15,·5,·2,·4,·1,·1,·7,·1,·0,·0,·1,·0,·1,·1,·0,·0,·0,·0,·0,·0,·0,·0,·1,50 o10·=·(21,·9,·11,·5,·3,·2,·4,·1,·2,·2,·1,·0,·0,·0,·1,·0,·1,·0,·2,·1,·1,·0,·1,
51 ······-----------------------------------------------------------------------51 ······-----------------------------------------------------------------------
52 ······0,·3,·0,·0,·0,·0)52 ······0,·0,·0,·0,·1,·0)
  
53 o10·:·Sequence53 o10·:·Sequence
54 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*54 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
55 ····*·_\x8b_\x8u_\x8i_\x8l_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8F_\x8i_\x8l_\x8t_\x8e_\x8r·--·creates·the·appropriate·filter·string·for·use·with55 ····*·_\x8b_\x8u_\x8i_\x8l_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8F_\x8i_\x8l_\x8t_\x8e_\x8r·--·creates·the·appropriate·filter·string·for·use·with
56 ······filterGraphs·and·countGraphs56 ······filterGraphs·and·countGraphs
57 ····*·_\x8f_\x8i_\x8l_\x8t_\x8e_\x8r_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·filters·(i.e.,·selects)·graphs·in·a·list·for·given57 ····*·_\x8f_\x8i_\x8l_\x8t_\x8e_\x8r_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·filters·(i.e.,·selects)·graphs·in·a·list·for·given
58 ······properties58 ······properties
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96 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}96 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
97 o1·:·List</code></pre>97 o1·:·List</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomGraphs(5,·5)100 <td>··············<pre><code·class="language-macaulay2">i2·:·generateRandomGraphs(5,·5)
  
101 o2·=·{DyG,·DlW,·DaG,·Dlw,·DBg}101 o2·=·{Dnw,·Di?,·Dfg,·D^K,·Dm[}
  
102 o2·:·List</code></pre>102 o2·:·List</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)105 <td>··············<pre><code·class="language-macaulay2">i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
106 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}106 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
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31 i1·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)31 i1·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
32 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}32 o1·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
33 o1·:·List33 o1·:·List
34 i2·:·generateRandomGraphs(5,·5)34 i2·:·generateRandomGraphs(5,·5)
  
35 o2·=·{DyG,·DlW,·DaG,·Dlw,·DBg}35 o2·=·{Dnw,·Di?,·Dfg,·D^K,·Dm[}
  
36 o2·:·List36 o2·:·List
37 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)37 i3·:·generateRandomGraphs(5,·5,·RandomSeed·=>·314159)
  
38 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}38 o3·=·{DDO,·Dx_,·Dlw,·Dx{,·D_K}
  
39 o3·:·List39 o3·:·List
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./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
    
Offset 80, 15 lines modifiedOffset 80, 15 lines modified
80 ········<div>80 ········<div>
81 ··········<p>This·method·generates·a·specified·number·of·random·graphs·on·a·given·number·of·vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.</p>81 ··········<p>This·method·generates·a·specified·number·of·random·graphs·on·a·given·number·of·vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.</p>
82 ········</div>82 ········</div>
83 ········<table·class="examples">83 ········<table·class="examples">
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i1·:·generateRandomRegularGraphs(5,·3,·2)85 <td>··············<pre><code·class="language-macaulay2">i1·:·generateRandomRegularGraphs(5,·3,·2)
  
86 o1·=·{DYc,·Dbg,·D[S}86 o1·=·{Dhc,·DYc,·DMg}
  
87 o1·:·List</code></pre>87 o1·:·List</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
90 ······</div>90 ······</div>
91 ······<div>91 ······<div>
92 ········<h2>Caveat</h2>92 ········<h2>Caveat</h2>
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19 ····*·Outputs:19 ····*·Outputs:
20 ··········o·G,·a·_\x8l_\x8i_\x8s_\x8t,·the·randomly·generated·regular·graphs20 ··········o·G,·a·_\x8l_\x8i_\x8s_\x8t,·the·randomly·generated·regular·graphs
21 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*21 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
22 This·method·generates·a·specified·number·of·random·graphs·on·a·given·number·of22 This·method·generates·a·specified·number·of·random·graphs·on·a·given·number·of
23 vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.23 vertices·with·a·given·regularity.·Note·that·some·graphs·may·be·isomorphic.
24 i1·:·generateRandomRegularGraphs(5,·3,·2)24 i1·:·generateRandomRegularGraphs(5,·3,·2)
  
25 o1·=·{DYc,·Dbg,·D[S}25 o1·=·{Dhc,·DYc,·DMg}
  
26 o1·:·List26 o1·:·List
27 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
28 The·number·of·vertices·$n$·must·be·positive·as·nauty·cannot·handle·graphs·with28 The·number·of·vertices·$n$·must·be·positive·as·nauty·cannot·handle·graphs·with
29 zero·vertices.29 zero·vertices.
30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
31 ····*·_\x8g_\x8e_\x8n_\x8e_\x8r_\x8a_\x8t_\x8e_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·generates·random·graphs·on·a·given·number·of31 ····*·_\x8g_\x8e_\x8n_\x8e_\x8r_\x8a_\x8t_\x8e_\x8R_\x8a_\x8n_\x8d_\x8o_\x8m_\x8G_\x8r_\x8a_\x8p_\x8h_\x8s·--·generates·random·graphs·on·a·given·number·of
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109 ········</div>109 ········</div>
110 ········<table·class="examples">110 ········<table·class="examples">
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i3·:·G·=·generateBipartiteGraphs·7;</code></pre>112 <td>··············<pre><code·class="language-macaulay2">i3·:·G·=·generateBipartiteGraphs·7;</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i4·:·time·graphComplement·G;115 <td>··············<pre><code·class="language-macaulay2">i4·:·time·graphComplement·G;
116 ·--·used·0.000666844s·(cpu);·0.000554565s·(thread);·0s·(gc)</code></pre>116 ·--·used·0.000988884s·(cpu);·0.000620312s·(thread);·0s·(gc)</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·(graphComplement·\·G);119 <td>··············<pre><code·class="language-macaulay2">i5·:·time·(graphComplement·\·G);
120 ·--·used·0.110816s·(cpu);·0.0666706s·(thread);·0s·(gc)</code></pre>120 ·--·used·0.158517s·(cpu);·0.0842207s·(thread);·0s·(gc)</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ········</table>122 ········</table>
123 ······</div>123 ······</div>
124 ······<div·class="waystouse">124 ······<div·class="waystouse">
125 ········<h2>Ways·to·use·<span·class="tt">graphComplement</span>:</h2>125 ········<h2>Ways·to·use·<span·class="tt">graphComplement</span>:</h2>
126 ········<ul>126 ········<ul>
127 ··········<li>127 ··········<li>
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Offset 39, 16 lines modifiedOffset 39, 16 lines modified
  
39 o2·:·Graph39 o2·:·Graph
40 Batch·calls·can·be·performed·considerably·faster·when·using·the·List·input40 Batch·calls·can·be·performed·considerably·faster·when·using·the·List·input
41 format.·However,·care·should·be·taken·as·the·returned·list·is·entirely·in41 format.·However,·care·should·be·taken·as·the·returned·list·is·entirely·in
42 Graph6·or·Sparse6·format.42 Graph6·or·Sparse6·format.
43 i3·:·G·=·generateBipartiteGraphs·7;43 i3·:·G·=·generateBipartiteGraphs·7;
44 i4·:·time·graphComplement·G;44 i4·:·time·graphComplement·G;
45 ·--·used·0.000666844s·(cpu);·0.000554565s·(thread);·0s·(gc)45 ·--·used·0.000988884s·(cpu);·0.000620312s·(thread);·0s·(gc)
46 i5·:·time·(graphComplement·\·G);46 i5·:·time·(graphComplement·\·G);
47 ·--·used·0.110816s·(cpu);·0.0666706s·(thread);·0s·(gc)47 ·--·used·0.158517s·(cpu);·0.0842207s·(thread);·0s·(gc)
48 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8gr\x8ra\x8ap\x8ph\x8hC\x8Co\x8om\x8mp\x8pl\x8le\x8em\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*48 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·g\x8gr\x8ra\x8ap\x8ph\x8hC\x8Co\x8om\x8mp\x8pl\x8le\x8em\x8me\x8en\x8nt\x8t:\x8:·*\x8**\x8**\x8**\x8**\x8*
49 ····*·graphComplement(Graph)49 ····*·graphComplement(Graph)
50 ····*·graphComplement(List)50 ····*·graphComplement(List)
51 ····*·graphComplement(String)51 ····*·graphComplement(String)
52 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*52 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
53 The·object·_\x8g_\x8r_\x8a_\x8p_\x8h_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.53 The·object·_\x8g_\x8r_\x8a_\x8p_\x8h_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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777 B
./usr/share/doc/Macaulay2/NoetherNormalization/dump/rawdocumentation.dump
    
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768 B
./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump
    
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647 B
./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out
    
Offset 47, 15 lines modifiedOffset 47, 15 lines modified
47 o4·:·Ideal·of·R47 o4·:·Ideal·of·R
  
48 i5·:·isPrimary·Q48 i5·:·isPrimary·Q
  
49 o5·=·true49 o5·=·true
  
50 i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·"PunctualQuot")50 i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·"PunctualQuot")
51 ·--·.08397s·elapsed51 ·--·.213762s·elapsed
  
52 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|52 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|
53 ·····------------------------------------------------------------------------53 ·····------------------------------------------------------------------------
54 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}54 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}
  
55 o6·:·List55 o6·:·List
  
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1.32 KB
./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html
    
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102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i5·:·isPrimary·Q103 <td>··············<pre><code·class="language-macaulay2">i5·:·isPrimary·Q
  
104 o5·=·true</code></pre>104 o5·=·true</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·&quot;PunctualQuot&quot;)107 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·&quot;PunctualQuot&quot;)
108 ·--·.08397s·elapsed108 ·--·.213762s·elapsed
  
109 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|109 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
111 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}111 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}
  
112 o6·:·List</code></pre>112 o6·:·List</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
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524 B
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51 ·······1·2·3····2·351 ·······1·2·3····2·3
  
52 o4·:·Ideal·of·R52 o4·:·Ideal·of·R
53 i5·:·isPrimary·Q53 i5·:·isPrimary·Q
  
54 o5·=·true54 o5·=·true
55 i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·"PunctualQuot")55 i6·:·elapsedTime·noetherianOperators(Q,·Strategy·=>·"PunctualQuot")
56 ·--·.08397s·elapsed56 ·--·.213762s·elapsed
  
57 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|57 o6·=·{|·1·|,·|·dx_1·|,·|·dx_2·|,·|·dx_1^2·|,·|·dx_1dx_2·|,·|·dx_2^2·|,·|
58 ·····------------------------------------------------------------------------58 ·····------------------------------------------------------------------------
59 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}59 ·····2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3·|}
  
60 o6·:·List60 o6·:·List
61 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*61 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
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751 B
./usr/share/doc/Macaulay2/NonminimalComplexes/dump/rawdocumentation.dump
    
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749 B
./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump
    
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1.77 KB
./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out
    
Offset 78, 15 lines modifiedOffset 78, 15 lines modified
78 i13·:·for·i·to·dim·X·list·hilbertFunction·(i,·A1)78 i13·:·for·i·to·dim·X·list·hilbertFunction·(i,·A1)
  
79 o13·=·{1,·2,·3,·3,·2,·1}79 o13·=·{1,·2,·3,·3,·2,·1}
  
80 o13·:·List80 o13·:·List
  
81 i14·:·Y·=·time·smoothFanoToricVariety(5,100);81 i14·:·Y·=·time·smoothFanoToricVariety(5,100);
82 ·--·used·0.173308s·(cpu);·0.172346s·(thread);·0s·(gc)82 ·--·used·0.198161s·(cpu);·0.194777s·(thread);·0s·(gc)
  
83 i15·:·A2·=·intersectionRing·Y;83 i15·:·A2·=·intersectionRing·Y;
  
84 i16·:·assert·(#·rays·Y·===·numgens·A2)84 i16·:·assert·(#·rays·Y·===·numgens·A2)
  
85 i17·:·ideal·A285 i17·:·ideal·A2
  
Offset 110, 19 lines modifiedOffset 110, 19 lines modified
110 ········2·················2···2·······················2···2·····························2·············2······················2·····3······2110 ········2·················2···2·······················2···2·····························2·············2······················2·····3······2
111 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t·t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)111 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t·t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)
112 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10····9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10112 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10····9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10
  
113 o18·:·QuotientRing113 o18·:·QuotientRing
  
114 i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)114 i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)
115 ·--·used·0.00108651s·(cpu);·0.00106029s·(thread);·0s·(gc)115 ·--·used·0.00141183s·(cpu);·0.00116904s·(thread);·0s·(gc)
116 ·--·used·2.0371e-05s·(cpu);·6.3576e-05s·(thread);·0s·(gc)116 ·--·used·2.1851e-05s·(cpu);·8.7474e-05s·(thread);·0s·(gc)
117 ·--·used·6.049e-06s·(cpu);·5.2388e-05s·(thread);·0s·(gc)117 ·--·used·1.0209e-05s·(cpu);·7.8007e-05s·(thread);·0s·(gc)
118 ·--·used·5.859e-06s·(cpu);·4.7772e-05s·(thread);·0s·(gc)118 ·--·used·8.596e-06s·(cpu);·7.5742e-05s·(thread);·0s·(gc)
 119 ·--·used·8.205e-06s·(cpu);·7.1234e-05s·(thread);·0s·(gc)
119 ·--·used·5.278e-06s·(cpu);·4.8573e-05s·(thread);·0s·(gc)120 ·--·used·9.487e-06s·(cpu);·7.1655e-05s·(thread);·0s·(gc)
120 ·--·used·5.759e-06s·(cpu);·4.4467e-05s·(thread);·0s·(gc) 
  
121 o19·=·{1,·6,·13,·13,·6,·1}121 o19·=·{1,·6,·13,·13,·6,·1}
  
122 o19·:·List122 o19·:·List
  
123 i20·:·123 i20·:·
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803 B
./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_is__Well__Defined_lp__Normal__Toric__Variety_rp.out
    
Offset 2, 27 lines modifiedOffset 2, 27 lines modified
  
2 i1·:·assert·all·(5,·d·->·isWellDefined·toricProjectiveSpace·(d+1))2 i1·:·assert·all·(5,·d·->·isWellDefined·toricProjectiveSpace·(d+1))
  
3 i2·:·setRandomSeed·(currentTime·());3 i2·:·setRandomSeed·(currentTime·());
  
4 i3·:·a·=·sort·apply·(3,·i·->·random·(7))4 i3·:·a·=·sort·apply·(3,·i·->·random·(7))
  
5 o3·=·{4,·5,·6}5 o3·=·{1,·2,·3}
  
6 o3·:·List6 o3·:·List
  
7 i4·:·assert·isWellDefined·kleinschmidt·(4,a)7 i4·:·assert·isWellDefined·kleinschmidt·(4,a)
  
8 i5·:·q·=·sort·apply·(5,·j·->·random·(1,9));8 i5·:·q·=·sort·apply·(5,·j·->·random·(1,9));
  
9 i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j·->·random·(1,9));9 i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j·->·random·(1,9));
  
10 i7·:·q10 i7·:·q
  
11 o7·=·{2,·3,·4,·8,·9}11 o7·=·{1,·2,·5,·8,·8}
  
12 o7·:·List12 o7·:·List
  
13 i8·:·assert·isWellDefined·weightedProjectiveSpace·q13 i8·:·assert·isWellDefined·weightedProjectiveSpace·q
  
14 i9·:·X·=·new·MutableHashTable;14 i9·:·X·=·new·MutableHashTable;
  
⊟
1.82 KB
./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out
    
Offset 6, 61 lines modifiedOffset 6, 61 lines modified
  
6 o2·=·5*PP26 o2·=·5*PP2
7 ··········07 ··········0
  
8 o2·:·ToricDivisor·on·PP28 o2·:·ToricDivisor·on·PP2
  
9 i3·:·M1·=·elapsedTime·monomials·D19 i3·:·M1·=·elapsedTime·monomials·D1
10 ·--·.149902s·elapsed10 ·--·.15401s·elapsed
  
11 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···4···11 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···4···
12 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,12 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,
13 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·2·13 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·2·
14 ·····------------------------------------------------------------------------14 ·····------------------------------------------------------------------------
15 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····515 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····5
16 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}16 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}
17 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···017 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···0
  
18 o3·:·List18 o3·:·List
  
19 i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring·variety·D1))19 i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring·variety·D1))
20 ·--·.0010759s·elapsed20 ·--·.00455756s·elapsed
  
21 i5·:·FF2·=·hirzebruchSurface·2;21 i5·:·FF2·=·hirzebruchSurface·2;
  
22 i6·:·D2·=·2*FF2_0·+·3·*·FF2_122 i6·:·D2·=·2*FF2_0·+·3·*·FF2_1
  
23 o6·=·2*FF2··+·3*FF223 o6·=·2*FF2··+·3*FF2
24 ··········0········124 ··········0········1
  
25 o6·:·ToricDivisor·on·FF225 o6·:·ToricDivisor·on·FF2
  
26 i7·:·M2·=·elapsedTime·monomials·D226 i7·:·M2·=·elapsedTime·monomials·D2
27 ·--·.431127s·elapsed27 ·--·.31948s·elapsed
  
28 ·······2·····3·2·····3·····2·328 ·······2·····3·2·····3·····2·3
29 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}29 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}
30 ·······1·3···1·2···0·1·2···0·130 ·······1·3···1·2···0·1·2···0·1
  
31 o7·:·List31 o7·:·List
  
32 i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring·variety·D2))32 i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring·variety·D2))
33 ·--·.00103813s·elapsed33 ·--·.00111549s·elapsed
  
34 i9·:·X·=·kleinschmidt·(5,·{1,2,3});34 i9·:·X·=·kleinschmidt·(5,·{1,2,3});
  
35 i10·:·D3·=·3*X_0·+·5*X_135 i10·:·D3·=·3*X_0·+·5*X_1
  
36 o10·=·3*X··+·5*X36 o10·=·3*X··+·5*X
37 ·········0······137 ·········0······1
  
38 o10·:·ToricDivisor·on·X38 o10·:·ToricDivisor·on·X
  
39 i11·:·m3·=·elapsedTime·#·monomials·D339 i11·:·m3·=·elapsedTime·#·monomials·D3
40 ·--·34.4709s·elapsed40 ·--·36.6909s·elapsed
  
41 o11·=·790941 o11·=·7909
  
42 i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety·D3))42 i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety·D3))
43 ·--·.0246716s·elapsed43 ·--·.0231379s·elapsed
  
44 i13·:·44 i13·:·
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Offset 24, 19 lines modifiedOffset 24, 19 lines modified
24 o3·:·List24 o3·:·List
  
25 i4·:·X·=·normalToricVariety·F;25 i4·:·X·=·normalToricVariety·F;
  
26 i5·:·assert·(transpose·matrix·rays·X·==·rays·F·and·max·X·==·sort·maxCones·F)26 i5·:·assert·(transpose·matrix·rays·X·==·rays·F·and·max·X·==·sort·maxCones·F)
  
27 i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})27 i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})
28 ·--·used·0.000753955s·(cpu);·1.7266e-05s·(thread);·0s·(gc)28 ·--·used·0.000219281s·(cpu);·2.4907e-05s·(thread);·0s·(gc)
  
29 o6·=·X129 o6·=·X1
  
30 o6·:·NormalToricVariety30 o6·:·NormalToricVariety
  
31 i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull·matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};31 i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull·matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};
32 ·--·used·0.0911707s·(cpu);·0.0503166s·(thread);·0s·(gc)32 ·--·used·0.112158s·(cpu);·0.0685298s·(thread);·0s·(gc)
  
33 i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)33 i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)
  
34 i9·:·34 i9·:·
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Offset 88, 15 lines modifiedOffset 88, 15 lines modified
88 o18·=·|·0·1·0·|88 o18·=·|·0·1·0·|
89 ······|·0·0·1·|89 ······|·0·0·1·|
  
90 ···············2·······390 ···············2·······3
91 o18·:·Matrix·ZZ··<--·ZZ91 o18·:·Matrix·ZZ··<--·ZZ
  
92 i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);92 i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);
93 ·--·used·0.0721583s·(cpu);·0.0340272s·(thread);·0s·(gc)93 ·--·used·0.0912587s·(cpu);·0.0400754s·(thread);·0s·(gc)
  
94 i20·:·X2·=·time·normalToricVariety·vertMatrix;94 i20·:·X2·=·time·normalToricVariety·vertMatrix;
95 ·--·used·0.00112943s·(cpu);·0.00195081s·(thread);·0s·(gc)95 ·--·used·0.00203453s·(cpu);·0.00291772s·(thread);·0s·(gc)
  
96 i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)96 i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)
  
97 i22·:·97 i22·:·
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Offset 181, 15 lines modifiedOffset 181, 15 lines modified
181 ········</table>181 ········</table>
182 ········<div>182 ········<div>
183 ··········<p>We·end·with·a·slightly·larger·example.</p>183 ··········<p>We·end·with·a·slightly·larger·example.</p>
184 ········</div>184 ········</div>
185 ········<table·class="examples">185 ········<table·class="examples">
186 ··········<tr>186 ··········<tr>
187 <td>··············<pre><code·class="language-macaulay2">i14·:·Y·=·time·smoothFanoToricVariety(5,100);187 <td>··············<pre><code·class="language-macaulay2">i14·:·Y·=·time·smoothFanoToricVariety(5,100);
188 ·--·used·0.173308s·(cpu);·0.172346s·(thread);·0s·(gc)</code></pre>188 ·--·used·0.198161s·(cpu);·0.194777s·(thread);·0s·(gc)</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ··········<tr>190 ··········<tr>
191 <td>··············<pre><code·class="language-macaulay2">i15·:·A2·=·intersectionRing·Y;</code></pre>191 <td>··············<pre><code·class="language-macaulay2">i15·:·A2·=·intersectionRing·Y;</code></pre>
192 </td>··········</tr>192 </td>··········</tr>
193 ··········<tr>193 ··········<tr>
194 <td>··············<pre><code·class="language-macaulay2">i16·:·assert·(#·rays·Y·===·numgens·A2)</code></pre>194 <td>··············<pre><code·class="language-macaulay2">i16·:·assert·(#·rays·Y·===·numgens·A2)</code></pre>
195 </td>··········</tr>195 </td>··········</tr>
Offset 218, 20 lines modifiedOffset 218, 20 lines modified
218 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t·t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)218 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t·t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)
219 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10····9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10219 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10····9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10
  
220 o18·:·QuotientRing</code></pre>220 o18·:·QuotientRing</code></pre>
221 </td>··········</tr>221 </td>··········</tr>
222 ··········<tr>222 ··········<tr>
223 <td>··············<pre><code·class="language-macaulay2">i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)223 <td>··············<pre><code·class="language-macaulay2">i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)
224 ·--·used·0.00108651s·(cpu);·0.00106029s·(thread);·0s·(gc)224 ·--·used·0.00141183s·(cpu);·0.00116904s·(thread);·0s·(gc)
225 ·--·used·2.0371e-05s·(cpu);·6.3576e-05s·(thread);·0s·(gc)225 ·--·used·2.1851e-05s·(cpu);·8.7474e-05s·(thread);·0s·(gc)
226 ·--·used·6.049e-06s·(cpu);·5.2388e-05s·(thread);·0s·(gc)226 ·--·used·1.0209e-05s·(cpu);·7.8007e-05s·(thread);·0s·(gc)
227 ·--·used·5.859e-06s·(cpu);·4.7772e-05s·(thread);·0s·(gc)227 ·--·used·8.596e-06s·(cpu);·7.5742e-05s·(thread);·0s·(gc)
 228 ·--·used·8.205e-06s·(cpu);·7.1234e-05s·(thread);·0s·(gc)
228 ·--·used·5.278e-06s·(cpu);·4.8573e-05s·(thread);·0s·(gc)229 ·--·used·9.487e-06s·(cpu);·7.1655e-05s·(thread);·0s·(gc)
229 ·--·used·5.759e-06s·(cpu);·4.4467e-05s·(thread);·0s·(gc) 
  
230 o19·=·{1,·6,·13,·13,·6,·1}230 o19·=·{1,·6,·13,·13,·6,·1}
  
231 o19·:·List</code></pre>231 o19·:·List</code></pre>
232 </td>··········</tr>232 </td>··········</tr>
233 ········</table>233 ········</table>
234 ······</div>234 ······</div>
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html2text {}
    
Offset 97, 15 lines modifiedOffset 97, 15 lines modified
97 i13·:·for·i·to·dim·X·list·hilbertFunction·(i,·A1)97 i13·:·for·i·to·dim·X·list·hilbertFunction·(i,·A1)
  
98 o13·=·{1,·2,·3,·3,·2,·1}98 o13·=·{1,·2,·3,·3,·2,·1}
  
99 o13·:·List99 o13·:·List
100 We·end·with·a·slightly·larger·example.100 We·end·with·a·slightly·larger·example.
101 i14·:·Y·=·time·smoothFanoToricVariety(5,100);101 i14·:·Y·=·time·smoothFanoToricVariety(5,100);
102 ·--·used·0.173308s·(cpu);·0.172346s·(thread);·0s·(gc)102 ·--·used·0.198161s·(cpu);·0.194777s·(thread);·0s·(gc)
103 i15·:·A2·=·intersectionRing·Y;103 i15·:·A2·=·intersectionRing·Y;
104 i16·:·assert·(#·rays·Y·===·numgens·A2)104 i16·:·assert·(#·rays·Y·===·numgens·A2)
105 i17·:·ideal·A2105 i17·:·ideal·A2
  
106 o17·=·ideal·(t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·t··,106 o17·=·ideal·(t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·,·t·t·t··,
107 ··············2·3···2·5···4·5···3·6···4·6···1·7···7·9···8·9···0·1·10107 ··············2·3···2·5···4·5···3·6···4·6···1·7···7·9···8·9···0·1·10
108 ······-----------------------------------------------------------------------108 ······-----------------------------------------------------------------------
Offset 130, 20 lines modifiedOffset 130, 20 lines modified
130 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t130 ······(t··+·t·t·,·t·t··+·t·,·t··+·t·t·,·t·t·,·t·t··+·t·,·t··-·t·t··-·3t·t···+·t
131 t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)131 t···+·2t··,·-·t·t··+·t··+·2t·t··,·t·t·,·-·t·t···+·t··,·t·t··)
132 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10132 ········3····3·5···3·5····5···5····5·6···3·6···5·6····6···8····8·9·····8·10
133 9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10133 9·10·····10·····8·9····9·····9·10···8·9·····8·10····10···8·10
  
134 o18·:·QuotientRing134 o18·:·QuotientRing
135 i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)135 i19·:·for·i·to·dim·Y·list·time·hilbertFunction·(i,·A2)
136 ·--·used·0.00108651s·(cpu);·0.00106029s·(thread);·0s·(gc)136 ·--·used·0.00141183s·(cpu);·0.00116904s·(thread);·0s·(gc)
137 ·--·used·2.0371e-05s·(cpu);·6.3576e-05s·(thread);·0s·(gc)137 ·--·used·2.1851e-05s·(cpu);·8.7474e-05s·(thread);·0s·(gc)
138 ·--·used·6.049e-06s·(cpu);·5.2388e-05s·(thread);·0s·(gc)138 ·--·used·1.0209e-05s·(cpu);·7.8007e-05s·(thread);·0s·(gc)
139 ·--·used·5.859e-06s·(cpu);·4.7772e-05s·(thread);·0s·(gc)139 ·--·used·8.596e-06s·(cpu);·7.5742e-05s·(thread);·0s·(gc)
 140 ·--·used·8.205e-06s·(cpu);·7.1234e-05s·(thread);·0s·(gc)
140 ·--·used·5.278e-06s·(cpu);·4.8573e-05s·(thread);·0s·(gc)141 ·--·used·9.487e-06s·(cpu);·7.1655e-05s·(thread);·0s·(gc)
141 ·--·used·5.759e-06s·(cpu);·4.4467e-05s·(thread);·0s·(gc) 
  
142 o19·=·{1,·6,·13,·13,·6,·1}142 o19·=·{1,·6,·13,·13,·6,·1}
  
143 o19·:·List143 o19·:·List
144 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*144 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
145 ····*·_\x8w_\x8o_\x8r_\x8k_\x8i_\x8n_\x8g_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8s_\x8h_\x8e_\x8a_\x8v_\x8e_\x8s·--·information·about·coherent·sheaves·and·total145 ····*·_\x8w_\x8o_\x8r_\x8k_\x8i_\x8n_\x8g_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8s_\x8h_\x8e_\x8a_\x8v_\x8e_\x8s·--·information·about·coherent·sheaves·and·total
146 ······coordinate·rings·(a.k.a.·Cox·rings)146 ······coordinate·rings·(a.k.a.·Cox·rings)
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Offset 100, 15 lines modifiedOffset 100, 15 lines modified
100 ········<table·class="examples">100 ········<table·class="examples">
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i2·:·setRandomSeed·(currentTime·());</code></pre>102 <td>··············<pre><code·class="language-macaulay2">i2·:·setRandomSeed·(currentTime·());</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i3·:·a·=·sort·apply·(3,·i·->·random·(7))105 <td>··············<pre><code·class="language-macaulay2">i3·:·a·=·sort·apply·(3,·i·->·random·(7))
  
106 o3·=·{4,·5,·6}106 o3·=·{1,·2,·3}
  
107 o3·:·List</code></pre>107 o3·:·List</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i4·:·assert·isWellDefined·kleinschmidt·(4,a)</code></pre>110 <td>··············<pre><code·class="language-macaulay2">i4·:·assert·isWellDefined·kleinschmidt·(4,a)</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ········</table>112 ········</table>
Offset 118, 15 lines modifiedOffset 118, 15 lines modified
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j·->·random·(1,9));</code></pre>120 <td>··············<pre><code·class="language-macaulay2">i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j·->·random·(1,9));</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i7·:·q123 <td>··············<pre><code·class="language-macaulay2">i7·:·q
  
124 o7·=·{2,·3,·4,·8,·9}124 o7·=·{1,·2,·5,·8,·8}
  
125 o7·:·List</code></pre>125 o7·:·List</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i8·:·assert·isWellDefined·weightedProjectiveSpace·q</code></pre>128 <td>··············<pre><code·class="language-macaulay2">i8·:·assert·isWellDefined·weightedProjectiveSpace·q</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ········</table>130 ········</table>
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31 The·first·examples·illustrate·that·small·projective·spaces·are·well-defined.31 The·first·examples·illustrate·that·small·projective·spaces·are·well-defined.
32 i1·:·assert·all·(5,·d·->·isWellDefined·toricProjectiveSpace·(d+1))32 i1·:·assert·all·(5,·d·->·isWellDefined·toricProjectiveSpace·(d+1))
33 The·second·examples·show·that·a·randomly·selected·Kleinschmidt·toric·variety33 The·second·examples·show·that·a·randomly·selected·Kleinschmidt·toric·variety
34 and·a·weighted·projective·space·are·also·well-defined.34 and·a·weighted·projective·space·are·also·well-defined.
35 i2·:·setRandomSeed·(currentTime·());35 i2·:·setRandomSeed·(currentTime·());
36 i3·:·a·=·sort·apply·(3,·i·->·random·(7))36 i3·:·a·=·sort·apply·(3,·i·->·random·(7))
  
37 o3·=·{4,·5,·6}37 o3·=·{1,·2,·3}
  
38 o3·:·List38 o3·:·List
39 i4·:·assert·isWellDefined·kleinschmidt·(4,a)39 i4·:·assert·isWellDefined·kleinschmidt·(4,a)
40 i5·:·q·=·sort·apply·(5,·j·->·random·(1,9));40 i5·:·q·=·sort·apply·(5,·j·->·random·(1,9));
41 i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j41 i6·:·while·not·all·(subsets·(q,#q-1),·s·->·gcd·s·===·1)·do·q·=·sort·apply·(5,·j
42 ->·random·(1,9));42 ->·random·(1,9));
43 i7·:·q43 i7·:·q
  
44 o7·=·{2,·3,·4,·8,·9}44 o7·=·{1,·2,·5,·8,·8}
  
45 o7·:·List45 o7·:·List
46 i8·:·assert·isWellDefined·weightedProjectiveSpace·q46 i8·:·assert·isWellDefined·weightedProjectiveSpace·q
47 The·next·ten·examples·illustrate·various·ways·that·two·lists·can·fail·to·define47 The·next·ten·examples·illustrate·various·ways·that·two·lists·can·fail·to·define
48 a·normal·toric·variety.·By·making·the·current·debugging·level·greater·than·one,48 a·normal·toric·variety.·By·making·the·current·debugging·level·greater·than·one,
49 one·gets·some·addition·information·about·the·nature·of·the·failure.49 one·gets·some·addition·information·about·the·nature·of·the·failure.
50 i9·:·X·=·new·MutableHashTable;50 i9·:·X·=·new·MutableHashTable;
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Offset 94, 29 lines modifiedOffset 94, 29 lines modified
94 o2·=·5*PP294 o2·=·5*PP2
95 ··········095 ··········0
  
96 o2·:·ToricDivisor·on·PP2</code></pre>96 o2·:·ToricDivisor·on·PP2</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·M1·=·elapsedTime·monomials·D199 <td>··············<pre><code·class="language-macaulay2">i3·:·M1·=·elapsedTime·monomials·D1
100 ·--·.149902s·elapsed100 ·--·.15401s·elapsed
  
101 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···4···101 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···4···
102 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,102 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,
103 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·2·103 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·2·
104 ·····------------------------------------------------------------------------104 ·····------------------------------------------------------------------------
105 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····5105 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····5
106 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}106 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}
107 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···0107 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···0
  
108 o3·:·List</code></pre>108 o3·:·List</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring·variety·D1))111 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring·variety·D1))
112 ·--·.0010759s·elapsed</code></pre>112 ·--·.00455756s·elapsed</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ········</table>114 ········</table>
115 ········<div>115 ········<div>
116 ··········<p>Toric·varieties·of·Picard-rank·2·are·slightly·more·interesting.</p>116 ··········<p>Toric·varieties·of·Picard-rank·2·are·slightly·more·interesting.</p>
117 ········</div>117 ········</div>
118 ········<table·class="examples">118 ········<table·class="examples">
119 ··········<tr>119 ··········<tr>
Offset 128, 46 lines modifiedOffset 128, 46 lines modified
128 o6·=·2*FF2··+·3*FF2128 o6·=·2*FF2··+·3*FF2
129 ··········0········1129 ··········0········1
  
130 o6·:·ToricDivisor·on·FF2</code></pre>130 o6·:·ToricDivisor·on·FF2</code></pre>
131 </td>··········</tr>131 </td>··········</tr>
132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i7·:·M2·=·elapsedTime·monomials·D2133 <td>··············<pre><code·class="language-macaulay2">i7·:·M2·=·elapsedTime·monomials·D2
134 ·--·.431127s·elapsed134 ·--·.31948s·elapsed
  
135 ·······2·····3·2·····3·····2·3135 ·······2·····3·2·····3·····2·3
136 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}136 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}
137 ·······1·3···1·2···0·1·2···0·1137 ·······1·3···1·2···0·1·2···0·1
  
138 o7·:·List</code></pre>138 o7·:·List</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring·variety·D2))141 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring·variety·D2))
142 ·--·.00103813s·elapsed</code></pre>142 ·--·.00111549s·elapsed</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i9·:·X·=·kleinschmidt·(5,·{1,2,3});</code></pre>145 <td>··············<pre><code·class="language-macaulay2">i9·:·X·=·kleinschmidt·(5,·{1,2,3});</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i10·:·D3·=·3*X_0·+·5*X_1148 <td>··············<pre><code·class="language-macaulay2">i10·:·D3·=·3*X_0·+·5*X_1
  
149 o10·=·3*X··+·5*X149 o10·=·3*X··+·5*X
150 ·········0······1150 ·········0······1
  
151 o10·:·ToricDivisor·on·X</code></pre>151 o10·:·ToricDivisor·on·X</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i11·:·m3·=·elapsedTime·#·monomials·D3154 <td>··············<pre><code·class="language-macaulay2">i11·:·m3·=·elapsedTime·#·monomials·D3
155 ·--·34.4709s·elapsed155 ·--·36.6909s·elapsed
  
156 o11·=·7909</code></pre>156 o11·=·7909</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ··········<tr>158 ··········<tr>
159 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety·D3))159 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety·D3))
160 ·--·.0246716s·elapsed</code></pre>160 ·--·.0231379s·elapsed</code></pre>
161 </td>··········</tr>161 </td>··········</tr>
162 ········</table>162 ········</table>
163 ········<div>163 ········<div>
164 ··········<p>By·exploiting·<a·title="computes·the·lattice·points·of·a·polytope"·href="../../Polyhedra/html/_lattice__Points.html">latticePoints</a>,·this·method·function·avoids·using·the·<a·title="basis·or·generating·set·of·all·or·part·of·a·ring,·ideal·or·module"·href="../../Macaulay2Doc/html/_basis.html">basis</a>·function.</p>164 ··········<p>By·exploiting·<a·title="computes·the·lattice·points·of·a·polytope"·href="../../Polyhedra/html/_lattice__Points.html">latticePoints</a>,·this·method·function·avoids·using·the·<a·title="basis·or·generating·set·of·all·or·part·of·a·ring,·ideal·or·module"·href="../../Macaulay2Doc/html/_basis.html">basis</a>·function.</p>
165 ········</div>165 ········</div>
166 ······</div>166 ······</div>
167 ······<div>167 ······<div>
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28 i2·:·D1·=·5*PP2_028 i2·:·D1·=·5*PP2_0
  
29 o2·=·5*PP229 o2·=·5*PP2
30 ··········030 ··········0
  
31 o2·:·ToricDivisor·on·PP231 o2·:·ToricDivisor·on·PP2
32 i3·:·M1·=·elapsedTime·monomials·D132 i3·:·M1·=·elapsedTime·monomials·D1
33 ·--·.149902s·elapsed33 ·--·.15401s·elapsed
  
34 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···434 ·······5·····4·····4···2·3·······3···2·3···3·2·····2·2···2···2···3·2···4
35 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,35 o3·=·{x·,·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·,·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·x·,
36 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·236 ·······2···1·2···0·2···1·2···0·1·2···0·2···1·2···0·1·2···0·1·2···0·2···1·2
37 ·····------------------------------------------------------------------------37 ·····------------------------------------------------------------------------
38 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····538 ········3·····2·2·····3·······4·····5·····4···2·3···3·2···4·····5
39 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}39 ·····x·x·x·,·x·x·x·,·x·x·x·,·x·x·,·x·,·x·x·,·x·x·,·x·x·,·x·x·,·x·}
40 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···040 ······0·1·2···0·1·2···0·1·2···0·2···1···0·1···0·1···0·1···0·1···0
  
41 o3·:·List41 o3·:·List
42 i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring42 i4·:·elapsedTime·assert·(set·M1·===·set·first·entries·basis(degree·D1,·ring
43 variety·D1))43 variety·D1))
44 ·--·.0010759s·elapsed44 ·--·.00455756s·elapsed
45 Toric·varieties·of·Picard-rank·2·are·slightly·more·interesting.45 Toric·varieties·of·Picard-rank·2·are·slightly·more·interesting.
46 i5·:·FF2·=·hirzebruchSurface·2;46 i5·:·FF2·=·hirzebruchSurface·2;
47 i6·:·D2·=·2*FF2_0·+·3·*·FF2_147 i6·:·D2·=·2*FF2_0·+·3·*·FF2_1
  
48 o6·=·2*FF2··+·3*FF248 o6·=·2*FF2··+·3*FF2
49 ··········0········149 ··········0········1
  
50 o6·:·ToricDivisor·on·FF250 o6·:·ToricDivisor·on·FF2
51 i7·:·M2·=·elapsedTime·monomials·D251 i7·:·M2·=·elapsedTime·monomials·D2
52 ·--·.431127s·elapsed52 ·--·.31948s·elapsed
  
53 ·······2·····3·2·····3·····2·353 ·······2·····3·2·····3·····2·3
54 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}54 o7·=·{x·x·,·x·x·,·x·x·x·,·x·x·}
55 ·······1·3···1·2···0·1·2···0·155 ·······1·3···1·2···0·1·2···0·1
  
56 o7·:·List56 o7·:·List
57 i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring57 i8·:·elapsedTime·assert·(set·M2·===·set·first·entries·basis·(degree·D2,·ring
58 variety·D2))58 variety·D2))
59 ·--·.00103813s·elapsed59 ·--·.00111549s·elapsed
60 i9·:·X·=·kleinschmidt·(5,·{1,2,3});60 i9·:·X·=·kleinschmidt·(5,·{1,2,3});
61 i10·:·D3·=·3*X_0·+·5*X_161 i10·:·D3·=·3*X_0·+·5*X_1
  
62 o10·=·3*X··+·5*X62 o10·=·3*X··+·5*X
63 ·········0······163 ·········0······1
  
64 o10·:·ToricDivisor·on·X64 o10·:·ToricDivisor·on·X
65 i11·:·m3·=·elapsedTime·#·monomials·D365 i11·:·m3·=·elapsedTime·#·monomials·D3
66 ·--·34.4709s·elapsed66 ·--·36.6909s·elapsed
  
67 o11·=·790967 o11·=·7909
68 i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety68 i12·:·elapsedTime·assert·(m3·===·#first·entries·basis·(degree·D3,·ring·variety
69 D3))69 D3))
70 ·--·.0246716s·elapsed70 ·--·.0231379s·elapsed
71 By·exploiting·_\x8l_\x8a_\x8t_\x8t_\x8i_\x8c_\x8e_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s,·this·method·function·avoids·using·the·_\x8b_\x8a_\x8s_\x8i_\x8s71 By·exploiting·_\x8l_\x8a_\x8t_\x8t_\x8i_\x8c_\x8e_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s,·this·method·function·avoids·using·the·_\x8b_\x8a_\x8s_\x8i_\x8s
72 function.72 function.
73 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*73 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
74 ····*·_\x8w_\x8o_\x8r_\x8k_\x8i_\x8n_\x8g_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8d_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r_\x8s·--·information·about·toric·divisors·and·their74 ····*·_\x8w_\x8o_\x8r_\x8k_\x8i_\x8n_\x8g_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8d_\x8i_\x8v_\x8i_\x8s_\x8o_\x8r_\x8s·--·information·about·toric·divisors·and·their
75 ······related·groups75 ······related·groups
76 ····*·_\x8r_\x8i_\x8n_\x8g_\x8(_\x8N_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·make·the·total·coordinate·ring·(a.k.a.·Cox76 ····*·_\x8r_\x8i_\x8n_\x8g_\x8(_\x8N_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·make·the·total·coordinate·ring·(a.k.a.·Cox
77 ······ring)77 ······ring)
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120 ········</table>120 ········</table>
121 ········<div>121 ········<div>
122 ··········<p>The·recommended·method·for·creating·a·<a·title="the·class·of·all·normal·toric·varieties"·href="___Normal__Toric__Variety.html">NormalToricVariety</a>·from·a·fan·is·<a·title="make·a·normal·toric·variety"·href="_normal__Toric__Variety_lp__List_cm__List_rp.html">normalToricVariety(List,List)</a>.··In·fact,·this·package·avoids·using·objects·from·the·<a·title="for·computations·with·convex·polyhedra,·cones,·and·fans"·href="../../Polyhedra/html/index.html">Polyhedra</a>·package·whenever·possible.···Here·is·a·trivial·example,·namely·projective·2-space,·illustrating·the·substantial·increase·in·time·resulting·from·the·use·of·a·<a·title="for·computations·with·convex·polyhedra,·cones,·and·fans"·href="../../Polyhedra/html/index.html">Polyhedra</a>·fan.</p>122 ··········<p>The·recommended·method·for·creating·a·<a·title="the·class·of·all·normal·toric·varieties"·href="___Normal__Toric__Variety.html">NormalToricVariety</a>·from·a·fan·is·<a·title="make·a·normal·toric·variety"·href="_normal__Toric__Variety_lp__List_cm__List_rp.html">normalToricVariety(List,List)</a>.··In·fact,·this·package·avoids·using·objects·from·the·<a·title="for·computations·with·convex·polyhedra,·cones,·and·fans"·href="../../Polyhedra/html/index.html">Polyhedra</a>·package·whenever·possible.···Here·is·a·trivial·example,·namely·projective·2-space,·illustrating·the·substantial·increase·in·time·resulting·from·the·use·of·a·<a·title="for·computations·with·convex·polyhedra,·cones,·and·fans"·href="../../Polyhedra/html/index.html">Polyhedra</a>·fan.</p>
123 ········</div>123 ········</div>
124 ········<table·class="examples">124 ········<table·class="examples">
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})126 <td>··············<pre><code·class="language-macaulay2">i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})
127 ·--·used·0.000753955s·(cpu);·1.7266e-05s·(thread);·0s·(gc)127 ·--·used·0.000219281s·(cpu);·2.4907e-05s·(thread);·0s·(gc)
  
128 o6·=·X1128 o6·=·X1
  
129 o6·:·NormalToricVariety</code></pre>129 o6·:·NormalToricVariety</code></pre>
130 </td>··········</tr>130 </td>··········</tr>
131 ··········<tr>131 ··········<tr>
132 <td>··············<pre><code·class="language-macaulay2">i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull·matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};132 <td>··············<pre><code·class="language-macaulay2">i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull·matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};
133 ·--·used·0.0911707s·(cpu);·0.0503166s·(thread);·0s·(gc)</code></pre>133 ·--·used·0.112158s·(cpu);·0.0685298s·(thread);·0s·(gc)</code></pre>
134 </td>··········</tr>134 </td>··········</tr>
135 ··········<tr>135 ··········<tr>
136 <td>··············<pre><code·class="language-macaulay2">i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)</code></pre>136 <td>··············<pre><code·class="language-macaulay2">i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ········</table>138 ········</table>
139 ······</div>139 ······</div>
140 ······<div>140 ······<div>
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49 i5·:·assert·(transpose·matrix·rays·X·==·rays·F·and·max·X·==·sort·maxCones·F)49 i5·:·assert·(transpose·matrix·rays·X·==·rays·F·and·max·X·==·sort·maxCones·F)
50 The·recommended·method·for·creating·a·_\x8N_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·from·a·fan·is50 The·recommended·method·for·creating·a·_\x8N_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·from·a·fan·is
51 _\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8L_\x8i_\x8s_\x8t_\x8,_\x8L_\x8i_\x8s_\x8t_\x8).·In·fact,·this·package·avoids·using·objects·from51 _\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8L_\x8i_\x8s_\x8t_\x8,_\x8L_\x8i_\x8s_\x8t_\x8).·In·fact,·this·package·avoids·using·objects·from
52 the·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·package·whenever·possible.·Here·is·a·trivial·example,·namely52 the·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·package·whenever·possible.·Here·is·a·trivial·example,·namely
53 projective·2-space,·illustrating·the·substantial·increase·in·time·resulting53 projective·2-space,·illustrating·the·substantial·increase·in·time·resulting
54 from·the·use·of·a·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·fan.54 from·the·use·of·a·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·fan.
55 i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})55 i6·:·X1·=·time·normalToricVariety·({{-1,-1},{1,0},{0,1}},·{{0,1},{1,2},{0,2}})
56 ·--·used·0.000753955s·(cpu);·1.7266e-05s·(thread);·0s·(gc)56 ·--·used·0.000219281s·(cpu);·2.4907e-05s·(thread);·0s·(gc)
  
57 o6·=·X157 o6·=·X1
  
58 o6·:·NormalToricVariety58 o6·:·NormalToricVariety
59 i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull59 i7·:·X2·=·time·normalToricVariety·fan·{posHull·matrix·{{-1,1},{-1,0}},·posHull
60 matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};60 matrix·{{1,0},{0,1}},·posHull·matrix{{-1,0},{-1,1}}};
61 ·--·used·0.0911707s·(cpu);·0.0503166s·(thread);·0s·(gc)61 ·--·used·0.112158s·(cpu);·0.0685298s·(thread);·0s·(gc)
62 i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)62 i8·:·assert·(sort·rays·X1·==·sort·rays·X2·and·max·X1·==·max·X2)
63 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*63 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
64 ····*·_\x8m_\x8a_\x8k_\x8i_\x8n_\x8g_\x8·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8·_\x8t_\x8o_\x8r_\x8i_\x8c_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s·--·information·about·the·basic·constructors64 ····*·_\x8m_\x8a_\x8k_\x8i_\x8n_\x8g_\x8·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8·_\x8t_\x8o_\x8r_\x8i_\x8c_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s·--·information·about·the·basic·constructors
65 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·make·a·normal·toric·variety65 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·make·a·normal·toric·variety
66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
67 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8F_\x8a_\x8n_\x8)·--·make·a·normal·toric·variety·from·a·'Polyhedra'67 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8F_\x8a_\x8n_\x8)·--·make·a·normal·toric·variety·from·a·'Polyhedra'
68 ······fan68 ······fan
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202 ······|·0·0·1·|202 ······|·0·0·1·|
  
203 ···············2·······3203 ···············2·······3
204 o18·:·Matrix·ZZ··&lt;--·ZZ</code></pre>204 o18·:·Matrix·ZZ··&lt;--·ZZ</code></pre>
205 </td>··········</tr>205 </td>··········</tr>
206 ··········<tr>206 ··········<tr>
207 <td>··············<pre><code·class="language-macaulay2">i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);207 <td>··············<pre><code·class="language-macaulay2">i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);
208 ·--·used·0.0721583s·(cpu);·0.0340272s·(thread);·0s·(gc)</code></pre>208 ·--·used·0.0912587s·(cpu);·0.0400754s·(thread);·0s·(gc)</code></pre>
209 </td>··········</tr>209 </td>··········</tr>
210 ··········<tr>210 ··········<tr>
211 <td>··············<pre><code·class="language-macaulay2">i20·:·X2·=·time·normalToricVariety·vertMatrix;211 <td>··············<pre><code·class="language-macaulay2">i20·:·X2·=·time·normalToricVariety·vertMatrix;
212 ·--·used·0.00112943s·(cpu);·0.00195081s·(thread);·0s·(gc)</code></pre>212 ·--·used·0.00203453s·(cpu);·0.00291772s·(thread);·0s·(gc)</code></pre>
213 </td>··········</tr>213 </td>··········</tr>
214 ··········<tr>214 ··········<tr>
215 <td>··············<pre><code·class="language-macaulay2">i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)</code></pre>215 <td>··············<pre><code·class="language-macaulay2">i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)</code></pre>
216 </td>··········</tr>216 </td>··········</tr>
217 ········</table>217 ········</table>
218 ······</div>218 ······</div>
219 ······<div>219 ······<div>
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103 o18·=·|·0·1·0·|103 o18·=·|·0·1·0·|
104 ······|·0·0·1·|104 ······|·0·0·1·|
  
105 ···············2·······3105 ···············2·······3
106 o18·:·Matrix·ZZ··<--·ZZ106 o18·:·Matrix·ZZ··<--·ZZ
107 i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);107 i19·:·X1·=·time·normalToricVariety·convexHull·(vertMatrix);
108 ·--·used·0.0721583s·(cpu);·0.0340272s·(thread);·0s·(gc)108 ·--·used·0.0912587s·(cpu);·0.0400754s·(thread);·0s·(gc)
109 i20·:·X2·=·time·normalToricVariety·vertMatrix;109 i20·:·X2·=·time·normalToricVariety·vertMatrix;
110 ·--·used·0.00112943s·(cpu);·0.00195081s·(thread);·0s·(gc)110 ·--·used·0.00203453s·(cpu);·0.00291772s·(thread);·0s·(gc)
111 i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)111 i21·:·assert·(set·rays·X2·===·set·rays·X1·and·max·X1·===·max·X2)
112 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*112 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
113 ····*·_\x8m_\x8a_\x8k_\x8i_\x8n_\x8g_\x8·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8·_\x8t_\x8o_\x8r_\x8i_\x8c_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s·--·information·about·the·basic·constructors113 ····*·_\x8m_\x8a_\x8k_\x8i_\x8n_\x8g_\x8·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8·_\x8t_\x8o_\x8r_\x8i_\x8c_\x8·_\x8v_\x8a_\x8r_\x8i_\x8e_\x8t_\x8i_\x8e_\x8s·--·information·about·the·basic·constructors
114 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·make·a·normal·toric·variety·from·a·polytope114 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·make·a·normal·toric·variety·from·a·polytope
115 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*115 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
116 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8o_\x8n_\x8)·--·make·a·normal·toric·variety·from·a116 ····*·_\x8n_\x8o_\x8r_\x8m_\x8a_\x8l_\x8T_\x8o_\x8r_\x8i_\x8c_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8(_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8o_\x8n_\x8)·--·make·a·normal·toric·variety·from·a
117 ······'Polyhedra'·polyhedron117 ······'Polyhedra'·polyhedron
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18 o3·:·R18 o3·:·R
  
19 i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},·Software=>BERTINI)·else·"need·to·install·Bertini·to·run·these·lines"19 i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},·Software=>BERTINI)·else·"need·to·install·Bertini·to·run·these·lines"
20 sh:·1:·bertini:·not·found20 sh:·line·1:·bertini:·command·not·found
21 true21 true
  
  
22 o4·=·need·to·install·Bertini·to·run·these·lines22 o4·=·need·to·install·Bertini·to·run·these·lines
  
23 i5·:·23 i5·:·
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./usr/share/doc/Macaulay2/NumericalAlgebraicGeometry/html/_parameter__Homotopy_lp__List_cm__List_cm__List_rp.html
    
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102 o3·:·R</code></pre>102 o3·:·R</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},·Software=>BERTINI)·else·&quot;need·to·install·Bertini·to·run·these·lines&quot;105 <td>··············<pre><code·class="language-macaulay2">i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},·Software=>BERTINI)·else·&quot;need·to·install·Bertini·to·run·these·lines&quot;
106 sh:·1:·bertini:·not·found106 sh:·line·1:·bertini:·command·not·found
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108 o4·=·need·to·install·Bertini·to·run·these·lines</code></pre>108 o4·=·need·to·install·Bertini·to·run·these·lines</code></pre>
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37 o3·:·R37 o3·:·R
38 i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},38 i4·:·try·parameterHomotopy({f1,f2},{u1,u2,u3},{{1,0,0},{0,1+2*ii,0}},
39 Software=>BERTINI)·else·"need·to·install·Bertini·to·run·these·lines"39 Software=>BERTINI)·else·"need·to·install·Bertini·to·run·these·lines"
40 sh:·1:·bertini:·not·found40 sh:·line·1:·bertini:·command·not·found
41 true41 true
  
  
42 o4·=·need·to·install·Bertini·to·run·these·lines42 o4·=·need·to·install·Bertini·to·run·these·lines
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23 --·warning:·experimental·computation·over·inexact·field·begun23 --·warning:·experimental·computation·over·inexact·field·begun
24 --··········results·not·reliable·(one·warning·given·per·session)24 --··········results·not·reliable·(one·warning·given·per·session)
  
25 o4·=·true25 o4·=·true
  
26 i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)26 i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)
27 Sampling·image·points·...27 Sampling·image·points·...
28 ·····--·used·.00399981·seconds28 ·····--·used·.0116347·seconds
29 Creating·interpolation·matrix·...29 Creating·interpolation·matrix·...
30 ·····--·used·.00400961·seconds30 ·····--·used·.0080006·seconds
31 Performing·normalization·preconditioning·...31 Performing·normalization·preconditioning·...
32 ·····--·used·.000833194·seconds32 ·····--·used·.00400396·seconds
33 Computing·numerical·kernel·...33 Computing·numerical·kernel·...
34 ·····--·used·.00182452·seconds34 ·····--·used·0·seconds
  
35 o5·=·a·"numerical·interpolation·table",·indicating35 o5·=·a·"numerical·interpolation·table",·indicating
36 ·····the·space·of·degree·3·forms·in·the·ideal·of·the·image·has·dimension·336 ·····the·space·of·degree·3·forms·in·the·ideal·of·the·image·has·dimension·3
  
37 o5·:·NumericalInterpolationTable37 o5·:·NumericalInterpolationTable
  
38 i6·:·extractImageEquations(T,·AttemptZZ·=>·true)38 i6·:·extractImageEquations(T,·AttemptZZ·=>·true)
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./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_extract__Image__Equations.out
    
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13 o2·=·|·s3·s2t·st2·t3·|13 o2·=·|·s3·s2t·st2·t3·|
  
14 ·············1······414 ·············1······4
15 o2·:·Matrix·R··<--·R15 o2·:·Matrix·R··<--·R
  
16 i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)16 i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)
17 Sampling·image·points·...17 Sampling·image·points·...
18 ·····--·used·0·seconds18 ·····--·used·.00399549·seconds
19 Creating·interpolation·matrix·...19 Creating·interpolation·matrix·...
20 ·····--·used·0·seconds20 ·····--·used·.00403205·seconds
21 Performing·normalization·preconditioning·...21 Performing·normalization·preconditioning·...
22 ·····--·used·0·seconds22 ·····--·used·0·seconds
23 Computing·numerical·kernel·...23 Computing·numerical·kernel·...
24 ·····--·used·0·seconds24 ·····--·used·0·seconds
  
25 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|25 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|
  
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./usr/share/doc/Macaulay2/NumericalImplicitization/example-output/_numerical__Hilbert__Function.out
    
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13 o2·=·|·s3·s2t·st2·t3·|13 o2·=·|·s3·s2t·st2·t3·|
  
14 ·············1······414 ·············1······4
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21 Performing·normalization·preconditioning·...21 Performing·normalization·preconditioning·...
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23 Computing·numerical·kernel·...23 Computing·numerical·kernel·...
24 ·····--·used·0·seconds24 ·····--·used·0·seconds
  
25 o3·=·a·"numerical·interpolation·table",·indicating25 o3·=·a·"numerical·interpolation·table",·indicating
26 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·2226 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·22
  
27 o3·:·NumericalInterpolationTable27 o3·:·NumericalInterpolationTable
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34 i5·:·F·=·(minors(2,·genericMatrix(R,·2,·4)))_*;34 i5·:·F·=·(minors(2,·genericMatrix(R,·2,·4)))_*;
  
35 i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);35 i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);
  
36 i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)36 i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)
37 Creating·interpolation·matrix·...37 Creating·interpolation·matrix·...
38 ·····--·used·0·seconds38 ·····--·used·.000801253·seconds
39 Performing·normalization·preconditioning·...39 Performing·normalization·preconditioning·...
40 ·····--·used·.00800582·seconds40 ·····--·used·.00799824·seconds
41 Computing·numerical·kernel·...41 Computing·numerical·kernel·...
42 ·····--·used·0·seconds42 ·····--·used·.00399985·seconds
  
43 o7·=·a·"numerical·interpolation·table",·indicating43 o7·=·a·"numerical·interpolation·table",·indicating
44 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·144 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·1
  
45 o7·:·NumericalInterpolationTable45 o7·:·NumericalInterpolationTable
  
46 i8·:·46 i8·:·
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22 --·warning:·experimental·computation·over·inexact·field·begun22 --·warning:·experimental·computation·over·inexact·field·begun
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24 ·············1······7024 ·············1······70
25 o8·:·Matrix·R··<--·R25 o8·:·Matrix·R··<--·R
  
26 i9·:·time·numericalImageDim(F,·ideal·0_R)26 i9·:·time·numericalImageDim(F,·ideal·0_R)
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28 o9·=·6928 o9·=·69
  
29 i10·:·29 i10·:·
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31 o5·:·Ideal·of·R31 o5·:·Ideal·of·R
  
32 i6·:·I·=·I1·+·I2;32 i6·:·I·=·I1·+·I2;
  
33 o6·:·Ideal·of·R33 o6·:·Ideal·of·R
  
34 i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)34 i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)
35 ·--·.620537s·elapsed35 ·--·.637533s·elapsed
  
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./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html
    
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89 --··········results·not·reliable·(one·warning·given·per·session)89 --··········results·not·reliable·(one·warning·given·per·session)
  
90 o4·=·true</code></pre>90 o4·=·true</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)93 <td>··············<pre><code·class="language-macaulay2">i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)
94 Sampling·image·points·...94 Sampling·image·points·...
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101 ·····--·used·.00182452·seconds101 ·····--·used·0·seconds
  
102 o5·=·a·&quot;numerical·interpolation·table&quot;,·indicating102 o5·=·a·&quot;numerical·interpolation·table&quot;,·indicating
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104 o5·:·NumericalInterpolationTable</code></pre>104 o5·:·NumericalInterpolationTable</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
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36 ==·036 ==·0
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38 --··········results·not·reliable·(one·warning·given·per·session)38 --··········results·not·reliable·(one·warning·given·per·session)
  
39 o4·=·true39 o4·=·true
40 i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)40 i5·:·T·=·numericalHilbertFunction(F,·I,·3,·ConvertToCone·=>·true)
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49 o5·=·a·"numerical·interpolation·table",·indicating49 o5·=·a·"numerical·interpolation·table",·indicating
50 ·····the·space·of·degree·3·forms·in·the·ideal·of·the·image·has·dimension·350 ·····the·space·of·degree·3·forms·in·the·ideal·of·the·image·has·dimension·3
  
51 o5·:·NumericalInterpolationTable51 o5·:·NumericalInterpolationTable
52 i6·:·extractImageEquations(T,·AttemptZZ·=>·true)52 i6·:·extractImageEquations(T,·AttemptZZ·=>·true)
  
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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html
    
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105 ·············1······4105 ·············1······4
106 o2·:·Matrix·R··&lt;--·R</code></pre>106 o2·:·Matrix·R··&lt;--·R</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)109 <td>··············<pre><code·class="language-macaulay2">i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)
110 Sampling·image·points·...110 Sampling·image·points·...
111 ·····--·used·0·seconds111 ·····--·used·.00399549·seconds
112 Creating·interpolation·matrix·...112 Creating·interpolation·matrix·...
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114 Performing·normalization·preconditioning·...114 Performing·normalization·preconditioning·...
115 ·····--·used·0·seconds115 ·····--·used·0·seconds
116 Computing·numerical·kernel·...116 Computing·numerical·kernel·...
117 ·····--·used·0·seconds117 ·····--·used·0·seconds
  
118 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|118 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|
  
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41 o2·=·|·s3·s2t·st2·t3·|41 o2·=·|·s3·s2t·st2·t3·|
  
42 ·············1······442 ·············1······4
43 o2·:·Matrix·R··<--·R43 o2·:·Matrix·R··<--·R
44 i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)44 i3·:·extractImageEquations(F,·ideal·0_R,·2,·AttemptZZ·=>·true)
45 Sampling·image·points·...45 Sampling·image·points·...
46 ·····--·used·0·seconds46 ·····--·used·.00399549·seconds
47 Creating·interpolation·matrix·...47 Creating·interpolation·matrix·...
48 ·····--·used·0·seconds48 ·····--·used·.00403205·seconds
49 Performing·normalization·preconditioning·...49 Performing·normalization·preconditioning·...
50 ·····--·used·0·seconds50 ·····--·used·0·seconds
51 Computing·numerical·kernel·...51 Computing·numerical·kernel·...
52 ·····--·used·0·seconds52 ·····--·used·0·seconds
  
53 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|53 o3·=·|·y_1^2-y_0y_2·y_1y_2-y_0y_3·y_2^2-y_1y_3·|
  
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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html
    
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114 ·············1······4114 ·············1······4
115 o2·:·Matrix·R··&lt;--·R</code></pre>115 o2·:·Matrix·R··&lt;--·R</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i3·:·numericalHilbertFunction(F,·ideal·0_R,·4)118 <td>··············<pre><code·class="language-macaulay2">i3·:·numericalHilbertFunction(F,·ideal·0_R,·4)
119 Sampling·image·points·...119 Sampling·image·points·...
120 ·····--·used·.00826644·seconds120 ·····--·used·.0159557·seconds
121 Creating·interpolation·matrix·...121 Creating·interpolation·matrix·...
122 ·····--·used·.00784787·seconds122 ·····--·used·.0160061·seconds
123 Performing·normalization·preconditioning·...123 Performing·normalization·preconditioning·...
124 ·····--·used·.00437966·seconds124 ·····--·used·.00799573·seconds
125 Computing·numerical·kernel·...125 Computing·numerical·kernel·...
126 ·····--·used·0·seconds126 ·····--·used·0·seconds
  
127 o3·=·a·&quot;numerical·interpolation·table&quot;,·indicating127 o3·=·a·&quot;numerical·interpolation·table&quot;,·indicating
128 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·22128 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·22
  
129 o3·:·NumericalInterpolationTable</code></pre>129 o3·:·NumericalInterpolationTable</code></pre>
Offset 146, 19 lines modifiedOffset 146, 19 lines modified
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);</code></pre>148 <td>··············<pre><code·class="language-macaulay2">i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);</code></pre>
149 </td>··········</tr>149 </td>··········</tr>
150 ··········<tr>150 ··········<tr>
151 <td>··············<pre><code·class="language-macaulay2">i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)151 <td>··············<pre><code·class="language-macaulay2">i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)
152 Creating·interpolation·matrix·...152 Creating·interpolation·matrix·...
153 ·····--·used·0·seconds153 ·····--·used·.000801253·seconds
154 Performing·normalization·preconditioning·...154 Performing·normalization·preconditioning·...
155 ·····--·used·.00800582·seconds155 ·····--·used·.00799824·seconds
156 Computing·numerical·kernel·...156 Computing·numerical·kernel·...
157 ·····--·used·0·seconds157 ·····--·used·.00399985·seconds
  
158 o7·=·a·&quot;numerical·interpolation·table&quot;,·indicating158 o7·=·a·&quot;numerical·interpolation·table&quot;,·indicating
159 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·1159 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·1
  
160 o7·:·NumericalInterpolationTable</code></pre>160 o7·:·NumericalInterpolationTable</code></pre>
161 </td>··········</tr>161 </td>··········</tr>
162 ········</table>162 ········</table>
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html2text {}
    
Offset 60, 19 lines modifiedOffset 60, 19 lines modified
  
60 o2·=·|·s3·s2t·st2·t3·|60 o2·=·|·s3·s2t·st2·t3·|
  
61 ·············1······461 ·············1······4
62 o2·:·Matrix·R··<--·R62 o2·:·Matrix·R··<--·R
63 i3·:·numericalHilbertFunction(F,·ideal·0_R,·4)63 i3·:·numericalHilbertFunction(F,·ideal·0_R,·4)
64 Sampling·image·points·...64 Sampling·image·points·...
65 ·····--·used·.00826644·seconds65 ·····--·used·.0159557·seconds
66 Creating·interpolation·matrix·...66 Creating·interpolation·matrix·...
67 ·····--·used·.00784787·seconds67 ·····--·used·.0160061·seconds
68 Performing·normalization·preconditioning·...68 Performing·normalization·preconditioning·...
69 ·····--·used·.00437966·seconds69 ·····--·used·.00799573·seconds
70 Computing·numerical·kernel·...70 Computing·numerical·kernel·...
71 ·····--·used·0·seconds71 ·····--·used·0·seconds
  
72 o3·=·a·"numerical·interpolation·table",·indicating72 o3·=·a·"numerical·interpolation·table",·indicating
73 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·2273 ·····the·space·of·degree·4·forms·in·the·ideal·of·the·image·has·dimension·22
  
74 o3·:·NumericalInterpolationTable74 o3·:·NumericalInterpolationTable
Offset 80, 19 lines modifiedOffset 80, 19 lines modified
80 defining·ideal·of·the·Grassmannian·$Gr(2,4)$·of·$P^1$'s·in·$P^3$,·in·the80 defining·ideal·of·the·Grassmannian·$Gr(2,4)$·of·$P^1$'s·in·$P^3$,·in·the
81 ambient·space·$P^5$.81 ambient·space·$P^5$.
82 i4·:·R·=·CC[x_(1,1)..x_(2,4)];82 i4·:·R·=·CC[x_(1,1)..x_(2,4)];
83 i5·:·F·=·(minors(2,·genericMatrix(R,·2,·4)))_*;83 i5·:·F·=·(minors(2,·genericMatrix(R,·2,·4)))_*;
84 i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);84 i6·:·S·=·numericalImageSample(F,·ideal·0_R,·60);
85 i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)85 i7·:·numericalHilbertFunction(F,·ideal·0_R,·S,·2,·UseSLP·=>·true)
86 Creating·interpolation·matrix·...86 Creating·interpolation·matrix·...
87 ·····--·used·0·seconds87 ·····--·used·.000801253·seconds
88 Performing·normalization·preconditioning·...88 Performing·normalization·preconditioning·...
89 ·····--·used·.00800582·seconds89 ·····--·used·.00799824·seconds
90 Computing·numerical·kernel·...90 Computing·numerical·kernel·...
91 ·····--·used·0·seconds91 ·····--·used·.00399985·seconds
  
92 o7·=·a·"numerical·interpolation·table",·indicating92 o7·=·a·"numerical·interpolation·table",·indicating
93 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·193 ·····the·space·of·degree·2·forms·in·the·ideal·of·the·image·has·dimension·1
  
94 o7·:·NumericalInterpolationTable94 o7·:·NumericalInterpolationTable
95 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*95 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
96 ····*·_\x8N_\x8u_\x8m_\x8e_\x8r_\x8i_\x8c_\x8a_\x8l_\x8I_\x8n_\x8t_\x8e_\x8r_\x8p_\x8o_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8T_\x8a_\x8b_\x8l_\x8e·--·the·class·of·all96 ····*·_\x8N_\x8u_\x8m_\x8e_\x8r_\x8i_\x8c_\x8a_\x8l_\x8I_\x8n_\x8t_\x8e_\x8r_\x8p_\x8o_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8T_\x8a_\x8b_\x8l_\x8e·--·the·class·of·all
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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html
    
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129 --··········results·not·reliable·(one·warning·given·per·session)129 --··········results·not·reliable·(one·warning·given·per·session)
  
130 ·············1······70130 ·············1······70
131 o8·:·Matrix·R··&lt;--·R</code></pre>131 o8·:·Matrix·R··&lt;--·R</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i9·:·time·numericalImageDim(F,·ideal·0_R)134 <td>··············<pre><code·class="language-macaulay2">i9·:·time·numericalImageDim(F,·ideal·0_R)
135 ·--·used·0.056214s·(cpu);·0.0565068s·(thread);·0s·(gc)135 ·--·used·0.0644144s·(cpu);·0.0634717s·(thread);·0s·(gc)
  
136 o9·=·69</code></pre>136 o9·=·69</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ········</table>138 ········</table>
139 ······</div>139 ······</div>
140 ······<div·class="waystouse">140 ······<div·class="waystouse">
141 ········<h2>Ways·to·use·<span·class="tt">numericalImageDim</span>:</h2>141 ········<h2>Ways·to·use·<span·class="tt">numericalImageDim</span>:</h2>
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Offset 46, 15 lines modifiedOffset 46, 15 lines modified
46 i8·:·F·=·sum(1..14,·i·->·basis(4,·R,·Variables=>toList(a_(i,1)..a_(i,5))));46 i8·:·F·=·sum(1..14,·i·->·basis(4,·R,·Variables=>toList(a_(i,1)..a_(i,5))));
47 --·warning:·experimental·computation·over·inexact·field·begun47 --·warning:·experimental·computation·over·inexact·field·begun
48 --··········results·not·reliable·(one·warning·given·per·session)48 --··········results·not·reliable·(one·warning·given·per·session)
  
49 ·············1······7049 ·············1······70
50 o8·:·Matrix·R··<--·R50 o8·:·Matrix·R··<--·R
51 i9·:·time·numericalImageDim(F,·ideal·0_R)51 i9·:·time·numericalImageDim(F,·ideal·0_R)
52 ·--·used·0.056214s·(cpu);·0.0565068s·(thread);·0s·(gc)52 ·--·used·0.0644144s·(cpu);·0.0634717s·(thread);·0s·(gc)
  
53 o9·=·6953 o9·=·69
54 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·n\x8nu\x8um\x8me\x8er\x8ri\x8ic\x8ca\x8al\x8lI\x8Im\x8ma\x8ag\x8ge\x8eD\x8Di\x8im\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*54 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·n\x8nu\x8um\x8me\x8er\x8ri\x8ic\x8ca\x8al\x8lI\x8Im\x8ma\x8ag\x8ge\x8eD\x8Di\x8im\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*
55 ····*·numericalImageDim(List,Ideal)55 ····*·numericalImageDim(List,Ideal)
56 ····*·numericalImageDim(List,Ideal,Point)56 ····*·numericalImageDim(List,Ideal,Point)
57 ····*·numericalImageDim(Matrix,Ideal)57 ····*·numericalImageDim(Matrix,Ideal)
58 ····*·numericalImageDim(Matrix,Ideal,Point)58 ····*·numericalImageDim(Matrix,Ideal,Point)
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./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html
    
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123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·I1·+·I2;124 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·I1·+·I2;
  
125 o6·:·Ideal·of·R</code></pre>125 o6·:·Ideal·of·R</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ··········<tr>127 ··········<tr>
128 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)128 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)
129 ·--·.620537s·elapsed129 ·--·.637533s·elapsed
  
130 o7·=·p130 o7·=·p
  
131 o7·:·Point</code></pre>131 o7·:·Point</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i8·:·matrix·pack(5,·p#Coordinates)134 <td>··············<pre><code·class="language-macaulay2">i8·:·matrix·pack(5,·p#Coordinates)
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50 i5·:·I2·=·ideal·apply(entries·transpose·A,·row·->·sum(row,·v·->·v^2)·-·1);50 i5·:·I2·=·ideal·apply(entries·transpose·A,·row·->·sum(row,·v·->·v^2)·-·1);
  
51 o5·:·Ideal·of·R51 o5·:·Ideal·of·R
52 i6·:·I·=·I1·+·I2;52 i6·:·I·=·I1·+·I2;
  
53 o6·:·Ideal·of·R53 o6·:·Ideal·of·R
54 i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)54 i7·:·elapsedTime·p·=·realPoint(I,·Iterations·=>·100)
55 ·--·.620537s·elapsed55 ·--·.637533s·elapsed
  
56 o7·=·p56 o7·=·p
  
57 o7·:·Point57 o7·:·Point
58 i8·:·matrix·pack(5,·p#Coordinates)58 i8·:·matrix·pack(5,·p#Coordinates)
  
59 o8·=·|·.722359··.289465··-.295808··.591752··-.454678·|59 o8·=·|·.722359··.289465··-.295808··.591752··-.454678·|
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./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/___Lab__Book__Protocol.out
    
Offset 14, 35 lines modifiedOffset 14, 35 lines modified
  
14 i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a14 i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a
  
15 o5·=·215 o5·=·2
  
16 i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0,Verbose=>true))16 i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0,Verbose=>true))
17 unfolding17 unfolding
18 ·--·.0704745s·elapsed18 ·--·.110088s·elapsed
19 flatteningRelations19 flatteningRelations
20 ·--·.0708185s·elapsed20 ·--·.0998481s·elapsed
21 next·gb21 next·gb
22 ·--·.000789972s·elapsed22 ·--·.000700193s·elapsed
23 true23 true
24 unfolding24 unfolding
25 ·--·.0617568s·elapsed25 ·--·.116186s·elapsed
26 flatteningRelations26 flatteningRelations
27 ·--·.0576035s·elapsed27 ·--·.0920172s·elapsed
28 next·gb28 next·gb
29 ·--·.000485693s·elapsed29 ·--·.000812553s·elapsed
30 true30 true
31 ·--·.78134s·elapsed31 ·--·1.3037s·elapsed
  
32 o6·=·{}32 o6·=·{}
  
33 o6·:·List33 o6·:·List
  
34 i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))34 i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))
35 ·--·.788254s·elapsed35 ·--·1.32832s·elapsed
  
36 o7·=·{}36 o7·=·{}
  
37 o7·:·List37 o7·:·List
  
38 i8·:·LL7b=={}38 i8·:·LL7b=={}
  
Offset 75, 22 lines modifiedOffset 75, 22 lines modified
  
75 o10·:·Sequence75 o10·:·Sequence
  
76 i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)76 i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)
77 (13,·1)77 (13,·1)
78 {5,·8,·11,·12}78 {5,·8,·11,·12}
79 unfolding79 unfolding
80 ·--·.129026s·elapsed80 ·--·.188457s·elapsed
81 flatteningRelations81 flatteningRelations
82 ·--·.0900135s·elapsed82 ·--·.129432s·elapsed
83 next·gb83 next·gb
84 ·--·.000640066s·elapsed84 ·--·.000922349s·elapsed
85 true85 true
86 ·--·.354599s·elapsed86 ·--·.546217s·elapsed
87 (5,·8,··all·semigroups·are·smoothable)·--·.385985s·elapsed87 (5,·8,··all·semigroups·are·smoothable)·--·.609911s·elapsed
  
  
88 o11·=·{}88 o11·=·{}
  
89 o11·:·List89 o11·:·List
  
90 i12·:·L={6,8,9,11}90 i12·:·L={6,8,9,11}
Offset 100, 22 lines modifiedOffset 100, 22 lines modified
100 o12·:·List100 o12·:·List
  
101 i13·:·genus·L101 i13·:·genus·L
  
102 o13·=·8102 o13·=·8
  
103 i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)103 i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)
104 ·--·.044793s·elapsed104 ·--·.0564691s·elapsed
105 6105 6
106 false106 false
107 5107 5
108 false108 false
109 4109 4
110 decompose110 decompose
111 ·--·.192843s·elapsed111 ·--·.319679s·elapsed
112 number·of·components:·2112 number·of·components:·2
113 support·c,·codim·c:·{(1,·1),·(16,·3)}113 support·c,·codim·c:·{(1,·1),·(16,·3)}
114 {0,·-1}114 {0,·-1}
  
115 o14·=·true115 o14·=·true
  
116 i15·:·116 i15·:·
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Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i3·:·setRandomSeed·"some·singular·and·some·smooth·curves";9 i3·:·setRandomSeed·"some·singular·and·some·smooth·curves";
  
10 i4·:·elapsedTime·tally·apply(10,i->·(10 i4·:·elapsedTime·tally·apply(10,i->·(
11 ·············c=minors(2,random(S^2,S^{3:-2}));11 ·············c=minors(2,random(S^2,S^{3:-2}));
12 ·············c=sub(c,x_0=>1);12 ·············c=sub(c,x_0=>1);
13 ·············R=kk[support·c];c=sub(c,R);13 ·············R=kk[support·c];c=sub(c,R);
14 ·············heuristicSmoothness·c))14 ·············heuristicSmoothness·c))
15 ·--·1.95527s·elapsed15 ·--·3.32953s·elapsed
  
16 o4·=·Tally{false·=>·6}16 o4·=·Tally{false·=>·6}
17 ···········true·=>·417 ···········true·=>·4
  
18 o4·:·Tally18 o4·:·Tally
  
19 i5·:·19 i5·:·
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./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_is__Smoothable__Semigroup.out
    
Offset 7, 17 lines modifiedOffset 7, 17 lines modified
7 o1·:·List7 o1·:·List
  
8 i2·:·genus·L8 i2·:·genus·L
  
9 o2·=·89 o2·=·8
  
10 i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)10 i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)
11 ·--·.523728s·elapsed11 ·--·.800086s·elapsed
  
12 o3·=·false12 o3·=·false
  
13 i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)13 i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)
14 ·--·2.23245s·elapsed14 ·--·3.39097s·elapsed
  
15 o4·=·true15 o4·=·true
  
16 i5·:·16 i5·:·
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Offset 7, 12 lines modifiedOffset 7, 12 lines modified
7 o1·:·List7 o1·:·List
  
8 i2·:·genus·L8 i2·:·genus·L
  
9 o2·=·89 o2·=·8
  
10 i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)10 i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)
11 ·--·2.57853s·elapsed11 ·--·5.19504s·elapsed
  
12 o3·=·true12 o3·=·true
  
13 i4·:·13 i4·:·
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./usr/share/doc/Macaulay2/NumericalSemigroups/example-output/_non__Weierstrass__Semigroups.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·68609965328516315561 --·-*-·M2-comint·-*-·hash:·6860996532851631556
  
2 i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)2 i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)
3 (6,·7,··all·semigroups·are·smoothable)·--·.879778s·elapsed3 (6,·7,··all·semigroups·are·smoothable)·--·1.85222s·elapsed
  
  
4 o1·=·{}4 o1·=·{}
  
5 o1·:·List5 o1·:·List
  
6 i2·:·LLdifficult={{6,·8,·9,·11}}6 i2·:·LLdifficult={{6,·8,·9,·11}}
Offset 14, 61 lines modifiedOffset 14, 61 lines modified
  
14 o2·:·List14 o2·:·List
  
15 i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)15 i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
16 (17,·5)16 (17,·5)
17 {6,·7,·8,·17}17 {6,·7,·8,·17}
18 unfolding18 unfolding
19 ·--·.262451s·elapsed19 ·--·.626872s·elapsed
20 flatteningRelations20 flatteningRelations
21 ·--·.0990406s·elapsed21 ·--·.194757s·elapsed
22 next·gb22 next·gb
23 ·--·.00139547s·elapsed23 ·--·.00130453s·elapsed
24 true24 true
25 ·--·.631436s·elapsed25 ·--·1.34662s·elapsed
26 {6,·7,·9,·17}26 {6,·7,·9,·17}
27 unfolding27 unfolding
28 ·--·.269116s·elapsed28 ·--·.611882s·elapsed
29 flatteningRelations29 flatteningRelations
30 ·--·.102708s·elapsed30 ·--·.259928s·elapsed
31 next·gb31 next·gb
32 ·--·.00198154s·elapsed32 ·--·.0102483s·elapsed
33 decompose33 decompose
34 ·--·.131151s·elapsed34 ·--·.305591s·elapsed
35 number·of·components:·235 number·of·components:·2
36 support·c,·codim·c:·{(2,·2),·(5,·2)}36 support·c,·codim·c:·{(2,·2),·(5,·2)}
37 {0,·-1}37 {0,·-1}
38 ·--·1.91249s·elapsed38 ·--·3.74559s·elapsed
39 {6,·8,·9,·10}39 {6,·8,·9,·10}
40 unfolding40 unfolding
41 ·--·.0624948s·elapsed41 ·--·.141687s·elapsed
42 flatteningRelations42 flatteningRelations
43 ·--·.0858256s·elapsed43 ·--·.126728s·elapsed
44 next·gb44 next·gb
45 ·--·.000483891s·elapsed45 ·--·.00442511s·elapsed
46 true46 true
47 ·--·.465824s·elapsed47 ·--·.838739s·elapsed
48 {6,·8,·10,·11,·13}48 {6,·8,·10,·11,·13}
49 unfolding49 unfolding
50 ·--·.342213s·elapsed50 ·--·.690592s·elapsed
51 flatteningRelations51 flatteningRelations
52 ·--·.129273s·elapsed52 ·--·.315878s·elapsed
53 next·gb53 next·gb
54 ·--·.0028742s·elapsed54 ·--·.007027s·elapsed
55 decompose55 decompose
56 ·--·.580045s·elapsed56 ·--·1.07767s·elapsed
57 number·of·components:·157 number·of·components:·1
58 support·c,·codim·c:·{(5,·1)}58 support·c,·codim·c:·{(5,·1)}
59 {-1}59 {-1}
 60 ·--·3.45799s·elapsed
60 ·--·1.70163s·elapsed61 ·--·9.38906s·elapsed
61 ·--·4.71154s·elapsed 
62 062 0
  
63 ·--·.000002664s·elapsed63 ·--·.000003808s·elapsed
64 ·--·4.74234s·elapsed64 ·--·9.42074s·elapsed
65 {}65 {}
  
66 o3·=·{{6,·8,·9,·11}}66 o3·=·{{6,·8,·9,·11}}
  
67 o3·:·List67 o3·:·List
  
68 i4·:·68 i4·:·
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Offset 89, 36 lines modifiedOffset 89, 36 lines modified
89 <td>··············<pre><code·class="language-macaulay2">i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a89 <td>··············<pre><code·class="language-macaulay2">i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a
  
90 o5·=·2</code></pre>90 o5·=·2</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0,Verbose=>true))93 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0,Verbose=>true))
94 unfolding94 unfolding
95 ·--·.0704745s·elapsed95 ·--·.110088s·elapsed
96 flatteningRelations96 flatteningRelations
97 ·--·.0708185s·elapsed97 ·--·.0998481s·elapsed
98 next·gb98 next·gb
99 ·--·.000789972s·elapsed99 ·--·.000700193s·elapsed
100 true100 true
101 unfolding101 unfolding
102 ·--·.0617568s·elapsed102 ·--·.116186s·elapsed
103 flatteningRelations103 flatteningRelations
104 ·--·.0576035s·elapsed104 ·--·.0920172s·elapsed
105 next·gb105 next·gb
106 ·--·.000485693s·elapsed106 ·--·.000812553s·elapsed
107 true107 true
108 ·--·.78134s·elapsed108 ·--·1.3037s·elapsed
  
109 o6·=·{}109 o6·=·{}
  
110 o6·:·List</code></pre>110 o6·:·List</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ··········<tr>112 ··········<tr>
113 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))113 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))
114 ·--·.788254s·elapsed114 ·--·1.32832s·elapsed
  
115 o7·=·{}115 o7·=·{}
  
116 o7·:·List</code></pre>116 o7·:·List</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i8·:·LL7b=={}119 <td>··············<pre><code·class="language-macaulay2">i8·:·LL7b=={}
Offset 167, 22 lines modifiedOffset 167, 22 lines modified
167 o10·:·Sequence</code></pre>167 o10·:·Sequence</code></pre>
168 </td>··········</tr>168 </td>··········</tr>
169 ··········<tr>169 ··········<tr>
170 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)170 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)
171 (13,·1)171 (13,·1)
172 {5,·8,·11,·12}172 {5,·8,·11,·12}
173 unfolding173 unfolding
174 ·--·.129026s·elapsed174 ·--·.188457s·elapsed
175 flatteningRelations175 flatteningRelations
176 ·--·.0900135s·elapsed176 ·--·.129432s·elapsed
177 next·gb177 next·gb
178 ·--·.000640066s·elapsed178 ·--·.000922349s·elapsed
179 true179 true
180 ·--·.354599s·elapsed180 ·--·.546217s·elapsed
181 (5,·8,··all·semigroups·are·smoothable)·--·.385985s·elapsed181 (5,·8,··all·semigroups·are·smoothable)·--·.609911s·elapsed
  
  
182 o11·=·{}182 o11·=·{}
  
183 o11·:·List</code></pre>183 o11·:·List</code></pre>
184 </td>··········</tr>184 </td>··········</tr>
185 ········</table>185 ········</table>
Offset 200, 22 lines modifiedOffset 200, 22 lines modified
200 ··········<tr>200 ··········<tr>
201 <td>··············<pre><code·class="language-macaulay2">i13·:·genus·L201 <td>··············<pre><code·class="language-macaulay2">i13·:·genus·L
  
202 o13·=·8</code></pre>202 o13·=·8</code></pre>
203 </td>··········</tr>203 </td>··········</tr>
204 ··········<tr>204 ··········<tr>
205 <td>··············<pre><code·class="language-macaulay2">i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)205 <td>··············<pre><code·class="language-macaulay2">i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)
206 ·--·.044793s·elapsed206 ·--·.0564691s·elapsed
207 6207 6
208 false208 false
209 5209 5
210 false210 false
211 4211 4
212 decompose212 decompose
213 ·--·.192843s·elapsed213 ·--·.319679s·elapsed
214 number·of·components:·2214 number·of·components:·2
215 support·c,·codim·c:·{(1,·1),·(16,·3)}215 support·c,·codim·c:·{(1,·1),·(16,·3)}
216 {0,·-1}216 {0,·-1}
  
217 o14·=·true</code></pre>217 o14·=·true</code></pre>
218 </td>··········</tr>218 </td>··········</tr>
219 ········</table>219 ········</table>
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Offset 27, 34 lines modifiedOffset 27, 34 lines modified
27 o3·=·3927 o3·=·39
28 i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a28 i4·:·LL7a=select(LL7,L->not·knownExample·L);#LL7a
  
29 o5·=·229 o5·=·2
30 i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup30 i6·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup
31 (L,0.25,0,Verbose=>true))31 (L,0.25,0,Verbose=>true))
32 unfolding32 unfolding
33 ·--·.0704745s·elapsed33 ·--·.110088s·elapsed
34 flatteningRelations34 flatteningRelations
35 ·--·.0708185s·elapsed35 ·--·.0998481s·elapsed
36 next·gb36 next·gb
37 ·--·.000789972s·elapsed37 ·--·.000700193s·elapsed
38 true38 true
39 unfolding39 unfolding
40 ·--·.0617568s·elapsed40 ·--·.116186s·elapsed
41 flatteningRelations41 flatteningRelations
42 ·--·.0576035s·elapsed42 ·--·.0920172s·elapsed
43 next·gb43 next·gb
44 ·--·.000485693s·elapsed44 ·--·.000812553s·elapsed
45 true45 true
46 ·--·.78134s·elapsed46 ·--·1.3037s·elapsed
  
47 o6·=·{}47 o6·=·{}
  
48 o6·:·List48 o6·:·List
49 i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))49 i7·:·elapsedTime·LL7b=select(LL7a,L->not·isSmoothableSemigroup(L,0.25,0))
50 ·--·.788254s·elapsed50 ·--·1.32832s·elapsed
  
51 o7·=·{}51 o7·=·{}
  
52 o7·:·List52 o7·:·List
53 i8·:·LL7b=={}53 i8·:·LL7b=={}
  
54 o8·=·true54 o8·=·true
Offset 93, 22 lines modifiedOffset 93, 22 lines modified
93 o10·=·(5,·8)93 o10·=·(5,·8)
  
94 o10·:·Sequence94 o10·:·Sequence
95 i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)95 i11·:·elapsedTime·nonWeierstrassSemigroups(m,g,Verbose=>true)
96 (13,·1)96 (13,·1)
97 {5,·8,·11,·12}97 {5,·8,·11,·12}
98 unfolding98 unfolding
99 ·--·.129026s·elapsed99 ·--·.188457s·elapsed
100 flatteningRelations100 flatteningRelations
101 ·--·.0900135s·elapsed101 ·--·.129432s·elapsed
102 next·gb102 next·gb
103 ·--·.000640066s·elapsed103 ·--·.000922349s·elapsed
104 true104 true
105 ·--·.354599s·elapsed105 ·--·.546217s·elapsed
106 (5,·8,··all·semigroups·are·smoothable)·--·.385985s·elapsed106 (5,·8,··all·semigroups·are·smoothable)·--·.609911s·elapsed
  
  
107 o11·=·{}107 o11·=·{}
  
108 o11·:·List108 o11·:·List
109 In·the·verbose·mode·we·get·timings·of·various·computation·steps·and·further109 In·the·verbose·mode·we·get·timings·of·various·computation·steps·and·further
110 information.·The·first·line,·(13,1),·indicates·that·there·13·semigroups·of110 information.·The·first·line,·(13,1),·indicates·that·there·13·semigroups·of
Offset 121, 22 lines modifiedOffset 121, 22 lines modified
121 o12·=·{6,·8,·9,·11}121 o12·=·{6,·8,·9,·11}
  
122 o12·:·List122 o12·:·List
123 i13·:·genus·L123 i13·:·genus·L
  
124 o13·=·8124 o13·=·8
125 i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)125 i14·:·isWeierstrassSemigroup(L,0.2,Verbose=>true)
126 ·--·.044793s·elapsed126 ·--·.0564691s·elapsed
127 6127 6
128 false128 false
129 5129 5
130 false130 false
131 4131 4
132 decompose132 decompose
133 ·--·.192843s·elapsed133 ·--·.319679s·elapsed
134 number·of·components:·2134 number·of·components:·2
135 support·c,·codim·c:·{(1,·1),·(16,·3)}135 support·c,·codim·c:·{(1,·1),·(16,·3)}
136 {0,·-1}136 {0,·-1}
  
137 o14·=·true137 o14·=·true
138 The·first·integer,·6,·tells·that·in·this·attempt·deformation·parameters·of138 The·first·integer,·6,·tells·that·in·this·attempt·deformation·parameters·of
139 degree·>=·6·were·used·and·no·smooth·fiber·was·found.·Finally·with·all139 degree·>=·6·were·used·and·no·smooth·fiber·was·found.·Finally·with·all
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Offset 91, 15 lines modifiedOffset 91, 15 lines modified
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·tally·apply(10,i->·(93 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·tally·apply(10,i->·(
94 ·············c=minors(2,random(S^2,S^{3:-2}));94 ·············c=minors(2,random(S^2,S^{3:-2}));
95 ·············c=sub(c,x_0=>1);95 ·············c=sub(c,x_0=>1);
96 ·············R=kk[support·c];c=sub(c,R);96 ·············R=kk[support·c];c=sub(c,R);
97 ·············heuristicSmoothness·c))97 ·············heuristicSmoothness·c))
98 ·--·1.95527s·elapsed98 ·--·3.32953s·elapsed
  
99 o4·=·Tally{false·=>·6}99 o4·=·Tally{false·=>·6}
100 ···········true·=>·4100 ···········true·=>·4
  
101 o4·:·Tally</code></pre>101 o4·:·Tally</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ········</table>103 ········</table>
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26 o2·:·PolynomialRing26 o2·:·PolynomialRing
27 i3·:·setRandomSeed·"some·singular·and·some·smooth·curves";27 i3·:·setRandomSeed·"some·singular·and·some·smooth·curves";
28 i4·:·elapsedTime·tally·apply(10,i->·(28 i4·:·elapsedTime·tally·apply(10,i->·(
29 ·············c=minors(2,random(S^2,S^{3:-2}));29 ·············c=minors(2,random(S^2,S^{3:-2}));
30 ·············c=sub(c,x_0=>1);30 ·············c=sub(c,x_0=>1);
31 ·············R=kk[support·c];c=sub(c,R);31 ·············R=kk[support·c];c=sub(c,R);
32 ·············heuristicSmoothness·c))32 ·············heuristicSmoothness·c))
33 ·--·1.95527s·elapsed33 ·--·3.32953s·elapsed
  
34 o4·=·Tally{false·=>·6}34 o4·=·Tally{false·=>·6}
35 ···········true·=>·435 ···········true·=>·4
  
36 o4·:·Tally36 o4·:·Tally
37 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8he\x8eu\x8ur\x8ri\x8is\x8st\x8ti\x8ic\x8cS\x8Sm\x8mo\x8oo\x8ot\x8th\x8hn\x8ne\x8es\x8ss\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·h\x8he\x8eu\x8ur\x8ri\x8is\x8st\x8ti\x8ic\x8cS\x8Sm\x8mo\x8oo\x8ot\x8th\x8hn\x8ne\x8es\x8ss\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
38 ····*·heuristicSmoothness(Ideal)38 ····*·heuristicSmoothness(Ideal)
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Offset 95, 21 lines modifiedOffset 95, 21 lines modified
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i2·:·genus·L96 <td>··············<pre><code·class="language-macaulay2">i2·:·genus·L
  
97 o2·=·8</code></pre>97 o2·=·8</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)100 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)
101 ·--·.523728s·elapsed101 ·--·.800086s·elapsed
  
102 o3·=·false</code></pre>102 o3·=·false</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)105 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)
106 ·--·2.23245s·elapsed106 ·--·3.39097s·elapsed
  
107 o4·=·true</code></pre>107 o4·=·true</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ········</table>109 ········</table>
110 ······</div>110 ······</div>
111 ······<div>111 ······<div>
112 ········<h2>See·also</h2>112 ········<h2>See·also</h2>
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Offset 30, 19 lines modifiedOffset 30, 19 lines modified
30 o1·=·{6,·8,·9,·11}30 o1·=·{6,·8,·9,·11}
  
31 o1·:·List31 o1·:·List
32 i2·:·genus·L32 i2·:·genus·L
  
33 o2·=·833 o2·=·8
34 i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)34 i3·:·elapsedTime·isSmoothableSemigroup(L,0.30,0)
35 ·--·.523728s·elapsed35 ·--·.800086s·elapsed
  
36 o3·=·false36 o3·=·false
37 i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)37 i4·:·elapsedTime·isSmoothableSemigroup(L,0.14,0)
38 ·--·2.23245s·elapsed38 ·--·3.39097s·elapsed
  
39 o4·=·true39 o4·=·true
40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*40 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
41 ····*·_\x8m_\x8a_\x8k_\x8e_\x8U_\x8n_\x8f_\x8o_\x8l_\x8d_\x8i_\x8n_\x8g·--·Makes·the·universal·homogeneous·unfolding·of·an·ideal41 ····*·_\x8m_\x8a_\x8k_\x8e_\x8U_\x8n_\x8f_\x8o_\x8l_\x8d_\x8i_\x8n_\x8g·--·Makes·the·universal·homogeneous·unfolding·of·an·ideal
42 ······with·positive·degree·parameters42 ······with·positive·degree·parameters
43 ····*·_\x8f_\x8l_\x8a_\x8t_\x8t_\x8e_\x8n_\x8i_\x8n_\x8g_\x8R_\x8e_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·Compute·the·flattening·relations·of·an·unfolding43 ····*·_\x8f_\x8l_\x8a_\x8t_\x8t_\x8e_\x8n_\x8i_\x8n_\x8g_\x8R_\x8e_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·Compute·the·flattening·relations·of·an·unfolding
44 ····*·_\x8g_\x8e_\x8t_\x8F_\x8l_\x8a_\x8t_\x8F_\x8a_\x8m_\x8i_\x8l_\x8y·--·Compute·the·flat·family·depending·on·a·subset·of44 ····*·_\x8g_\x8e_\x8t_\x8F_\x8l_\x8a_\x8t_\x8F_\x8a_\x8m_\x8i_\x8l_\x8y·--·Compute·the·flat·family·depending·on·a·subset·of
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93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·genus·L94 <td>··············<pre><code·class="language-macaulay2">i2·:·genus·L
  
95 o2·=·8</code></pre>95 o2·=·8</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)98 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)
99 ·--·2.57853s·elapsed99 ·--·5.19504s·elapsed
  
100 o3·=·true</code></pre>100 o3·=·true</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ········</table>102 ········</table>
103 ······</div>103 ······</div>
104 ······<div>104 ······<div>
105 ········<h2>See·also</h2>105 ········<h2>See·also</h2>
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30 o1·=·{6,·8,·9,·11}30 o1·=·{6,·8,·9,·11}
  
31 o1·:·List31 o1·:·List
32 i2·:·genus·L32 i2·:·genus·L
  
33 o2·=·833 o2·=·8
34 i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)34 i3·:·elapsedTime·isWeierstrassSemigroup(L,0.15)
35 ·--·2.57853s·elapsed35 ·--·5.19504s·elapsed
  
36 o3·=·true36 o3·=·true
37 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*37 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
38 ····*·_\x8m_\x8a_\x8k_\x8e_\x8U_\x8n_\x8f_\x8o_\x8l_\x8d_\x8i_\x8n_\x8g·--·Makes·the·universal·homogeneous·unfolding·of·an·ideal38 ····*·_\x8m_\x8a_\x8k_\x8e_\x8U_\x8n_\x8f_\x8o_\x8l_\x8d_\x8i_\x8n_\x8g·--·Makes·the·universal·homogeneous·unfolding·of·an·ideal
39 ······with·positive·degree·parameters39 ······with·positive·degree·parameters
40 ····*·_\x8f_\x8l_\x8a_\x8t_\x8t_\x8e_\x8n_\x8i_\x8n_\x8g_\x8R_\x8e_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·Compute·the·flattening·relations·of·an·unfolding40 ····*·_\x8f_\x8l_\x8a_\x8t_\x8t_\x8e_\x8n_\x8i_\x8n_\x8g_\x8R_\x8e_\x8l_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·--·Compute·the·flattening·relations·of·an·unfolding
41 ····*·_\x8g_\x8e_\x8t_\x8F_\x8l_\x8a_\x8t_\x8F_\x8a_\x8m_\x8i_\x8l_\x8y·--·Compute·the·flat·family·depending·on·a·subset·of41 ····*·_\x8g_\x8e_\x8t_\x8F_\x8l_\x8a_\x8t_\x8F_\x8a_\x8m_\x8i_\x8l_\x8y·--·Compute·the·flat·family·depending·on·a·subset·of
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82 ········<h2>Description</h2>82 ········<h2>Description</h2>
83 ········<div>83 ········<div>
84 ··········<p>We·test·which·semigroups·of·multiplicity·m·and·genus·g·are·smoothable.·If·no·smoothing·was·found·then·L·is·a·candidate·for·a·non·Weierstrass·semigroup.·In·this·search·certain·semigroups·L·in·LLdifficult,·where·the·computation·is·particular·heavy·are·excluded.</p>84 ··········<p>We·test·which·semigroups·of·multiplicity·m·and·genus·g·are·smoothable.·If·no·smoothing·was·found·then·L·is·a·candidate·for·a·non·Weierstrass·semigroup.·In·this·search·certain·semigroups·L·in·LLdifficult,·where·the·computation·is·particular·heavy·are·excluded.</p>
85 ········</div>85 ········</div>
86 ········<table·class="examples">86 ········<table·class="examples">
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)88 <td>··············<pre><code·class="language-macaulay2">i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)
89 (6,·7,··all·semigroups·are·smoothable)·--·.879778s·elapsed89 (6,·7,··all·semigroups·are·smoothable)·--·1.85222s·elapsed
  
  
90 o1·=·{}90 o1·=·{}
  
91 o1·:·List</code></pre>91 o1·:·List</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
Offset 101, 61 lines modifiedOffset 101, 61 lines modified
101 o2·:·List</code></pre>101 o2·:·List</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)104 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
105 (17,·5)105 (17,·5)
106 {6,·7,·8,·17}106 {6,·7,·8,·17}
107 unfolding107 unfolding
108 ·--·.262451s·elapsed108 ·--·.626872s·elapsed
109 flatteningRelations109 flatteningRelations
110 ·--·.0990406s·elapsed110 ·--·.194757s·elapsed
111 next·gb111 next·gb
112 ·--·.00139547s·elapsed112 ·--·.00130453s·elapsed
113 true113 true
114 ·--·.631436s·elapsed114 ·--·1.34662s·elapsed
115 {6,·7,·9,·17}115 {6,·7,·9,·17}
116 unfolding116 unfolding
117 ·--·.269116s·elapsed117 ·--·.611882s·elapsed
118 flatteningRelations118 flatteningRelations
119 ·--·.102708s·elapsed119 ·--·.259928s·elapsed
120 next·gb120 next·gb
121 ·--·.00198154s·elapsed121 ·--·.0102483s·elapsed
122 decompose122 decompose
123 ·--·.131151s·elapsed123 ·--·.305591s·elapsed
124 number·of·components:·2124 number·of·components:·2
125 support·c,·codim·c:·{(2,·2),·(5,·2)}125 support·c,·codim·c:·{(2,·2),·(5,·2)}
126 {0,·-1}126 {0,·-1}
127 ·--·1.91249s·elapsed127 ·--·3.74559s·elapsed
128 {6,·8,·9,·10}128 {6,·8,·9,·10}
129 unfolding129 unfolding
130 ·--·.0624948s·elapsed130 ·--·.141687s·elapsed
131 flatteningRelations131 flatteningRelations
132 ·--·.0858256s·elapsed132 ·--·.126728s·elapsed
133 next·gb133 next·gb
134 ·--·.000483891s·elapsed134 ·--·.00442511s·elapsed
135 true135 true
136 ·--·.465824s·elapsed136 ·--·.838739s·elapsed
137 {6,·8,·10,·11,·13}137 {6,·8,·10,·11,·13}
138 unfolding138 unfolding
139 ·--·.342213s·elapsed139 ·--·.690592s·elapsed
140 flatteningRelations140 flatteningRelations
141 ·--·.129273s·elapsed141 ·--·.315878s·elapsed
142 next·gb142 next·gb
143 ·--·.0028742s·elapsed143 ·--·.007027s·elapsed
144 decompose144 decompose
145 ·--·.580045s·elapsed145 ·--·1.07767s·elapsed
146 number·of·components:·1146 number·of·components:·1
147 support·c,·codim·c:·{(5,·1)}147 support·c,·codim·c:·{(5,·1)}
148 {-1}148 {-1}
 149 ·--·3.45799s·elapsed
149 ·--·1.70163s·elapsed150 ·--·9.38906s·elapsed
150 ·--·4.71154s·elapsed 
151 0151 0
  
152 ·--·.000002664s·elapsed152 ·--·.000003808s·elapsed
153 ·--·4.74234s·elapsed153 ·--·9.42074s·elapsed
154 {}154 {}
  
155 o3·=·{{6,·8,·9,·11}}155 o3·=·{{6,·8,·9,·11}}
  
156 o3·:·List</code></pre>156 o3·:·List</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ········</table>158 ········</table>
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22 ············LLdifficult22 ············LLdifficult
23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
24 We·test·which·semigroups·of·multiplicity·m·and·genus·g·are·smoothable.·If·no24 We·test·which·semigroups·of·multiplicity·m·and·genus·g·are·smoothable.·If·no
25 smoothing·was·found·then·L·is·a·candidate·for·a·non·Weierstrass·semigroup.·In25 smoothing·was·found·then·L·is·a·candidate·for·a·non·Weierstrass·semigroup.·In
26 this·search·certain·semigroups·L·in·LLdifficult,·where·the·computation·is26 this·search·certain·semigroups·L·in·LLdifficult,·where·the·computation·is
27 particular·heavy·are·excluded.27 particular·heavy·are·excluded.
28 i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)28 i1·:·elapsedTime·nonWeierstrassSemigroups(6,7)
29 (6,·7,··all·semigroups·are·smoothable)·--·.879778s·elapsed29 (6,·7,··all·semigroups·are·smoothable)·--·1.85222s·elapsed
  
  
30 o1·=·{}30 o1·=·{}
  
31 o1·:·List31 o1·:·List
32 i2·:·LLdifficult={{6,·8,·9,·11}}32 i2·:·LLdifficult={{6,·8,·9,·11}}
  
33 o2·=·{{6,·8,·9,·11}}33 o2·=·{{6,·8,·9,·11}}
  
34 o2·:·List34 o2·:·List
35 i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)35 i3·:·elapsedTime·nonWeierstrassSemigroups(6,8,LLdifficult,Verbose=>true)
36 (17,·5)36 (17,·5)
37 {6,·7,·8,·17}37 {6,·7,·8,·17}
38 unfolding38 unfolding
39 ·--·.262451s·elapsed39 ·--·.626872s·elapsed
40 flatteningRelations40 flatteningRelations
41 ·--·.0990406s·elapsed41 ·--·.194757s·elapsed
42 next·gb42 next·gb
43 ·--·.00139547s·elapsed43 ·--·.00130453s·elapsed
44 true44 true
45 ·--·.631436s·elapsed45 ·--·1.34662s·elapsed
46 {6,·7,·9,·17}46 {6,·7,·9,·17}
47 unfolding47 unfolding
48 ·--·.269116s·elapsed48 ·--·.611882s·elapsed
49 flatteningRelations49 flatteningRelations
50 ·--·.102708s·elapsed50 ·--·.259928s·elapsed
51 next·gb51 next·gb
52 ·--·.00198154s·elapsed52 ·--·.0102483s·elapsed
53 decompose53 decompose
54 ·--·.131151s·elapsed54 ·--·.305591s·elapsed
55 number·of·components:·255 number·of·components:·2
56 support·c,·codim·c:·{(2,·2),·(5,·2)}56 support·c,·codim·c:·{(2,·2),·(5,·2)}
57 {0,·-1}57 {0,·-1}
58 ·--·1.91249s·elapsed58 ·--·3.74559s·elapsed
59 {6,·8,·9,·10}59 {6,·8,·9,·10}
60 unfolding60 unfolding
61 ·--·.0624948s·elapsed61 ·--·.141687s·elapsed
62 flatteningRelations62 flatteningRelations
63 ·--·.0858256s·elapsed63 ·--·.126728s·elapsed
64 next·gb64 next·gb
65 ·--·.000483891s·elapsed65 ·--·.00442511s·elapsed
66 true66 true
67 ·--·.465824s·elapsed67 ·--·.838739s·elapsed
68 {6,·8,·10,·11,·13}68 {6,·8,·10,·11,·13}
69 unfolding69 unfolding
70 ·--·.342213s·elapsed70 ·--·.690592s·elapsed
71 flatteningRelations71 flatteningRelations
72 ·--·.129273s·elapsed72 ·--·.315878s·elapsed
73 next·gb73 next·gb
74 ·--·.0028742s·elapsed74 ·--·.007027s·elapsed
75 decompose75 decompose
76 ·--·.580045s·elapsed76 ·--·1.07767s·elapsed
77 number·of·components:·177 number·of·components:·1
78 support·c,·codim·c:·{(5,·1)}78 support·c,·codim·c:·{(5,·1)}
79 {-1}79 {-1}
 80 ·--·3.45799s·elapsed
80 ·--·1.70163s·elapsed81 ·--·9.38906s·elapsed
81 ·--·4.71154s·elapsed 
82 082 0
  
83 ·--·.000002664s·elapsed83 ·--·.000003808s·elapsed
84 ·--·4.74234s·elapsed84 ·--·9.42074s·elapsed
85 {}85 {}
  
86 o3·=·{{6,·8,·9,·11}}86 o3·=·{{6,·8,·9,·11}}
  
87 o3·:·List87 o3·:·List
88 In·the·verbose·mode·we·get·timings·of·various·computation·steps·and·further88 In·the·verbose·mode·we·get·timings·of·various·computation·steps·and·further
89 information.·The·first·line,·(17,5),·indicates·that·there·17·semigroups·of89 information.·The·first·line,·(17,5),·indicates·that·there·17·semigroups·of
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11 YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g11 YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g
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5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·1)8 i5·:·time·C·=·oiRes({b},·1)
9 ·--·used·0.103546s·(cpu);·0.0740379s·(thread);·0s·(gc)9 ·--·used·0.110006s·(cpu);·0.0787769s·(thread);·0s·(gc)
  
10 o5·=·0:·(e0,·{2},·{-2})10 o5·=·0:·(e0,·{2},·{-2})
11 ·····1:·(e1,·{4,·4},·{-4,·-4})11 ·····1:·(e1,·{4,·4},·{-4,·-4})
  
12 o5·:·OIResolution12 o5·:·OIResolution
  
13 i6·:·C.dd_013 i6·:·C.dd_0
⊟
649 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___O__I__Resolution_sp_us_sp__Z__Z.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·1);8 i5·:·time·C·=·oiRes({b},·1);
9 ·--·used·0.110122s·(cpu);·0.0808499s·(thread);·0s·(gc)9 ·--·used·0.125476s·(cpu);·0.0871367s·(thread);·0s·(gc)
  
10 i6·:·C_010 i6·:·C_0
  
11 o6·=·Basis·symbol:·e011 o6·=·Basis·symbol:·e0
12 ·····Basis·element·widths:·{2}12 ·····Basis·element·widths:·{2}
13 ·····Degree·shifts:·{-2}13 ·····Degree·shifts:·{-2}
14 ·····Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)14 ·····Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
⊟
621 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Top__Nonminimal.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)8 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
9 ·--·used·0.241952s·(cpu);·0.199122s·(thread);·0s·(gc)9 ·--·used·0.306312s·(cpu);·0.209568s·(thread);·0s·(gc)
  
10 o5·=·0:·(e0,·{2},·{-2})10 o5·=·0:·(e0,·{2},·{-2})
11 ·····1:·(e1,·{4},·{-4})11 ·····1:·(e1,·{4},·{-4})
12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
13 o5·:·OIResolution13 o5·:·OIResolution
  
⊟
666 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe__Full.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·1);8 i5·:·time·C·=·oiRes({b},·1);
9 ·--·used·0.110527s·(cpu);·0.0800811s·(thread);·0s·(gc)9 ·--·used·0.125311s·(cpu);·0.0816457s·(thread);·0s·(gc)
  
10 i6·:·describeFull·C10 i6·:·describeFull·C
  
11 o6·=·0:·Module:·Basis·symbol:·e011 o6·=·0:·Module:·Basis·symbol:·e0
12 ················Basis·element·widths:·{2}12 ················Basis·element·widths:·{2}
13 ················Degree·shifts:·{-2}13 ················Degree·shifts:·{-2}
14 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)14 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
⊟
698 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__O__I__Resolution_rp.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·1);8 i5·:·time·C·=·oiRes({b},·1);
9 ·--·used·0.103644s·(cpu);·0.0762393s·(thread);·0s·(gc)9 ·--·used·0.124109s·(cpu);·0.0827685s·(thread);·0s·(gc)
  
10 i6·:·describe·C10 i6·:·describe·C
  
11 o6·=·0:·Module:·Basis·symbol:·e011 o6·=·0:·Module:·Basis·symbol:·e0
12 ················Basis·element·widths:·{2}12 ················Basis·element·widths:·{2}
13 ················Degree·shifts:·{-2}13 ················Degree·shifts:·{-2}
14 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)14 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
⊟
611 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__Complex.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)8 i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)
9 ·--·used·0.245218s·(cpu);·0.188024s·(thread);·0s·(gc)9 ·--·used·0.26879s·(cpu);·0.19978s·(thread);·0s·(gc)
  
10 o5·=·0:·(e0,·{2},·{-2})10 o5·=·0:·(e0,·{2},·{-2})
11 ·····1:·(e1,·{4},·{-4})11 ·····1:·(e1,·{4},·{-4})
12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
13 o5·:·OIResolution13 o5·:·OIResolution
  
⊟
789 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_is__O__I__G__B.out
    
Offset 15, 15 lines modifiedOffset 15, 15 lines modified
15 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);15 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
  
16 i10·:·isOIGB·{b1,·b2}16 i10·:·isOIGB·{b1,·b2}
  
17 o10·=·false17 o10·=·false
  
18 i11·:·time·B·=·oiGB·{b1,·b2}18 i11·:·time·B·=·oiGB·{b1,·b2}
19 ·--·used·0.0159958s·(cpu);·0.0172255s·(thread);·0s·(gc)19 ·--·used·0.0214608s·(cpu);·0.0194736s·(thread);·0s·(gc)
  
20 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,20 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
21 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·21 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
22 ······-----------------------------------------------------------------------22 ······-----------------------------------------------------------------------
23 ······x···x···x···e···········-·x···x···x···e··········}23 ······x···x···x···e···········-·x···x···x···e··········}
24 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},324 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
⊟
1.81 KB
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_minimize__O__I__G__B.out
    
Offset 11, 15 lines modifiedOffset 11, 15 lines modified
11 i5·:·installGeneratorsInWidth(F,·3);11 i5·:·installGeneratorsInWidth(F,·3);
  
12 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);12 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
  
13 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);13 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
  
14 i10·:·time·B·=·oiGB·{b1,·b2}14 i10·:·time·B·=·oiGB·{b1,·b2}
15 ·--·used·0.0238484s·(cpu);·0.0208946s·(thread);·0s·(gc)15 ·--·used·0.0213984s·(cpu);·0.0220702s·(thread);·0s·(gc)
  
16 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,16 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
17 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·17 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
18 ······-----------------------------------------------------------------------18 ······-----------------------------------------------------------------------
19 ······x···x···x···e···········-·x···x···x···e··········}19 ······x···x···x···e···········-·x···x···x···e··········}
20 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},320 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
Offset 41, 18 lines modifiedOffset 41, 18 lines modified
41 ···········-·x···x···e··········}41 ···········-·x···x···e··········}
42 ······3},3····2,1·1,2·3,{1,·3},342 ······3},3····2,1·1,2·3,{1,·3},3
  
43 o13·:·List43 o13·:·List
  
44 i14·:·minimizeOIGB·C·--·an·element·gets·removed44 i14·:·minimizeOIGB·C·--·an·element·gets·removed
  
45 ···········································································45 ········································································
46 o14·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········, 
47 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·46 o14·=·{x···x···e········+·x···x···e··········,·x···x···x···e···········-
 47 ········1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3···2,3·2,2·1,1·3,{2,·3},3··
48 ······-----------------------------------------------------------------------48 ······-----------------------------------------------------------------------
49 ·····································249 ···········2
50 ······x···x···x···e···········-·x···x···e··········}50 ······x···x···e··········,·x···e········+·x···e·······}
51 ·······2,3·2,2·1,1·3,{2,·3},3····2,1·1,2·3,{1,·3},351 ·······2,1·1,2·3,{1,·3},3···1,1·1,{1},1····2,1·1,{1},2
  
52 o14·:·List52 o14·:·List
  
53 i15·:·53 i15·:·
⊟
572 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·C·=·oiRes({b},·1);8 i5·:·time·C·=·oiRes({b},·1);
9 ·--·used·0.110635s·(cpu);·0.0767381s·(thread);·0s·(gc)9 ·--·used·0.112295s·(cpu);·0.0832699s·(thread);·0s·(gc)
  
10 i6·:·net·C10 i6·:·net·C
  
11 o6·=·0:·(e0,·{2},·{-2})11 o6·=·0:·(e0,·{2},·{-2})
12 ·····1:·(e1,·{4,·4},·{-4,·-4})12 ·····1:·(e1,·{4,·4},·{-4,·-4})
  
13 i7·:·13 i7·:·
⊟
828 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__G__B.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i4·:·installGeneratorsInWidth(F,·2);9 i4·:·installGeneratorsInWidth(F,·2);
  
10 i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);10 i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
  
11 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);11 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
  
12 i9·:·time·oiGB·{b1,·b2}12 i9·:·time·oiGB·{b1,·b2}
13 ·--·used·0.0187842s·(cpu);·0.0196717s·(thread);·0s·(gc)13 ·--·used·0.0278921s·(cpu);·0.025753s·(thread);·0s·(gc)
  
14 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,14 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
15 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·15 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
16 ·····------------------------------------------------------------------------16 ·····------------------------------------------------------------------------
17 ·····x···x···x···e···········-·x···x···x···e··········}17 ·····x···x···x···e···········-·x···x···x···e··········}
18 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},318 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
⊟
601 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_oi__Res.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);5 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
  
6 i3·:·installGeneratorsInWidth(F,·2);6 i3·:·installGeneratorsInWidth(F,·2);
  
7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);7 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
  
8 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)8 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
9 ·--·used·0.255787s·(cpu);·0.204527s·(thread);·0s·(gc)9 ·--·used·0.287897s·(cpu);·0.213173s·(thread);·0s·(gc)
  
10 o5·=·0:·(e0,·{2},·{-2})10 o5·=·0:·(e0,·{2},·{-2})
11 ·····1:·(e1,·{4},·{-4})11 ·····1:·(e1,·{4},·{-4})
12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})12 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
13 o5·:·OIResolution13 o5·:·OIResolution
  
⊟
898 B
./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i4·:·installGeneratorsInWidth(F,·2);9 i4·:·installGeneratorsInWidth(F,·2);
  
10 i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);10 i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
  
11 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);11 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
  
12 i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)12 i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)
13 ·--·used·0.1244s·(cpu);·0.0983777s·(thread);·0s·(gc)13 ·--·used·0.146288s·(cpu);·0.103883s·(thread);·0s·(gc)
  
14 ·······································································14 ·······································································
15 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,15 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,
16 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},2·16 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},2·
17 ·····------------------------------------------------------------------------17 ·····------------------------------------------------------------------------
18 ······2··················218 ······2··················2
19 ·····x···x···e········-·x···x···e·······}19 ·····x···x···e········-·x···x···e·······}
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58 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>58 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
59 </td>··········</tr>59 </td>··········</tr>
60 ··········<tr>60 ··········<tr>
61 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>61 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
62 </td>··········</tr>62 </td>··········</tr>
63 ··········<tr>63 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1)64 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1)
65 ·--·used·0.103546s·(cpu);·0.0740379s·(thread);·0s·(gc)65 ·--·used·0.110006s·(cpu);·0.0787769s·(thread);·0s·(gc)
  
66 o5·=·0:·(e0,·{2},·{-2})66 o5·=·0:·(e0,·{2},·{-2})
67 ·····1:·(e1,·{4,·4},·{-4,·-4})67 ·····1:·(e1,·{4,·4},·{-4,·-4})
  
68 o5·:·OIResolution</code></pre>68 o5·:·OIResolution</code></pre>
69 </td>··········</tr>69 </td>··········</tr>
70 ··········<tr>70 ··········<tr>
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11 complex,·use·_\x8i_\x8s_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x.·To·get·the·$n$th·differential·in·an·OI-resolution·C,11 complex,·use·_\x8i_\x8s_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x.·To·get·the·$n$th·differential·in·an·OI-resolution·C,
12 use·C.dd_n.12 use·C.dd_n.
13 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);13 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
14 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);14 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
15 i3·:·installGeneratorsInWidth(F,·2);15 i3·:·installGeneratorsInWidth(F,·2);
16 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);16 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
17 i5·:·time·C·=·oiRes({b},·1)17 i5·:·time·C·=·oiRes({b},·1)
18 ·--·used·0.103546s·(cpu);·0.0740379s·(thread);·0s·(gc)18 ·--·used·0.110006s·(cpu);·0.0787769s·(thread);·0s·(gc)
  
19 o5·=·0:·(e0,·{2},·{-2})19 o5·=·0:·(e0,·{2},·{-2})
20 ·····1:·(e1,·{4,·4},·{-4,·-4})20 ·····1:·(e1,·{4,·4},·{-4,·-4})
  
21 o5·:·OIResolution21 o5·:·OIResolution
22 i6·:·C.dd_022 i6·:·C.dd_0
  
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85 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>85 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>88 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);91 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);
92 ·--·used·0.110122s·(cpu);·0.0808499s·(thread);·0s·(gc)</code></pre>92 ·--·used·0.125476s·(cpu);·0.0871367s·(thread);·0s·(gc)</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i6·:·C_095 <td>··············<pre><code·class="language-macaulay2">i6·:·C_0
  
96 o6·=·Basis·symbol:·e096 o6·=·Basis·symbol:·e0
97 ·····Basis·element·widths:·{2}97 ·····Basis·element·widths:·{2}
98 ·····Degree·shifts:·{-2}98 ·····Degree·shifts:·{-2}
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17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 Returns·the·free·OI-module·of·$C$·in·homological·degree·$n$.18 Returns·the·free·OI-module·of·$C$·in·homological·degree·$n$.
19 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);19 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
20 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);20 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
21 i3·:·installGeneratorsInWidth(F,·2);21 i3·:·installGeneratorsInWidth(F,·2);
22 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);22 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
23 i5·:·time·C·=·oiRes({b},·1);23 i5·:·time·C·=·oiRes({b},·1);
24 ·--·used·0.110122s·(cpu);·0.0808499s·(thread);·0s·(gc)24 ·--·used·0.125476s·(cpu);·0.0871367s·(thread);·0s·(gc)
25 i6·:·C_025 i6·:·C_0
  
26 o6·=·Basis·symbol:·e026 o6·=·Basis·symbol:·e0
27 ·····Basis·element·widths:·{2}27 ·····Basis·element·widths:·{2}
28 ·····Degree·shifts:·{-2}28 ·····Degree·shifts:·{-2}
29 ·····Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)29 ·····Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
30 ·····Monomial·order:·Lex30 ·····Monomial·order:·Lex
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58 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>58 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
59 </td>··········</tr>59 </td>··········</tr>
60 ··········<tr>60 ··········<tr>
61 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>61 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
62 </td>··········</tr>62 </td>··········</tr>
63 ··········<tr>63 ··········<tr>
64 <td>··············<pre><code·class="language-macaulay2">i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)64 <td>··············<pre><code·class="language-macaulay2">i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
65 ·--·used·0.241952s·(cpu);·0.199122s·(thread);·0s·(gc)65 ·--·used·0.306312s·(cpu);·0.209568s·(thread);·0s·(gc)
  
66 o5·=·0:·(e0,·{2},·{-2})66 o5·=·0:·(e0,·{2},·{-2})
67 ·····1:·(e1,·{4},·{-4})67 ·····1:·(e1,·{4},·{-4})
68 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})68 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
69 o5·:·OIResolution</code></pre>69 o5·:·OIResolution</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
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11 homological·degree·$n-1$·to·be·minimized.·Therefore,·use·TopNonminimal·=>·true11 homological·degree·$n-1$·to·be·minimized.·Therefore,·use·TopNonminimal·=>·true
12 for·no·minimization·of·the·basis·in·degree·$n-1$.12 for·no·minimization·of·the·basis·in·degree·$n-1$.
13 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);13 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
14 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);14 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
15 i3·:·installGeneratorsInWidth(F,·2);15 i3·:·installGeneratorsInWidth(F,·2);
16 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);16 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
17 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)17 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
18 ·--·used·0.241952s·(cpu);·0.199122s·(thread);·0s·(gc)18 ·--·used·0.306312s·(cpu);·0.209568s·(thread);·0s·(gc)
  
19 o5·=·0:·(e0,·{2},·{-2})19 o5·=·0:·(e0,·{2},·{-2})
20 ·····1:·(e1,·{4},·{-4})20 ·····1:·(e1,·{4},·{-4})
21 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})21 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
22 o5·:·OIResolution22 o5·:·OIResolution
23 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·T\x8To\x8op\x8pN\x8No\x8on\x8nm\x8mi\x8in\x8ni\x8im\x8ma\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·T\x8To\x8op\x8pN\x8No\x8on\x8nm\x8mi\x8in\x8ni\x8im\x8ma\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*
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81 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>81 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
82 </td>··········</tr>82 </td>··········</tr>
83 ··········<tr>83 ··········<tr>
84 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>84 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);87 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);
88 ·--·used·0.110527s·(cpu);·0.0800811s·(thread);·0s·(gc)</code></pre>88 ·--·used·0.125311s·(cpu);·0.0816457s·(thread);·0s·(gc)</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i6·:·describeFull·C91 <td>··············<pre><code·class="language-macaulay2">i6·:·describeFull·C
  
92 o6·=·0:·Module:·Basis·symbol:·e092 o6·=·0:·Module:·Basis·symbol:·e0
93 ················Basis·element·widths:·{2}93 ················Basis·element·widths:·{2}
94 ················Degree·shifts:·{-2}94 ················Degree·shifts:·{-2}
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15 Displays·the·free·OI-modules·and·describes·the·differentials·of·an·OI-15 Displays·the·free·OI-modules·and·describes·the·differentials·of·an·OI-
16 resolution.16 resolution.
17 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);17 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
18 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);18 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
19 i3·:·installGeneratorsInWidth(F,·2);19 i3·:·installGeneratorsInWidth(F,·2);
20 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);20 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
21 i5·:·time·C·=·oiRes({b},·1);21 i5·:·time·C·=·oiRes({b},·1);
22 ·--·used·0.110527s·(cpu);·0.0800811s·(thread);·0s·(gc)22 ·--·used·0.125311s·(cpu);·0.0816457s·(thread);·0s·(gc)
23 i6·:·describeFull·C23 i6·:·describeFull·C
  
24 o6·=·0:·Module:·Basis·symbol:·e024 o6·=·0:·Module:·Basis·symbol:·e0
25 ················Basis·element·widths:·{2}25 ················Basis·element·widths:·{2}
26 ················Degree·shifts:·{-2}26 ················Degree·shifts:·{-2}
27 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)27 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
28 ················Monomial·order:·Lex28 ················Monomial·order:·Lex
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83 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>83 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>86 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);
90 ·--·used·0.103644s·(cpu);·0.0762393s·(thread);·0s·(gc)</code></pre>90 ·--·used·0.124109s·(cpu);·0.0827685s·(thread);·0s·(gc)</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i6·:·describe·C93 <td>··············<pre><code·class="language-macaulay2">i6·:·describe·C
  
94 o6·=·0:·Module:·Basis·symbol:·e094 o6·=·0:·Module:·Basis·symbol:·e0
95 ················Basis·element·widths:·{2}95 ················Basis·element·widths:·{2}
96 ················Degree·shifts:·{-2}96 ················Degree·shifts:·{-2}
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15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*15 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
16 Displays·the·free·OI-modules·and·differentials·of·an·OI-resolution.16 Displays·the·free·OI-modules·and·differentials·of·an·OI-resolution.
17 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);17 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
18 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);18 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
19 i3·:·installGeneratorsInWidth(F,·2);19 i3·:·installGeneratorsInWidth(F,·2);
20 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);20 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
21 i5·:·time·C·=·oiRes({b},·1);21 i5·:·time·C·=·oiRes({b},·1);
22 ·--·used·0.103644s·(cpu);·0.0762393s·(thread);·0s·(gc)22 ·--·used·0.124109s·(cpu);·0.0827685s·(thread);·0s·(gc)
23 i6·:·describe·C23 i6·:·describe·C
  
24 o6·=·0:·Module:·Basis·symbol:·e024 o6·=·0:·Module:·Basis·symbol:·e0
25 ················Basis·element·widths:·{2}25 ················Basis·element·widths:·{2}
26 ················Degree·shifts:·{-2}26 ················Degree·shifts:·{-2}
27 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)27 ················Polynomial·OI-algebra:·(2,·x,·QQ,·RowUpColUp)
28 ················Monomial·order:·Lex28 ················Monomial·order:·Lex
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87 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>87 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>90 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)93 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)
94 ·--·used·0.245218s·(cpu);·0.188024s·(thread);·0s·(gc)94 ·--·used·0.26879s·(cpu);·0.19978s·(thread);·0s·(gc)
  
95 o5·=·0:·(e0,·{2},·{-2})95 o5·=·0:·(e0,·{2},·{-2})
96 ·····1:·(e1,·{4},·{-4})96 ·····1:·(e1,·{4},·{-4})
97 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})97 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
98 o5·:·OIResolution</code></pre>98 o5·:·OIResolution</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
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18 option·must·be·either·true·or·false,·depending·on·whether·one·wants·debug18 option·must·be·either·true·or·false,·depending·on·whether·one·wants·debug
19 information·printed.19 information·printed.
20 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);20 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
21 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);21 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
22 i3·:·installGeneratorsInWidth(F,·2);22 i3·:·installGeneratorsInWidth(F,·2);
23 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);23 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
24 i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)24 i5·:·time·C·=·oiRes({b},·2,·TopNonminimal·=>·true)
25 ·--·used·0.245218s·(cpu);·0.188024s·(thread);·0s·(gc)25 ·--·used·0.26879s·(cpu);·0.19978s·(thread);·0s·(gc)
  
26 o5·=·0:·(e0,·{2},·{-2})26 o5·=·0:·(e0,·{2},·{-2})
27 ·····1:·(e1,·{4},·{-4})27 ·····1:·(e1,·{4},·{-4})
28 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})28 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
29 o5·:·OIResolution29 o5·:·OIResolution
30 i6·:·isComplex·C30 i6·:·isComplex·C
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101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i10·:·isOIGB·{b1,·b2}102 <td>··············<pre><code·class="language-macaulay2">i10·:·isOIGB·{b1,·b2}
  
103 o10·=·false</code></pre>103 o10·=·false</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i11·:·time·B·=·oiGB·{b1,·b2}106 <td>··············<pre><code·class="language-macaulay2">i11·:·time·B·=·oiGB·{b1,·b2}
107 ·--·used·0.0159958s·(cpu);·0.0172255s·(thread);·0s·(gc)107 ·--·used·0.0214608s·(cpu);·0.0194736s·(thread);·0s·(gc)
  
108 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,108 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
109 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·109 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
110 ······-----------------------------------------------------------------------110 ······-----------------------------------------------------------------------
111 ······x···x···x···e···········-·x···x···x···e··········}111 ······x···x···x···e···········-·x···x···x···e··········}
112 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3112 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
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25 i5·:·installGeneratorsInWidth(F,·3);25 i5·:·installGeneratorsInWidth(F,·3);
26 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);26 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
27 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);27 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
28 i10·:·isOIGB·{b1,·b2}28 i10·:·isOIGB·{b1,·b2}
  
29 o10·=·false29 o10·=·false
30 i11·:·time·B·=·oiGB·{b1,·b2}30 i11·:·time·B·=·oiGB·{b1,·b2}
31 ·--·used·0.0159958s·(cpu);·0.0172255s·(thread);·0s·(gc)31 ·--·used·0.0214608s·(cpu);·0.0194736s·(thread);·0s·(gc)
  
32 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,32 o11·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
33 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},333 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3
34 ······-----------------------------------------------------------------------34 ······-----------------------------------------------------------------------
35 ······x···x···x···e···········-·x···x···x···e··········}35 ······x···x···x···e···········-·x···x···x···e··········}
36 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},336 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
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96 <td>··············<pre><code·class="language-macaulay2">i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);</code></pre>96 <td>··············<pre><code·class="language-macaulay2">i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);</code></pre>99 <td>··············<pre><code·class="language-macaulay2">i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i10·:·time·B·=·oiGB·{b1,·b2}102 <td>··············<pre><code·class="language-macaulay2">i10·:·time·B·=·oiGB·{b1,·b2}
103 ·--·used·0.0238484s·(cpu);·0.0208946s·(thread);·0s·(gc)103 ·--·used·0.0213984s·(cpu);·0.0220702s·(thread);·0s·(gc)
  
104 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,104 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
105 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·105 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
106 ······-----------------------------------------------------------------------106 ······-----------------------------------------------------------------------
107 ······x···x···x···e···········-·x···x···x···e··········}107 ······x···x···x···e···········-·x···x···x···e··········}
108 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3108 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
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129 ······3},3····2,1·1,2·3,{1,·3},3129 ······3},3····2,1·1,2·3,{1,·3},3
  
130 o13·:·List</code></pre>130 o13·:·List</code></pre>
131 </td>··········</tr>131 </td>··········</tr>
132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i14·:·minimizeOIGB·C·--·an·element·gets·removed133 <td>··············<pre><code·class="language-macaulay2">i14·:·minimizeOIGB·C·--·an·element·gets·removed
  
134 ···········································································134 ········································································
135 o14·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········, 
136 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·135 o14·=·{x···x···e········+·x···x···e··········,·x···x···x···e···········-
 136 ········1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3···2,3·2,2·1,1·3,{2,·3},3··
137 ······-----------------------------------------------------------------------137 ······-----------------------------------------------------------------------
138 ·····································2138 ···········2
139 ······x···x···x···e···········-·x···x···e··········}139 ······x···x···e··········,·x···e········+·x···e·······}
140 ·······2,3·2,2·1,1·3,{2,·3},3····2,1·1,2·3,{1,·3},3140 ·······2,1·1,2·3,{1,·3},3···1,1·1,{1},1····2,1·1,{1},2
  
141 o14·:·List</code></pre>141 o14·:·List</code></pre>
142 </td>··········</tr>142 </td>··········</tr>
143 ········</table>143 ········</table>
144 ······</div>144 ······</div>
145 ······<div·class="waystouse">145 ······<div·class="waystouse">
146 ········<h2>Ways·to·use·<span·class="tt">minimizeOIGB</span>:</h2>146 ········<h2>Ways·to·use·<span·class="tt">minimizeOIGB</span>:</h2>
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22 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);22 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);
23 i3·:·installGeneratorsInWidth(F,·1);23 i3·:·installGeneratorsInWidth(F,·1);
24 i4·:·installGeneratorsInWidth(F,·2);24 i4·:·installGeneratorsInWidth(F,·2);
25 i5·:·installGeneratorsInWidth(F,·3);25 i5·:·installGeneratorsInWidth(F,·3);
26 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);26 i6·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
27 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);27 i8·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
28 i10·:·time·B·=·oiGB·{b1,·b2}28 i10·:·time·B·=·oiGB·{b1,·b2}
29 ·--·used·0.0238484s·(cpu);·0.0208946s·(thread);·0s·(gc)29 ·--·used·0.0213984s·(cpu);·0.0220702s·(thread);·0s·(gc)
  
30 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,30 o10·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
31 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},331 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3
32 ······-----------------------------------------------------------------------32 ······-----------------------------------------------------------------------
33 ······x···x···x···e···········-·x···x···x···e··········}33 ······x···x···x···e···········-·x···x···x···e··········}
34 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},334 ·······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
Offset 51, 19 lines modifiedOffset 51, 19 lines modified
51 ···········-·x···x···e··········}51 ···········-·x···x···e··········}
52 ······3},3····2,1·1,2·3,{1,·3},352 ······3},3····2,1·1,2·3,{1,·3},3
  
53 o13·:·List53 o13·:·List
54 i14·:·minimizeOIGB·C·--·an·element·gets·removed54 i14·:·minimizeOIGB·C·--·an·element·gets·removed
  
  
55 o14·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········, 
56 ········1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},355 o14·=·{x···x···e········+·x···x···e··········,·x···x···x···e···········-
 56 ········1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3···2,3·2,2·1,1·3,{2,·3},3
57 ······-----------------------------------------------------------------------57 ······-----------------------------------------------------------------------
58 ·····································258 ···········2
59 ······x···x···x···e···········-·x···x···e··········}59 ······x···x···e··········,·x···e········+·x···e·······}
60 ·······2,3·2,2·1,1·3,{2,·3},3····2,1·1,2·3,{1,·3},360 ·······2,1·1,2·3,{1,·3},3···1,1·1,{1},1····2,1·1,{1},2
  
61 o14·:·List61 o14·:·List
62 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8mi\x8in\x8ni\x8im\x8mi\x8iz\x8ze\x8eO\x8OI\x8IG\x8GB\x8B:\x8:·*\x8**\x8**\x8**\x8**\x8*62 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8mi\x8in\x8ni\x8im\x8mi\x8iz\x8ze\x8eO\x8OI\x8IG\x8GB\x8B:\x8:·*\x8**\x8**\x8**\x8**\x8*
63 ····*·minimizeOIGB(List)63 ····*·minimizeOIGB(List)
64 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*64 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
65 The·object·_\x8m_\x8i_\x8n_\x8i_\x8m_\x8i_\x8z_\x8e_\x8O_\x8I_\x8G_\x8B·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.65 The·object·_\x8m_\x8i_\x8n_\x8i_\x8m_\x8i_\x8z_\x8e_\x8O_\x8I_\x8G_\x8B·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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83 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>83 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>86 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·oiRes({b},·1);
90 ·--·used·0.110635s·(cpu);·0.0767381s·(thread);·0s·(gc)</code></pre>90 ·--·used·0.112295s·(cpu);·0.0832699s·(thread);·0s·(gc)</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i6·:·net·C93 <td>··············<pre><code·class="language-macaulay2">i6·:·net·C
  
94 o6·=·0:·(e0,·{2},·{-2})94 o6·=·0:·(e0,·{2},·{-2})
95 ·····1:·(e1,·{4,·4},·{-4,·-4})</code></pre>95 ·····1:·(e1,·{4,·4},·{-4,·-4})</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
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16 Displays·the·basis·element·widths·and·degree·shifts·of·the·free·OI-modules·in16 Displays·the·basis·element·widths·and·degree·shifts·of·the·free·OI-modules·in
17 an·OI-resolution.17 an·OI-resolution.
18 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);18 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
19 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);19 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
20 i3·:·installGeneratorsInWidth(F,·2);20 i3·:·installGeneratorsInWidth(F,·2);
21 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);21 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
22 i5·:·time·C·=·oiRes({b},·1);22 i5·:·time·C·=·oiRes({b},·1);
23 ·--·used·0.110635s·(cpu);·0.0767381s·(thread);·0s·(gc)23 ·--·used·0.112295s·(cpu);·0.0832699s·(thread);·0s·(gc)
24 i6·:·net·C24 i6·:·net·C
  
25 o6·=·0:·(e0,·{2},·{-2})25 o6·=·0:·(e0,·{2},·{-2})
26 ·····1:·(e1,·{4,·4},·{-4,·-4})26 ·····1:·(e1,·{4,·4},·{-4,·-4})
27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*27 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
28 ····*·_\x8n_\x8e_\x8t_\x8(_\x8O_\x8I_\x8R_\x8e_\x8s_\x8o_\x8l_\x8u_\x8t_\x8i_\x8o_\x8n_\x8)·--·display·an·OI-resolution28 ····*·_\x8n_\x8e_\x8t_\x8(_\x8O_\x8I_\x8R_\x8e_\x8s_\x8o_\x8l_\x8u_\x8t_\x8i_\x8o_\x8n_\x8)·--·display·an·OI-resolution
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104 <td>··············<pre><code·class="language-macaulay2">i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);</code></pre>104 <td>··············<pre><code·class="language-macaulay2">i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i9·:·time·oiGB·{b1,·b2}110 <td>··············<pre><code·class="language-macaulay2">i9·:·time·oiGB·{b1,·b2}
111 ·--·used·0.0187842s·(cpu);·0.0196717s·(thread);·0s·(gc)111 ·--·used·0.0278921s·(cpu);·0.025753s·(thread);·0s·(gc)
  
112 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,112 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
113 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·113 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3·
114 ·····------------------------------------------------------------------------114 ·····------------------------------------------------------------------------
115 ·····x···x···x···e···········-·x···x···x···e··········}115 ·····x···x···x···e···········-·x···x···x···e··········}
116 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3116 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
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29 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);29 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
30 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);30 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);
31 i3·:·installGeneratorsInWidth(F,·1);31 i3·:·installGeneratorsInWidth(F,·1);
32 i4·:·installGeneratorsInWidth(F,·2);32 i4·:·installGeneratorsInWidth(F,·2);
33 i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);33 i5·:·use·F_1;·b1·=·x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
34 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);34 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
35 i9·:·time·oiGB·{b1,·b2}35 i9·:·time·oiGB·{b1,·b2}
36 ·--·used·0.0187842s·(cpu);·0.0196717s·(thread);·0s·(gc)36 ·--·used·0.0278921s·(cpu);·0.025753s·(thread);·0s·(gc)
  
37 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,37 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e··········,
38 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},338 ·······1,1·1,{1},1····2,1·1,{1},2···1,2·1,1·2,{2},2····2,2·2,1·2,{1,·2},3
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ·····x···x···x···e···········-·x···x···x···e··········}40 ·····x···x···x···e···········-·x···x···x···e··········}
41 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},341 ······2,3·2,2·1,1·3,{2,·3},3····2,3·2,1·1,2·3,{1,·3},3
  
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105 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>105 <td>··············<pre><code·class="language-macaulay2">i3·:·installGeneratorsInWidth(F,·2);</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>108 <td>··············<pre><code·class="language-macaulay2">i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)111 <td>··············<pre><code·class="language-macaulay2">i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
112 ·--·used·0.255787s·(cpu);·0.204527s·(thread);·0s·(gc)112 ·--·used·0.287897s·(cpu);·0.213173s·(thread);·0s·(gc)
  
113 o5·=·0:·(e0,·{2},·{-2})113 o5·=·0:·(e0,·{2},·{-2})
114 ·····1:·(e1,·{4},·{-4})114 ·····1:·(e1,·{4},·{-4})
115 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})115 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
116 o5·:·OIResolution</code></pre>116 o5·:·OIResolution</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
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34 Therefore,·use·TopNonminimal·=>·true·for·no·minimization·of·the·basis·in·degree34 Therefore,·use·TopNonminimal·=>·true·for·no·minimization·of·the·basis·in·degree
35 $n-1$.35 $n-1$.
36 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);36 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
37 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);37 i2·:·F·=·makeFreeOIModule(e,·{1,1},·P);
38 i3·:·installGeneratorsInWidth(F,·2);38 i3·:·installGeneratorsInWidth(F,·2);
39 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);39 i4·:·b·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
40 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)40 i5·:·time·oiRes({b},·2,·TopNonminimal·=>·true)
41 ·--·used·0.255787s·(cpu);·0.204527s·(thread);·0s·(gc)41 ·--·used·0.287897s·(cpu);·0.213173s·(thread);·0s·(gc)
  
42 o5·=·0:·(e0,·{2},·{-2})42 o5·=·0:·(e0,·{2},·{-2})
43 ·····1:·(e1,·{4},·{-4})43 ·····1:·(e1,·{4},·{-4})
44 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})44 ·····2:·(e2,·{4,·5,·5,·5,·5,·5},·{-4,·-5,·-5,·-5,·-5,·-5})
  
45 o5·:·OIResolution45 o5·:·OIResolution
46 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·o\x8oi\x8iR\x8Re\x8es\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*46 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·o\x8oi\x8iR\x8Re\x8es\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
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93 <td>··············<pre><code·class="language-macaulay2">i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);</code></pre>93 <td>··············<pre><code·class="language-macaulay2">i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);</code></pre>96 <td>··············<pre><code·class="language-macaulay2">i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)99 <td>··············<pre><code·class="language-macaulay2">i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)
100 ·--·used·0.1244s·(cpu);·0.0983777s·(thread);·0s·(gc)100 ·--·used·0.146288s·(cpu);·0.103883s·(thread);·0s·(gc)
  
101 ·······································································101 ·······································································
102 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,102 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,
103 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},2·103 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},2·
104 ·····------------------------------------------------------------------------104 ·····------------------------------------------------------------------------
105 ······2··················2105 ······2··················2
106 ·····x···x···e········-·x···x···e·······}106 ·····x···x···e········-·x···x···e·······}
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21 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);21 i1·:·P·=·makePolynomialOIAlgebra(2,·x,·QQ);
22 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);22 i2·:·F·=·makeFreeOIModule(e,·{1,1,2},·P);
23 i3·:·installGeneratorsInWidth(F,·1);23 i3·:·installGeneratorsInWidth(F,·1);
24 i4·:·installGeneratorsInWidth(F,·2);24 i4·:·installGeneratorsInWidth(F,·2);
25 i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);25 i5·:·use·F_1;·b1·=·x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2);
26 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);26 i7·:·use·F_2;·b2·=·x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
27 i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)27 i9·:·time·B·=·oiGB({b1,·b2},·Strategy·=>·FastNonminimal)
28 ·--·used·0.1244s·(cpu);·0.0983777s·(thread);·0s·(gc)28 ·--·used·0.146288s·(cpu);·0.103883s·(thread);·0s·(gc)
  
  
29 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,29 o9·=·{x···e········+·x···e·······,·x···x···e········+·x···x···e·······,
30 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},230 ·······2,1·1,{1},2····1,1·1,{1},2···1,2·1,1·2,{2},1····2,2·1,2·2,{2},2
31 ·····------------------------------------------------------------------------31 ·····------------------------------------------------------------------------
32 ······2··················232 ······2··················2
33 ·····x···x···e········-·x···x···e·······}33 ·····x···x···e········-·x···x···e·······}
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=87 #:len=8
8 bWF4Q29uZXM=8 bWF4Q29uZXM=
9 #:len=11209 #:len=1120
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg
11 Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gNTIzMiwgSW5wdXRzID0+IHtTUEFOe1RUeyJG11 Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gNTIzMiwgSW5wdXRzID0+IHtTUEFOe1RUeyJG
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39 o10·:·Fan39 o10·:·Fan
  
40 i11·:·isComplete·F40 i11·:·isComplete·F
  
41 o11·=·true41 o11·=·true
  
42 i12·:·isPolytopal·F42 i12·:·isPolytopal·F
43 {({ambient·dimension·=>·3···········},·{1,·2,·5}),·({ambient·dimension·=>·3···········},·{1,·2,·4}),·({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient·dimension·=>·3···········},·{2,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·3,·5}),·({ambient·dimension·=>·3···········},·{1,·3,·4})}43 {({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient·dimension·=>·3···········},·{2,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·3,·5}),·({ambient·dimension·=>·3···········},·{1,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·2,·5}),·({ambient·dimension·=>·3···········},·{1,·2,·4})}
44 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·044 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0
45 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·145 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1
46 ···number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·146 ···number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1
47 ···number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·147 ···number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1
48 v:·{{ambient·dimension·=>·3···········},·{ambient·dimension·=>·3···········}}48 v:·{{ambient·dimension·=>·3···········},·{ambient·dimension·=>·3···········}}
49 ·····dimension·of·lineality·space·=>·0····dimension·of·lineality·space·=>·049 ·····dimension·of·lineality·space·=>·0····dimension·of·lineality·space·=>·0
50 ·····dimension·of·the·cone·=>·3···········dimension·of·the·cone·=>·350 ·····dimension·of·the·cone·=>·3···········dimension·of·the·cone·=>·3
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28 ······top·dimension·of·the·cones·=>·228 ······top·dimension·of·the·cones·=>·2
  
29 o3·:·Fan29 o3·:·Fan
  
30 i4·:·apply(maxCones·F1,rays)30 i4·:·apply(maxCones·F1,rays)
  
31 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}31 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}
32 ······|·0·-1·|··|·1·-1·|··|·0·1·|··|·1·0·|··|·0·0·|··|·-1·0·|32 ······|·0·-1·|··|·1·-1·|··|·0·0·|··|·1·0·|··|·0·1·|··|·-1·0·|
33 ······|·0·-1·|··|·0·-1·|··|·0·0·|··|·0·1·|··|·0·1·|··|·-1·1·|33 ······|·0·-1·|··|·0·-1·|··|·0·1·|··|·0·1·|··|·0·0·|··|·-1·1·|
  
34 o4·:·List34 o4·:·List
  
35 i5·:·PC·=·polyhedralComplex·hypercube·335 i5·:·PC·=·polyhedralComplex·hypercube·3
  
36 o5·=·{ambient·dimension·=>·3·············}36 o5·=·{ambient·dimension·=>·3·············}
37 ······number·of·generating·polyhedra·=>·137 ······number·of·generating·polyhedra·=>·1
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51 ······number·of·generating·polyhedra·=>·651 ······number·of·generating·polyhedra·=>·6
52 ······top·dimension·of·the·polyhedra·=>·252 ······top·dimension·of·the·polyhedra·=>·2
  
53 o6·:·PolyhedralComplex53 o6·:·PolyhedralComplex
  
54 i7·:·apply(maxPolyhedra·PC1,vertices)54 i7·:·apply(maxPolyhedra·PC1,vertices)
  
55 o7·=·{|·-1·-1·-1·-1·|,·|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-155 o7·=·{|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-1·1··-1·1·|,·|·-1
56 ······|·-1·1··-1·1··|··|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-156 ······|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-1·-1·1··1·|··|·-1
57 ······|·-1·-1·1··1··|··|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·1·57 ······|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·1··1··1··1·|··|·-1
58 ·····------------------------------------------------------------------------58 ·····------------------------------------------------------------------------
59 ·····1··-1·1·|,·|·-1·1··-1·1··|}59 ·····1··-1·1··|,·|·-1·-1·-1·-1·|}
 60 ·····-1·1··1··|··|·-1·1··-1·1··|
60 ·····-1·1··1·|··|·-1·-1·1··1··|61 ·····-1·-1·-1·|··|·-1·-1·1··1··|
61 ·····1··1··1·|··|·-1·-1·-1·-1·| 
  
62 o7·:·List62 o7·:·List
  
63 i8·:·63 i8·:·
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104 o11·=·true</code></pre>104 o11·=·true</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ········</table>106 ········</table>
107 ········<p></p>107 ········<p></p>
108 For·a·complete·fan·we·can·check·if·it·is·projective:········<table·class="examples">108 For·a·complete·fan·we·can·check·if·it·is·projective:········<table·class="examples">
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i12·:·isPolytopal·F110 <td>··············<pre><code·class="language-macaulay2">i12·:·isPolytopal·F
111 {({ambient·dimension·=>·3···········},·{1,·2,·5}),·({ambient·dimension·=>·3···········},·{1,·2,·4}),·({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient·dimension·=>·3···········},·{2,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·3,·5}),·({ambient·dimension·=>·3···········},·{1,·3,·4})}111 {({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient·dimension·=>·3···········},·{2,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·3,·5}),·({ambient·dimension·=>·3···········},·{1,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·2,·5}),·({ambient·dimension·=>·3···········},·{1,·2,·4})}
112 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0112 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0·················dimension·of·lineality·space·=>·0
113 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1113 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1
114 ···number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1114 ···number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1·····························number·of·facets·=>·1
115 ···number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1115 ···number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1·······························number·of·rays·=>·1
116 v:·{{ambient·dimension·=>·3···········},·{ambient·dimension·=>·3···········}}116 v:·{{ambient·dimension·=>·3···········},·{ambient·dimension·=>·3···········}}
117 ·····dimension·of·lineality·space·=>·0····dimension·of·lineality·space·=>·0117 ·····dimension·of·lineality·space·=>·0····dimension·of·lineality·space·=>·0
118 ·····dimension·of·the·cone·=>·3···········dimension·of·the·cone·=>·3118 ·····dimension·of·the·cone·=>·3···········dimension·of·the·cone·=>·3
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37 o10·:·Fan37 o10·:·Fan
38 i11·:·isComplete·F38 i11·:·isComplete·F
  
39 o11·=·true39 o11·=·true
40 For·a·complete·fan·we·can·check·if·it·is·projective:40 For·a·complete·fan·we·can·check·if·it·is·projective:
41 i12·:·isPolytopal·F41 i12·:·isPolytopal·F
42 {({ambient·dimension·=>·3···········},·{1,·2,·5}),·({ambient·dimension·=>·342 {({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient·dimension·=>·3
43 },·{1,·2,·4}),·({ambient·dimension·=>·3···········},·{2,·3,·5}),·({ambient43 },·{2,·3,·4}),·({ambient·dimension·=>·3···········},·{1,·3,·5}),·({ambient
44 dimension·=>·3···········},·{2,·3,·4}),·({ambient·dimension·=>·3···········},44 dimension·=>·3···········},·{1,·3,·4}),·({ambient·dimension·=>·3···········},
45 {1,·3,·5}),·({ambient·dimension·=>·3···········},·{1,·3,·4})}45 {1,·2,·5}),·({ambient·dimension·=>·3···········},·{1,·2,·4})}
46 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality46 ···dimension·of·lineality·space·=>·0·················dimension·of·lineality
47 space·=>·0·················dimension·of·lineality·space·=>·047 space·=>·0·················dimension·of·lineality·space·=>·0
48 dimension·of·lineality·space·=>·0·················dimension·of·lineality·space48 dimension·of·lineality·space·=>·0·················dimension·of·lineality·space
49 =>·0·················dimension·of·lineality·space·=>·049 =>·0·················dimension·of·lineality·space·=>·0
50 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·150 ···dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1
51 dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·151 dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1
52 dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·152 dimension·of·the·cone·=>·1························dimension·of·the·cone·=>·1
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107 o3·:·Fan</code></pre>107 o3·:·Fan</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i4·:·apply(maxCones·F1,rays)110 <td>··············<pre><code·class="language-macaulay2">i4·:·apply(maxCones·F1,rays)
  
111 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}111 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}
112 ······|·0·-1·|··|·1·-1·|··|·0·1·|··|·1·0·|··|·0·0·|··|·-1·0·|112 ······|·0·-1·|··|·1·-1·|··|·0·0·|··|·1·0·|··|·0·1·|··|·-1·0·|
113 ······|·0·-1·|··|·0·-1·|··|·0·0·|··|·0·1·|··|·0·1·|··|·-1·1·|113 ······|·0·-1·|··|·0·-1·|··|·0·1·|··|·0·1·|··|·0·0·|··|·-1·1·|
  
114 o4·:·List</code></pre>114 o4·:·List</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ········</table>116 ········</table>
117 ········<p></p>117 ········<p></p>
118 For·a·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a><span·class="tt">·PC</span>·and·an·integer·<span·class="tt">k</span>·between·0·and·the·dimension·of·<span·class="tt">PC</span>,·<span·class="tt">skeleton</span>·computes·the·<span·class="tt">k</span>-skeleton·of·the·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a>·<span·class="tt">PC</span>,·i.e.·the·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a>·<span·class="tt">PC1</span>·generated·by·all·polyhedra·of·dimension·<span·class="tt">k</span>·in·<span·class="tt">PC</span>.········<table·class="examples">118 For·a·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a><span·class="tt">·PC</span>·and·an·integer·<span·class="tt">k</span>·between·0·and·the·dimension·of·<span·class="tt">PC</span>,·<span·class="tt">skeleton</span>·computes·the·<span·class="tt">k</span>-skeleton·of·the·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a>·<span·class="tt">PC</span>,·i.e.·the·<a·title="the·class·of·all·polyhedral·complexes"·href="___Polyhedral__Complex.html">PolyhedralComplex</a>·<span·class="tt">PC1</span>·generated·by·all·polyhedra·of·dimension·<span·class="tt">k</span>·in·<span·class="tt">PC</span>.········<table·class="examples">
119 ··········<tr>119 ··········<tr>
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136 ······top·dimension·of·the·polyhedra·=>·2136 ······top·dimension·of·the·polyhedra·=>·2
  
137 o6·:·PolyhedralComplex</code></pre>137 o6·:·PolyhedralComplex</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i7·:·apply(maxPolyhedra·PC1,vertices)140 <td>··············<pre><code·class="language-macaulay2">i7·:·apply(maxPolyhedra·PC1,vertices)
  
141 o7·=·{|·-1·-1·-1·-1·|,·|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-1141 o7·=·{|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-1·1··-1·1·|,·|·-1
142 ······|·-1·1··-1·1··|··|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-1142 ······|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-1·-1·1··1·|··|·-1
143 ······|·-1·-1·1··1··|··|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·1·143 ······|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·1··1··1··1·|··|·-1
144 ·····------------------------------------------------------------------------144 ·····------------------------------------------------------------------------
145 ·····1··-1·1·|,·|·-1·1··-1·1··|}145 ·····1··-1·1··|,·|·-1·-1·-1·-1·|}
 146 ·····-1·1··1··|··|·-1·1··-1·1··|
146 ·····-1·1··1·|··|·-1·-1·1··1··|147 ·····-1·-1·-1·|··|·-1·-1·1··1··|
147 ·····1··1··1·|··|·-1·-1·-1·-1·| 
  
148 o7·:·List</code></pre>148 o7·:·List</code></pre>
149 </td>··········</tr>149 </td>··········</tr>
150 ········</table>150 ········</table>
151 ······</div>151 ······</div>
152 ······<div·class="waystouse">152 ······<div·class="waystouse">
153 ········<h2>For·the·programmer</h2>153 ········<h2>For·the·programmer</h2>
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45 ······number·of·rays·=>·445 ······number·of·rays·=>·4
46 ······top·dimension·of·the·cones·=>·246 ······top·dimension·of·the·cones·=>·2
  
47 o3·:·Fan47 o3·:·Fan
48 i4·:·apply(maxCones·F1,rays)48 i4·:·apply(maxCones·F1,rays)
  
49 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}49 o4·=·{|·1·-1·|,·|·0·-1·|,·|·1·0·|,·|·0·0·|,·|·1·0·|,·|·-1·0·|}
50 ······|·0·-1·|··|·1·-1·|··|·0·1·|··|·1·0·|··|·0·0·|··|·-1·0·|50 ······|·0·-1·|··|·1·-1·|··|·0·0·|··|·1·0·|··|·0·1·|··|·-1·0·|
51 ······|·0·-1·|··|·0·-1·|··|·0·0·|··|·0·1·|··|·0·1·|··|·-1·1·|51 ······|·0·-1·|··|·0·-1·|··|·0·1·|··|·0·1·|··|·0·0·|··|·-1·1·|
  
52 o4·:·List52 o4·:·List
53 For·a·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC·and·an·integer·k·between·0·and·the·dimension·of·PC,53 For·a·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC·and·an·integer·k·between·0·and·the·dimension·of·PC,
54 skeleton·computes·the·k-skeleton·of·the·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC,·i.e.·the54 skeleton·computes·the·k-skeleton·of·the·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC,·i.e.·the
55 _\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC1·generated·by·all·polyhedra·of·dimension·k·in·PC.55 _\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a_\x8l_\x8C_\x8o_\x8m_\x8p_\x8l_\x8e_\x8x·PC1·generated·by·all·polyhedra·of·dimension·k·in·PC.
56 i5·:·PC·=·polyhedralComplex·hypercube·356 i5·:·PC·=·polyhedralComplex·hypercube·3
  
Offset 68, 18 lines modifiedOffset 68, 18 lines modified
68 o6·=·{ambient·dimension·=>·3·············}68 o6·=·{ambient·dimension·=>·3·············}
69 ······number·of·generating·polyhedra·=>·669 ······number·of·generating·polyhedra·=>·6
70 ······top·dimension·of·the·polyhedra·=>·270 ······top·dimension·of·the·polyhedra·=>·2
  
71 o6·:·PolyhedralComplex71 o6·:·PolyhedralComplex
72 i7·:·apply(maxPolyhedra·PC1,vertices)72 i7·:·apply(maxPolyhedra·PC1,vertices)
  
73 o7·=·{|·-1·-1·-1·-1·|,·|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-173 o7·=·{|·-1·1··-1·1·|,·|·1··1··1··1·|,·|·-1·1··-1·1··|,·|·-1·1··-1·1·|,·|·-1
74 ······|·-1·1··-1·1··|··|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-174 ······|·1··1··1··1·|··|·-1·1··-1·1·|··|·-1·-1·-1·-1·|··|·-1·-1·1··1·|··|·-1
75 ······|·-1·-1·1··1··|··|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·175 ······|·-1·-1·1··1·|··|·-1·-1·1··1·|··|·-1·-1·1··1··|··|·1··1··1··1·|··|·-1
76 ·····------------------------------------------------------------------------76 ·····------------------------------------------------------------------------
77 ·····1··-1·1·|,·|·-1·1··-1·1··|}77 ·····1··-1·1··|,·|·-1·-1·-1·-1·|}
 78 ·····-1·1··1··|··|·-1·1··-1·1··|
78 ·····-1·1··1·|··|·-1·-1·1··1··|79 ·····-1·-1·-1·|··|·-1·-1·1··1··|
79 ·····1··1··1·|··|·-1·-1·-1·-1·| 
  
80 o7·:·List80 o7·:·List
81 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*81 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
82 The·object·_\x8s_\x8k_\x8e_\x8l_\x8e_\x8t_\x8o_\x8n·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.82 The·object·_\x8s_\x8k_\x8e_\x8l_\x8e_\x8t_\x8o_\x8n·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n.
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760 B
./usr/share/doc/Macaulay2/OldToricVectorBundles/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/OnlineLookup/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
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712 B
./usr/share/doc/Macaulay2/OpenMath/dump/rawdocumentation.dump
    
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3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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747 B
./usr/share/doc/Macaulay2/PHCpack/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=347 #:len=34
8 ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ==8 ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ==
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742 B
./usr/share/doc/Macaulay2/PackageCitations/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=167 #:len=16
8 UGFja2FnZUNpdGF0aW9ucw==8 UGFja2FnZUNpdGF0aW9ucw==
9 #:len=19289 #:len=1928
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11 aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY4LCBTZWVBbHNv11 aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY4LCBTZWVBbHNv
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735 B
./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:38·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=147 #:len=14
8 c2Vjb25kRnVuY3Rpb24=8 c2Vjb25kRnVuY3Rpb24=
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740 B
./usr/share/doc/Macaulay2/Parametrization/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=167 #:len=16
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715 B
./usr/share/doc/Macaulay2/Parsing/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=127 #:len=12
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759 B
./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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1.35 KB
./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/___Lab__Book__Protocol.out
    
Offset 41, 15 lines modifiedOffset 41, 15 lines modified
41 i3·:·g=341 i3·:·g=3
  
42 o3·=·342 o3·=·3
  
43 i4·:·kk=·ZZ/101;43 i4·:·kk=·ZZ/101;
  
44 i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);44 i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);
45 ·--·1.15914s·elapsed45 ·--·2.19899s·elapsed
  
46 i6·:·M=cliffordModule(Mu1,Mu2,R)46 i6·:·M=cliffordModule(Mu1,Mu2,R)
  
47 o6·=·CliffordModule{...6...}47 o6·=·CliffordModule{...6...}
  
48 o6·:·CliffordModule48 o6·:·CliffordModule
  
Offset 67, 30 lines modifiedOffset 67, 30 lines modified
67 ··········m12=randomExtension(m1.yAction,m2.yAction);67 ··········m12=randomExtension(m1.yAction,m2.yAction);
68 ··········V·=·vectorBundleOnE·m12;68 ··········V·=·vectorBundleOnE·m12;
69 ··········Ul=tensorProduct(Mor,V);69 ··········Ul=tensorProduct(Mor,V);
70 ··········Ul1=tensorProduct(Mor1,V);70 ··········Ul1=tensorProduct(Mor1,V);
71 ··········d0=unique·degrees·target·Ul.yAction;71 ··········d0=unique·degrees·target·Ul.yAction;
72 ··········d1=unique·degrees·target·Ul1.yAction;72 ··········d1=unique·degrees·target·Ul1.yAction;
73 ··········#d1·>=3·or·#d0·>=3)·do·();73 ··········#d1·>=3·or·#d0·>=3)·do·();
74 ·--·.274643s·elapsed74 ·--·.78469s·elapsed
  
75 i12·:·betti·Ul.yAction,·betti·Ul1.yAction75 i12·:·betti·Ul.yAction,·betti·Ul1.yAction
  
76 ···············0··1··········0··176 ···············0··1··········0··1
77 o12·=·(total:·32·32,·total:·32·32)77 o12·=·(total:·32·32,·total:·32·32)
78 ··········-4:·16··.·····-2:·32··.78 ··········-4:·16··.·····-2:·32··.
79 ··········-3:·16··.·····-1:··.··.79 ··········-3:·16··.·····-1:··.··.
80 ··········-2:··.··.······0:··.··.80 ··········-2:··.··.······0:··.··.
81 ··········-1:··.·16······1:··.·3281 ··········-1:··.·16······1:··.·32
82 ···········0:··.·1682 ···········0:··.·16
  
83 o12·:·Sequence83 o12·:·Sequence
  
84 i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the·actions·of·generators84 i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the·actions·of·generators
85 ·--·13.4781s·elapsed85 ·--·24.7906s·elapsed
  
86 i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));86 i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
  
87 ··············32······3287 ··············32······32
88 o14·:·Matrix·S···<--·S88 o14·:·Matrix·S···<--·S
  
89 i15·:·r=289 i15·:·r=2
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./usr/share/doc/Macaulay2/PencilsOfQuadrics/example-output/_search__Ulrich.out
    
Offset 45, 30 lines modifiedOffset 45, 30 lines modified
45 i11·:·M=cliffordModule(Mu1,Mu2,R)45 i11·:·M=cliffordModule(Mu1,Mu2,R)
  
46 o11·=·CliffordModule{...6...}46 o11·=·CliffordModule{...6...}
  
47 o11·:·CliffordModule47 o11·:·CliffordModule
  
48 i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);48 i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);
49 ·--·.462128s·elapsed49 ·--·1.41673s·elapsed
  
50 i13·:·betti·res·Ulr50 i13·:·betti·res·Ulr
  
51 ·············0··1·251 ·············0··1·2
52 o13·=·total:·8·16·852 o13·=·total:·8·16·8
53 ··········0:·8·16·853 ··········0:·8·16·8
  
54 o13·:·BettiTally54 o13·:·BettiTally
  
55 i14·:·ann·Ulr·==·ideal·qs55 i14·:·ann·Ulr·==·ideal·qs
  
56 o14·=·true56 o14·=·true
  
57 i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);57 i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);
58 ·--·1.85269s·elapsed58 ·--·3.2734s·elapsed
  
59 i16·:·betti·res·Ulr359 i16·:·betti·res·Ulr3
  
60 ··············0··1··260 ··············0··1··2
61 o16·=·total:·12·24·1261 o16·=·total:·12·24·12
62 ··········0:·12·24·1262 ··········0:·12·24·12
  
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./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
    
Offset 120, 15 lines modifiedOffset 120, 15 lines modified
120 o3·=·3</code></pre>120 o3·=·3</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i4·:·kk=·ZZ/101;</code></pre>123 <td>··············<pre><code·class="language-macaulay2">i4·:·kk=·ZZ/101;</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);126 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);
127 ·--·1.15914s·elapsed</code></pre>127 ·--·2.19899s·elapsed</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ··········<tr>129 ··········<tr>
130 <td>··············<pre><code·class="language-macaulay2">i6·:·M=cliffordModule(Mu1,Mu2,R)130 <td>··············<pre><code·class="language-macaulay2">i6·:·M=cliffordModule(Mu1,Mu2,R)
  
131 o6·=·CliffordModule{...6...}131 o6·=·CliffordModule{...6...}
  
132 o6·:·CliffordModule</code></pre>132 o6·:·CliffordModule</code></pre>
Offset 152, 15 lines modifiedOffset 152, 15 lines modified
152 ··········m12=randomExtension(m1.yAction,m2.yAction);152 ··········m12=randomExtension(m1.yAction,m2.yAction);
153 ··········V·=·vectorBundleOnE·m12;153 ··········V·=·vectorBundleOnE·m12;
154 ··········Ul=tensorProduct(Mor,V);154 ··········Ul=tensorProduct(Mor,V);
155 ··········Ul1=tensorProduct(Mor1,V);155 ··········Ul1=tensorProduct(Mor1,V);
156 ··········d0=unique·degrees·target·Ul.yAction;156 ··········d0=unique·degrees·target·Ul.yAction;
157 ··········d1=unique·degrees·target·Ul1.yAction;157 ··········d1=unique·degrees·target·Ul1.yAction;
158 ··········#d1·>=3·or·#d0·>=3)·do·();158 ··········#d1·>=3·or·#d0·>=3)·do·();
159 ·--·.274643s·elapsed</code></pre>159 ·--·.78469s·elapsed</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
161 ··········<tr>161 ··········<tr>
162 <td>··············<pre><code·class="language-macaulay2">i12·:·betti·Ul.yAction,·betti·Ul1.yAction162 <td>··············<pre><code·class="language-macaulay2">i12·:·betti·Ul.yAction,·betti·Ul1.yAction
  
163 ···············0··1··········0··1163 ···············0··1··········0··1
164 o12·=·(total:·32·32,·total:·32·32)164 o12·=·(total:·32·32,·total:·32·32)
165 ··········-4:·16··.·····-2:·32··.165 ··········-4:·16··.·····-2:·32··.
Offset 169, 15 lines modifiedOffset 169, 15 lines modified
169 ··········-1:··.·16······1:··.·32169 ··········-1:··.·16······1:··.·32
170 ···········0:··.·16170 ···········0:··.·16
  
171 o12·:·Sequence</code></pre>171 o12·:·Sequence</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the·actions·of·generators174 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the·actions·of·generators
175 ·--·13.4781s·elapsed</code></pre>175 ·--·24.7906s·elapsed</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ··········<tr>177 ··········<tr>
178 <td>··············<pre><code·class="language-macaulay2">i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));178 <td>··············<pre><code·class="language-macaulay2">i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
  
179 ··············32······32179 ··············32······32
180 o14·:·Matrix·S···&lt;--·S</code></pre>180 o14·:·Matrix·S···&lt;--·S</code></pre>
181 </td>··········</tr>181 </td>··········</tr>
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Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 ············--·will·give·an·Ulrich·bundle,·with·betti·table56 ············--·will·give·an·Ulrich·bundle,·with·betti·table
57 ············--·16·32·1657 ············--·16·32·16
58 i3·:·g=358 i3·:·g=3
  
59 o3·=·359 o3·=·3
60 i4·:·kk=·ZZ/101;60 i4·:·kk=·ZZ/101;
61 i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);61 i5·:·elapsedTime·(S,qq,R,u,·M1,M2,·Mu1,·Mu2)=randomNicePencil(kk,g);
62 ·--·1.15914s·elapsed62 ·--·2.19899s·elapsed
63 i6·:·M=cliffordModule(Mu1,Mu2,R)63 i6·:·M=cliffordModule(Mu1,Mu2,R)
  
64 o6·=·CliffordModule{...6...}64 o6·=·CliffordModule{...6...}
  
65 o6·:·CliffordModule65 o6·:·CliffordModule
66 i7·:·Mor·=·vectorBundleOnE·M.evenCenter;66 i7·:·Mor·=·vectorBundleOnE·M.evenCenter;
67 i8·:·Mor1=·vectorBundleOnE·M.oddCenter;67 i8·:·Mor1=·vectorBundleOnE·M.oddCenter;
Offset 76, 29 lines modifiedOffset 76, 29 lines modified
76 ··········m12=randomExtension(m1.yAction,m2.yAction);76 ··········m12=randomExtension(m1.yAction,m2.yAction);
77 ··········V·=·vectorBundleOnE·m12;77 ··········V·=·vectorBundleOnE·m12;
78 ··········Ul=tensorProduct(Mor,V);78 ··········Ul=tensorProduct(Mor,V);
79 ··········Ul1=tensorProduct(Mor1,V);79 ··········Ul1=tensorProduct(Mor1,V);
80 ··········d0=unique·degrees·target·Ul.yAction;80 ··········d0=unique·degrees·target·Ul.yAction;
81 ··········d1=unique·degrees·target·Ul1.yAction;81 ··········d1=unique·degrees·target·Ul1.yAction;
82 ··········#d1·>=3·or·#d0·>=3)·do·();82 ··········#d1·>=3·or·#d0·>=3)·do·();
83 ·--·.274643s·elapsed83 ·--·.78469s·elapsed
84 i12·:·betti·Ul.yAction,·betti·Ul1.yAction84 i12·:·betti·Ul.yAction,·betti·Ul1.yAction
  
85 ···············0··1··········0··185 ···············0··1··········0··1
86 o12·=·(total:·32·32,·total:·32·32)86 o12·=·(total:·32·32,·total:·32·32)
87 ··········-4:·16··.·····-2:·32··.87 ··········-4:·16··.·····-2:·32··.
88 ··········-3:·16··.·····-1:··.··.88 ··········-3:·16··.·····-1:··.··.
89 ··········-2:··.··.······0:··.··.89 ··········-2:··.··.······0:··.··.
90 ··········-1:··.·16······1:··.·3290 ··········-1:··.·16······1:··.·32
91 ···········0:··.·1691 ···········0:··.·16
  
92 o12·:·Sequence92 o12·:·Sequence
93 i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the93 i13·:·elapsedTime·Ul·=·tensorProduct(M,V);·--·the·heaviest·part·computing·the
94 actions·of·generators94 actions·of·generators
95 ·--·13.4781s·elapsed95 ·--·24.7906s·elapsed
96 i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));96 i14·:·M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
  
97 ··············32······3297 ··············32······32
98 o14·:·Matrix·S···<--·S98 o14·:·Matrix·S···<--·S
99 i15·:·r=299 i15·:·r=2
  
100 o15·=·2100 o15·=·2
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139 o11·=·CliffordModule{...6...}139 o11·=·CliffordModule{...6...}
  
140 o11·:·CliffordModule</code></pre>140 o11·:·CliffordModule</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);143 <td>··············<pre><code·class="language-macaulay2">i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);
144 ·--·.462128s·elapsed</code></pre>144 ·--·1.41673s·elapsed</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i13·:·betti·res·Ulr147 <td>··············<pre><code·class="language-macaulay2">i13·:·betti·res·Ulr
  
148 ·············0··1·2148 ·············0··1·2
149 o13·=·total:·8·16·8149 o13·=·total:·8·16·8
150 ··········0:·8·16·8150 ··········0:·8·16·8
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157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i14·:·ann·Ulr·==·ideal·qs158 <td>··············<pre><code·class="language-macaulay2">i14·:·ann·Ulr·==·ideal·qs
  
159 o14·=·true</code></pre>159 o14·=·true</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
161 ··········<tr>161 ··········<tr>
162 <td>··············<pre><code·class="language-macaulay2">i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);162 <td>··············<pre><code·class="language-macaulay2">i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);
163 ·--·1.85269s·elapsed</code></pre>163 ·--·3.2734s·elapsed</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ··········<tr>165 ··········<tr>
166 <td>··············<pre><code·class="language-macaulay2">i16·:·betti·res·Ulr3166 <td>··············<pre><code·class="language-macaulay2">i16·:·betti·res·Ulr3
  
167 ··············0··1··2167 ··············0··1··2
168 o16·=·total:·12·24·12168 o16·=·total:·12·24·12
169 ··········0:·12·24·12169 ··········0:·12·24·12
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64 o10·:·Matrix·S··<--·S64 o10·:·Matrix·S··<--·S
65 i11·:·M=cliffordModule(Mu1,Mu2,R)65 i11·:·M=cliffordModule(Mu1,Mu2,R)
  
66 o11·=·CliffordModule{...6...}66 o11·=·CliffordModule{...6...}
  
67 o11·:·CliffordModule67 o11·:·CliffordModule
68 i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);68 i12·:·elapsedTime·Ulr·=·searchUlrich(M,S);
69 ·--·.462128s·elapsed69 ·--·1.41673s·elapsed
70 i13·:·betti·res·Ulr70 i13·:·betti·res·Ulr
  
71 ·············0··1·271 ·············0··1·2
72 o13·=·total:·8·16·872 o13·=·total:·8·16·8
73 ··········0:·8·16·873 ··········0:·8·16·8
  
74 o13·:·BettiTally74 o13·:·BettiTally
75 i14·:·ann·Ulr·==·ideal·qs75 i14·:·ann·Ulr·==·ideal·qs
  
76 o14·=·true76 o14·=·true
77 i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);77 i15·:·elapsedTime·Ulr3·=·searchUlrich(M,S,3);
78 ·--·1.85269s·elapsed78 ·--·3.2734s·elapsed
79 i16·:·betti·res·Ulr379 i16·:·betti·res·Ulr3
  
80 ··············0··1··280 ··············0··1··2
81 o16·=·total:·12·24·1281 o16·=·total:·12·24·12
82 ··········0:·12·24·1282 ··········0:·12·24·12
  
83 o16·:·BettiTally83 o16·:·BettiTally
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8 c3RhbmRhcmRUYWJsZWF1eChaWixMaXN0KQ==8 c3RhbmRhcmRUYWJsZWF1eChaWixMaXN0KQ==
9 #:len=2689 #:len=268
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTA1LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTA1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzdGFuZGFyZFRhYmxlYXV4LFpaLExpc3QpLCJzdGFu11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzdGFuZGFyZFRhYmxlYXV4LFpaLExpc3QpLCJzdGFu
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765 B
./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=277 #:len=27
8 YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp8 YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp
9 #:len=2959 #:len=295
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU5LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk
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729 B
./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=247 #:len=24
8 cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop8 cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop
9 #:len=2559 #:len=255
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv
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515 B
./usr/share/doc/Macaulay2/Points/example-output/_affine__Fat__Points.out
    
Offset 66, 17 lines modifiedOffset 66, 17 lines modified
66 i9·:·mults·=·{1,2,3,1,2,3,1,2,3,1,2,3}66 i9·:·mults·=·{1,2,3,1,2,3,1,2,3,1,2,3}
  
67 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}67 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}
  
68 o9·:·List68 o9·:·List
  
69 i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);69 i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);
70 ·--·2.0176s·elapsed70 ·--·2.21893s·elapsed
  
71 i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);71 i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);
72 ·--·3.49035s·elapsed72 ·--·3.96465s·elapsed
  
73 i12·:·G==H73 i12·:·G==H
  
74 o12·=·true74 o12·=·true
  
75 i13·:·75 i13·:·
⊟
1.59 KB
./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
    
Offset 162, 19 lines modifiedOffset 162, 19 lines modified
  
162 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}162 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}
  
163 o9·:·List</code></pre>163 o9·:·List</code></pre>
164 </td>··········</tr>164 </td>··········</tr>
165 ··········<tr>165 ··········<tr>
166 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);166 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);
167 ·--·2.0176s·elapsed</code></pre>167 ·--·2.21893s·elapsed</code></pre>
168 </td>··········</tr>168 </td>··········</tr>
169 ··········<tr>169 ··········<tr>
170 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);170 <td>··············<pre><code·class="language-macaulay2">i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);
171 ·--·3.49035s·elapsed</code></pre>171 ·--·3.96465s·elapsed</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i12·:·G==H174 <td>··············<pre><code·class="language-macaulay2">i12·:·G==H
  
175 o12·=·true</code></pre>175 o12·=·true</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ········</table>177 ········</table>
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781 B
html2text {}
    
Offset 82, 17 lines modifiedOffset 82, 17 lines modified
82 o8·:·Matrix·K··<--·K82 o8·:·Matrix·K··<--·K
83 i9·:·mults·=·{1,2,3,1,2,3,1,2,3,1,2,3}83 i9·:·mults·=·{1,2,3,1,2,3,1,2,3,1,2,3}
  
84 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}84 o9·=·{1,·2,·3,·1,·2,·3,·1,·2,·3,·1,·2,·3}
  
85 o9·:·List85 o9·:·List
86 i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);86 i10·:·elapsedTime·(Q,inG,G)·=·affineFatPoints(M,mults,R);
87 ·--·2.0176s·elapsed87 ·--·2.21893s·elapsed
88 i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);88 i11·:·elapsedTime·H·=·affineFatPointsByIntersection(M,mults,R);
89 ·--·3.49035s·elapsed89 ·--·3.96465s·elapsed
90 i12·:·G==H90 i12·:·G==H
  
91 o12·=·true91 o12·=·true
92 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*92 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
93 For·reduced·points,·this·function·may·be·a·bit·slower·than·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s.93 For·reduced·points,·this·function·may·be·a·bit·slower·than·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s.
94 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*94 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
95 ····*·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8F_\x8a_\x8t_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s_\x8B_\x8y_\x8I_\x8n_\x8t_\x8e_\x8r_\x8s_\x8e_\x8c_\x8t_\x8i_\x8o_\x8n_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8,_\x8L_\x8i_\x8s_\x8t_\x8,_\x8R_\x8i_\x8n_\x8g_\x8)·--·computes·ideal·of·fat95 ····*·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8F_\x8a_\x8t_\x8P_\x8o_\x8i_\x8n_\x8t_\x8s_\x8B_\x8y_\x8I_\x8n_\x8t_\x8e_\x8r_\x8s_\x8e_\x8c_\x8t_\x8i_\x8o_\x8n_\x8(_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8,_\x8L_\x8i_\x8s_\x8t_\x8,_\x8R_\x8i_\x8n_\x8g_\x8)·--·computes·ideal·of·fat
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715 B
./usr/share/doc/Macaulay2/Polyhedra/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 bWF4Q29uZXM=8 bWF4Q29uZXM=
9 #:len=11859 #:len=1185
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg
11 Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi11 Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi
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712 B
./usr/share/doc/Macaulay2/Polymake/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 UG9seW1ha2U=8 UG9seW1ha2U=
9 #:len=6109 #:len=610
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu
11 ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw11 ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw
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739 B
./usr/share/doc/Macaulay2/PolyominoIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=167 #:len=16
8 cG9seW9JZGVhbChMaXN0KQ==8 cG9seW9JZGVhbChMaXN0KQ==
9 #:len=2569 #:len=256
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwb2x5b0lkZWFsLExpc3QpLCJwb2x5b0lkZWFsKExp11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwb2x5b0lkZWFsLExpc3QpLCJwb2x5b0lkZWFsKExp
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920 B
./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/___Field.out
    
Offset 7, 16 lines modifiedOffset 7, 16 lines modified
7 o1·:·GaloisField7 o1·:·GaloisField
  
8 i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};8 i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
  
9 i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)9 i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)
  
10 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+10 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+
11 ···············3,1·2,3····3,3·2,1·····2,1·1,2····2,2·1,1·····3,1·1,2··11 ···············2,1·1,2····2,2·1,1·····3,1·1,2····3,2·1,1·····3,2·2,3··
12 ·····------------------------------------------------------------------------12 ·····------------------------------------------------------------------------
13 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)13 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)
14 ······3,2·1,1·····3,2·2,3····3,3·2,2·····3,1·2,2····3,2·2,114 ······3,3·2,2·····3,1·2,2····3,2·2,1·····3,1·2,3····3,3·2,1
  
15 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]15 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
16 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,116 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1
  
17 i4·:·17 i4·:·
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3.08 KB
./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/_polyo__Ideal.out
    
Offset 5, 44 lines modifiedOffset 5, 44 lines modified
5 o1·=·{{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{2,·2},·{3,·3}}}5 o1·=·{{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{2,·2},·{3,·3}}}
  
6 o1·:·List6 o1·:·List
  
7 i2·:·I·=·polyoIdeal·Q7 i2·:·I·=·polyoIdeal·Q
  
8 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,8 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
9 ·············3,3·2,1····3,1·2,3···3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,2·9 ·············3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,2···3,3·2,2····3,2·2,3·
10 ·····------------------------------------------------------------------------10 ·····------------------------------------------------------------------------
11 ·····x···x····-·x···x···,·x···x····-·x···x···)11 ·····x···x····-·x···x···,·x···x····-·x···x···)
12 ······3,3·2,2····3,2·2,3···3,2·1,1····3,1·1,212 ······3,2·1,1····3,1·1,2···3,3·2,1····3,1·2,3
  
13 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]13 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
14 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,114 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1
  
15 i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},·{{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};15 i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},·{{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};
  
16 i4·:·I·=·polyoIdeal·Q16 i4·:·I·=·polyoIdeal·Q
  
17 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,17 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
18 ·············2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1····3,1·1,2·18 ·············2,3·1,1····2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4·
19 ·····------------------------------------------------------------------------19 ·····------------------------------------------------------------------------
20 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···20 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
21 ······4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1····4,1·3,2···2,3·1,121 ······3,4·2,3····3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1
22 ·····------------------------------------------------------------------------22 ·····------------------------------------------------------------------------
23 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-23 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
24 ········2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4···3,4·2,3··24 ········2,1·1,4···4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3··
25 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
26 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,26 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
27 ······3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1····2,1·1,4·27 ······3,3·1,4···3,2·2,1····3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4·
28 ·····------------------------------------------------------------------------28 ·····------------------------------------------------------------------------
29 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···29 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
30 ······4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3····3,3·1,4···3,2·2,130 ······4,4·3,2····4,2·3,4···2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1
31 ·····------------------------------------------------------------------------31 ·····------------------------------------------------------------------------
32 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-32 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
33 ········3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4···4,4·3,2··33 ········3,1·1,2···4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1··
34 ·····------------------------------------------------------------------------34 ·····------------------------------------------------------------------------
35 ·····x···x···)35 ·····x···x···)
36 ······4,2·3,436 ······4,1·3,2
  
37 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]37 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
38 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,3···2,2···2,1···1,4···1,3···1,2···1,138 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,3···2,2···2,1···1,4···1,3···1,2···1,1
  
39 i5·:·39 i5·:·
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55 ··········<tr>55 ··········<tr>
56 <td>··············<pre><code·class="language-macaulay2">i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};</code></pre>56 <td>··············<pre><code·class="language-macaulay2">i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};</code></pre>
57 </td>··········</tr>57 </td>··········</tr>
58 ··········<tr>58 ··········<tr>
59 <td>··············<pre><code·class="language-macaulay2">i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)59 <td>··············<pre><code·class="language-macaulay2">i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)
  
60 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+60 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+
61 ···············3,1·2,3····3,3·2,1·····2,1·1,2····2,2·1,1·····3,1·1,2··61 ···············2,1·1,2····2,2·1,1·····3,1·1,2····3,2·1,1·····3,2·2,3··
62 ·····------------------------------------------------------------------------62 ·····------------------------------------------------------------------------
63 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)63 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)
64 ······3,2·1,1·····3,2·2,3····3,3·2,2·····3,1·2,2····3,2·2,164 ······3,3·2,2·····3,1·2,2····3,2·2,1·····3,1·2,3····3,3·2,1
  
65 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]65 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
66 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1</code></pre>66 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1</code></pre>
67 </td>··········</tr>67 </td>··········</tr>
68 ········</table>68 ········</table>
69 ······</div>69 ······</div>
70 ······<div>70 ······<div>
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12 o1·=·F12 o1·=·F
  
13 o1·:·GaloisField13 o1·:·GaloisField
14 i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};14 i2·:·Q={{{1,1},{2,2}},{{2,1},{3,2}},{{2,2},{3,3}}};
15 i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)15 i3·:·I·=·polyoIdeal(Q,Field=>·F,RingChoice=>1,TermOrder=>·GRevLex)
  
16 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+16 o3·=·ideal·(-·x···x····+·x···x···,·-·x···x····+·x···x···,·-·x···x····+
17 ···············3,1·2,3····3,3·2,1·····2,1·1,2····2,2·1,1·····3,1·1,217 ···············2,1·1,2····2,2·1,1·····3,1·1,2····3,2·1,1·····3,2·2,3
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)19 ·····x···x···,·-·x···x····+·x···x···,·-·x···x····+·x···x···)
20 ······3,2·1,1·····3,2·2,3····3,3·2,2·····3,1·2,2····3,2·2,120 ······3,3·2,2·····3,1·2,2····3,2·2,1·····3,1·2,3····3,3·2,1
  
21 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]21 o3·:·Ideal·of·F[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
22 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,122 ·················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1
23 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
24 ····*·_\x8p_\x8o_\x8l_\x8y_\x8o_\x8I_\x8d_\x8e_\x8a_\x8l·--·Ideal·of·inner·2-minors·of·a·collection·of·cells24 ····*·_\x8p_\x8o_\x8l_\x8y_\x8o_\x8I_\x8d_\x8e_\x8a_\x8l·--·Ideal·of·inner·2-minors·of·a·collection·of·cells
25 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·F\x8Fi\x8ie\x8el\x8ld\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8un\x8nc\x8ct\x8ti\x8io\x8on\x8ns\x8s·w\x8wi\x8it\x8th\x8h·o\x8op\x8pt\x8ti\x8io\x8on\x8na\x8al\x8l·a\x8ar\x8rg\x8gu\x8um\x8me\x8en\x8nt\x8t·n\x8na\x8am\x8me\x8ed\x8d·F\x8Fi\x8ie\x8el\x8ld\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
26 ····*·polyoIdeal(...,Field=>...)26 ····*·polyoIdeal(...,Field=>...)
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85 o1·:·List</code></pre>85 o1·:·List</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·I·=·polyoIdeal·Q88 <td>··············<pre><code·class="language-macaulay2">i2·:·I·=·polyoIdeal·Q
  
89 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,89 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
90 ·············3,3·2,1····3,1·2,3···3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,2·90 ·············3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,2···3,3·2,2····3,2·2,3·
91 ·····------------------------------------------------------------------------91 ·····------------------------------------------------------------------------
92 ·····x···x····-·x···x···,·x···x····-·x···x···)92 ·····x···x····-·x···x···,·x···x····-·x···x···)
93 ······3,3·2,2····3,2·2,3···3,2·1,1····3,1·1,293 ······3,2·1,1····3,1·1,2···3,3·2,1····3,1·2,3
  
94 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]94 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
95 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1</code></pre>95 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ········</table>97 ········</table>
98 <br><br>········<table·class="examples">98 <br><br>········<table·class="examples">
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},·{{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};</code></pre>100 <td>··············<pre><code·class="language-macaulay2">i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},·{{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i4·:·I·=·polyoIdeal·Q103 <td>··············<pre><code·class="language-macaulay2">i4·:·I·=·polyoIdeal·Q
  
104 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,104 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
105 ·············2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1····3,1·1,2·105 ·············2,3·1,1····2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4·
106 ·····------------------------------------------------------------------------106 ·····------------------------------------------------------------------------
107 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···107 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
108 ······4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1····4,1·3,2···2,3·1,1108 ······3,4·2,3····3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1
109 ·····------------------------------------------------------------------------109 ·····------------------------------------------------------------------------
110 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-110 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
111 ········2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4···3,4·2,3··111 ········2,1·1,4···4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3··
112 ·····------------------------------------------------------------------------112 ·····------------------------------------------------------------------------
113 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,113 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
114 ······3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1····2,1·1,4·114 ······3,3·1,4···3,2·2,1····3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4·
115 ·····------------------------------------------------------------------------115 ·····------------------------------------------------------------------------
116 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···116 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
117 ······4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3····3,3·1,4···3,2·2,1117 ······4,4·3,2····4,2·3,4···2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1
118 ·····------------------------------------------------------------------------118 ·····------------------------------------------------------------------------
119 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-119 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
120 ········3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4···4,4·3,2··120 ········3,1·1,2···4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1··
121 ·····------------------------------------------------------------------------121 ·····------------------------------------------------------------------------
122 ·····x···x···)122 ·····x···x···)
123 ······4,2·3,4123 ······4,1·3,2
  
124 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]124 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
125 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,3···2,2···2,1···1,4···1,3···1,2···1,1</code></pre>125 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,3···2,2···2,1···1,4···1,3···1,2···1,1</code></pre>
126 </td>··········</tr>126 </td>··········</tr>
127 ········</table>127 ········</table>
128 ······</div>128 ······</div>
129 ······<div·class="waystouse">129 ······<div·class="waystouse">
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32 o1·=·{{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{2,·2},·{3,·3}}}32 o1·=·{{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{2,·2},·{3,·3}}}
  
33 o1·:·List33 o1·:·List
34 i2·:·I·=·polyoIdeal·Q34 i2·:·I·=·polyoIdeal·Q
  
35 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,35 o2·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
36 ·············3,3·2,1····3,1·2,3···3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,236 ·············3,2·2,1····3,1·2,2···2,2·1,1····2,1·1,2···3,3·2,2····3,2·2,3
37 ·····------------------------------------------------------------------------37 ·····------------------------------------------------------------------------
38 ·····x···x····-·x···x···,·x···x····-·x···x···)38 ·····x···x····-·x···x···,·x···x····-·x···x···)
39 ······3,3·2,2····3,2·2,3···3,2·1,1····3,1·1,239 ······3,2·1,1····3,1·1,2···3,3·2,1····3,1·2,3
  
40 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]40 o2·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···]
41 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,141 ··················3,3···3,2···3,1···2,3···2,2···2,1···1,2···1,1
  
  
42 i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},42 i3·:·Q={{{1,·1},·{2,·2}},·{{2,·1},·{3,·2}},·{{3,·1},·{4,·2}},·{{3,·2},·{4,·3}},
43 {{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};43 {{3,·3},·{4,·4}},·{{2,·3},·{3,·4}},·{{1,·3},·{2,·4}},·{{1,·2},·{2,·3}}};
44 i4·:·I·=·polyoIdeal·Q44 i4·:·I·=·polyoIdeal·Q
  
45 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,45 o4·=·ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
46 ·············2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1····3,1·1,246 ·············2,3·1,1····2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
48 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x48 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x
49 ······4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1····4,1·3,2···2,3·1,149 ······3,4·2,3····3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1
50 ·····------------------------------------------------------------------------50 ·····------------------------------------------------------------------------
51 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-51 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
52 ········2,1·1,3···4,3·3,1····4,1·3,3···4,4·1,3····4,3·1,4···3,4·2,352 ········2,1·1,4···4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3
53 ·····------------------------------------------------------------------------53 ·····------------------------------------------------------------------------
54 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,54 ·····x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
55 ······3,3·2,4···4,2·2,1····4,1·2,2···2,3·1,2····2,2·1,3···2,4·1,1····2,1·1,455 ······3,3·1,4···3,2·2,1····3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4
56 ·····------------------------------------------------------------------------56 ·····------------------------------------------------------------------------
57 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x57 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x
58 ······4,4·3,1····4,1·3,4···4,3·3,2····4,2·3,3···3,4·1,3····3,3·1,4···3,2·2,158 ······4,4·3,2····4,2·3,4···2,4·1,3····2,3·1,4···4,4·3,3····4,3·3,4···3,2·1,1
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-60 ·····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x····-
61 ········3,1·2,2···4,2·1,1····4,1·1,2···2,4·1,2····2,2·1,4···4,4·3,261 ········3,1·1,2···4,4·2,3····4,3·2,4···2,2·1,1····2,1·1,2···4,2·3,1
62 ·····------------------------------------------------------------------------62 ·····------------------------------------------------------------------------
63 ·····x···x···)63 ·····x···x···)
64 ······4,2·3,464 ······4,1·3,2
  
65 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x65 o4·:·Ideal·of·QQ[x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x···,·x
66 ,·x···,·x···,·x···,·x···,·x···]66 ,·x···,·x···,·x···,·x···,·x···]
67 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,367 ··················4,4···4,3···4,2···4,1···3,4···3,3···3,2···3,1···2,4···2,3
68 2,2···2,1···1,4···1,3···1,2···1,168 2,2···2,1···1,4···1,3···1,2···1,1
69 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8po\x8ol\x8ly\x8yo\x8oI\x8Id\x8de\x8ea\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*69 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8po\x8ol\x8ly\x8yo\x8oI\x8Id\x8de\x8ea\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*
70 ····*·polyoIdeal(List)70 ····*·polyoIdeal(List)
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722 B
./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
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8 bWF4aW1hbEFudGljaGFpbnM=8 bWF4aW1hbEFudGljaGFpbnM=
9 #:len=11279 #:len=1127
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW5010 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW50
11 aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ4NzgsIElucHV0cyA9PiB7U1BBTntU11 aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ4NzgsIElucHV0cyA9PiB7U1BBTntU
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950 B
./usr/share/doc/Macaulay2/Posets/example-output/___Precompute.out
    
Offset 31, 27 lines modifiedOffset 31, 27 lines modified
31 o5·=·CacheTable{name·=>·P}31 o5·=·CacheTable{name·=>·P}
  
32 i6·:·C·==·P32 i6·:·C·==·P
  
33 o6·=·true33 o6·=·true
  
34 i7·:·time·isDistributive·C34 i7·:·time·isDistributive·C
35 ·--·used·0.000219641s·(cpu);·5.378e-06s·(thread);·0s·(gc)35 ·--·used·0.000336241s·(cpu);·6.642e-06s·(thread);·0s·(gc)
  
36 o7·=·true36 o7·=·true
  
37 i8·:·time·isDistributive·P37 i8·:·time·isDistributive·P
38 ·--·used·4.27254s·(cpu);·3.13024s·(thread);·0s·(gc)38 ·--·used·5.21091s·(cpu);·3.60995s·(thread);·0s·(gc)
  
39 o8·=·true39 o8·=·true
  
40 i9·:·C'·=·dual·C;40 i9·:·C'·=·dual·C;
  
41 i10·:·time·isDistributive·C'41 i10·:·time·isDistributive·C'
42 ·--·used·0.000211418s·(cpu);·4.347e-06s·(thread);·0s·(gc)42 ·--·used·0.000301546s·(cpu);·5.58e-06s·(thread);·0s·(gc)
  
43 o10·=·true43 o10·=·true
  
44 i11·:·peek·C'.cache44 i11·:·peek·C'.cache
  
45 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}······································}45 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}······································}
46 ·················coveringRelations·=>·{{1,·0},·{2,·1},·{3,·2},·{4,·3},·{5,·4},·{6,·5},·{7,·6},·{8,·7},·{9,·8}}46 ·················coveringRelations·=>·{{1,·0},·{2,·1},·{3,·2},·{4,·3},·{5,·4},·{6,·5},·{7,·6},·{8,·7},·{9,·8}}
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730 B
./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out
    
Offset 7, 22 lines modifiedOffset 7, 22 lines modified
7 o2·=·Partition{4,·2}7 o2·=·Partition{4,·2}
  
8 o2·:·Partition8 o2·:·Partition
  
9 i3·:·D·=·dominanceLattice·6;9 i3·:·D·=·dominanceLattice·6;
  
10 i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·"antichains")10 i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·"antichains")
11 ·--·used·0.209032s·(cpu);·0.165354s·(thread);·0s·(gc)11 ·--·used·0.322494s·(cpu);·0.18321s·(thread);·0s·(gc)
  
12 o4·=·Partition{9,·2}12 o4·=·Partition{9,·2}
  
13 o4·:·Partition13 o4·:·Partition
  
14 i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·"chains")14 i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·"chains")
15 ·--·used·0.00017254s·(cpu);·6.72e-06s·(thread);·0s·(gc)15 ·--·used·0.000204704s·(cpu);·9.127e-06s·(thread);·0s·(gc)
  
16 o5·=·Partition{9,·2}16 o5·=·Partition{9,·2}
  
17 o5·:·Partition17 o5·:·Partition
  
18 i6·:·greeneKleitmanPartition·chain·1018 i6·:·greeneKleitmanPartition·chain·10
  
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./usr/share/doc/Macaulay2/Posets/html/___Precompute.html
    
Offset 87, 35 lines modifiedOffset 87, 35 lines modified
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i6·:·C·==·P88 <td>··············<pre><code·class="language-macaulay2">i6·:·C·==·P
  
89 o6·=·true</code></pre>89 o6·=·true</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isDistributive·C92 <td>··············<pre><code·class="language-macaulay2">i7·:·time·isDistributive·C
93 ·--·used·0.000219641s·(cpu);·5.378e-06s·(thread);·0s·(gc)93 ·--·used·0.000336241s·(cpu);·6.642e-06s·(thread);·0s·(gc)
  
94 o7·=·true</code></pre>94 o7·=·true</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i8·:·time·isDistributive·P97 <td>··············<pre><code·class="language-macaulay2">i8·:·time·isDistributive·P
98 ·--·used·4.27254s·(cpu);·3.13024s·(thread);·0s·(gc)98 ·--·used·5.21091s·(cpu);·3.60995s·(thread);·0s·(gc)
  
99 o8·=·true</code></pre>99 o8·=·true</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ········</table>101 ········</table>
102 ········<div>102 ········<div>
103 ··········<p>We·also·know·that·the·dual·of·a·distributive·lattice·is·again·a·distributive·lattice.··Other·information·is·copied·when·possible.</p>103 ··········<p>We·also·know·that·the·dual·of·a·distributive·lattice·is·again·a·distributive·lattice.··Other·information·is·copied·when·possible.</p>
104 ········</div>104 ········</div>
105 ········<table·class="examples">105 ········<table·class="examples">
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i9·:·C'·=·dual·C;</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i9·:·C'·=·dual·C;</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isDistributive·C'110 <td>··············<pre><code·class="language-macaulay2">i10·:·time·isDistributive·C'
111 ·--·used·0.000211418s·(cpu);·4.347e-06s·(thread);·0s·(gc)111 ·--·used·0.000301546s·(cpu);·5.58e-06s·(thread);·0s·(gc)
  
112 o10·=·true</code></pre>112 o10·=·true</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i11·:·peek·C'.cache115 <td>··············<pre><code·class="language-macaulay2">i11·:·peek·C'.cache
  
116 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}······································}116 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}······································}
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html2text {}
    
Offset 41, 26 lines modifiedOffset 41, 26 lines modified
41 i5·:·peek·P.cache41 i5·:·peek·P.cache
  
42 o5·=·CacheTable{name·=>·P}42 o5·=·CacheTable{name·=>·P}
43 i6·:·C·==·P43 i6·:·C·==·P
  
44 o6·=·true44 o6·=·true
45 i7·:·time·isDistributive·C45 i7·:·time·isDistributive·C
46 ·--·used·0.000219641s·(cpu);·5.378e-06s·(thread);·0s·(gc)46 ·--·used·0.000336241s·(cpu);·6.642e-06s·(thread);·0s·(gc)
  
47 o7·=·true47 o7·=·true
48 i8·:·time·isDistributive·P48 i8·:·time·isDistributive·P
49 ·--·used·4.27254s·(cpu);·3.13024s·(thread);·0s·(gc)49 ·--·used·5.21091s·(cpu);·3.60995s·(thread);·0s·(gc)
  
50 o8·=·true50 o8·=·true
51 We·also·know·that·the·dual·of·a·distributive·lattice·is·again·a·distributive51 We·also·know·that·the·dual·of·a·distributive·lattice·is·again·a·distributive
52 lattice.·Other·information·is·copied·when·possible.52 lattice.·Other·information·is·copied·when·possible.
53 i9·:·C'·=·dual·C;53 i9·:·C'·=·dual·C;
54 i10·:·time·isDistributive·C'54 i10·:·time·isDistributive·C'
55 ·--·used·0.000211418s·(cpu);·4.347e-06s·(thread);·0s·(gc)55 ·--·used·0.000301546s·(cpu);·5.58e-06s·(thread);·0s·(gc)
  
56 o10·=·true56 o10·=·true
57 i11·:·peek·C'.cache57 i11·:·peek·C'.cache
  
58 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}58 o11·=·CacheTable{connectedComponents·=>·{{0,·1,·2,·3,·4,·5,·6,·7,·8,·9}}
59 }59 }
60 ·················coveringRelations·=>·{{1,·0},·{2,·1},·{3,·2},·{4,·3},·{5,·4},60 ·················coveringRelations·=>·{{1,·0},·{2,·1},·{3,·2},·{4,·3},·{5,·4},
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./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
    
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95 ········</div>95 ········</div>
96 ········<table·class="examples">96 ········<table·class="examples">
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i3·:·D·=·dominanceLattice·6;</code></pre>98 <td>··············<pre><code·class="language-macaulay2">i3·:·D·=·dominanceLattice·6;</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·&quot;antichains&quot;)101 <td>··············<pre><code·class="language-macaulay2">i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·&quot;antichains&quot;)
102 ·--·used·0.209032s·(cpu);·0.165354s·(thread);·0s·(gc)102 ·--·used·0.322494s·(cpu);·0.18321s·(thread);·0s·(gc)
  
103 o4·=·Partition{9,·2}103 o4·=·Partition{9,·2}
  
104 o4·:·Partition</code></pre>104 o4·:·Partition</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·&quot;chains&quot;)107 <td>··············<pre><code·class="language-macaulay2">i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·&quot;chains&quot;)
108 ·--·used·0.00017254s·(cpu);·6.72e-06s·(thread);·0s·(gc)108 ·--·used·0.000204704s·(cpu);·9.127e-06s·(thread);·0s·(gc)
  
109 o5·=·Partition{9,·2}109 o5·=·Partition{9,·2}
  
110 o5·:·Partition</code></pre>110 o5·:·Partition</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ········</table>112 ········</table>
113 ········<div>113 ········<div>
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Offset 29, 21 lines modifiedOffset 29, 21 lines modified
  
29 o2·:·Partition29 o2·:·Partition
30 The·conjugate·of·$l$·has·the·same·property,·but·with·chains·replaced·by30 The·conjugate·of·$l$·has·the·same·property,·but·with·chains·replaced·by
31 _\x8a_\x8n_\x8t_\x8i_\x8c_\x8h_\x8a_\x8i_\x8n_\x8s.·Because·of·this,·it·is·often·better·to·count·via·antichains·instead31 _\x8a_\x8n_\x8t_\x8i_\x8c_\x8h_\x8a_\x8i_\x8n_\x8s.·Because·of·this,·it·is·often·better·to·count·via·antichains·instead
32 of·chains.·This·can·be·done·by·passing·"antichains"·as·the·Strategy.32 of·chains.·This·can·be·done·by·passing·"antichains"·as·the·Strategy.
33 i3·:·D·=·dominanceLattice·6;33 i3·:·D·=·dominanceLattice·6;
34 i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·"antichains")34 i4·:·time·greeneKleitmanPartition(D,·Strategy·=>·"antichains")
35 ·--·used·0.209032s·(cpu);·0.165354s·(thread);·0s·(gc)35 ·--·used·0.322494s·(cpu);·0.18321s·(thread);·0s·(gc)
  
36 o4·=·Partition{9,·2}36 o4·=·Partition{9,·2}
  
37 o4·:·Partition37 o4·:·Partition
38 i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·"chains")38 i5·:·time·greeneKleitmanPartition(D,·Strategy·=>·"chains")
39 ·--·used·0.00017254s·(cpu);·6.72e-06s·(thread);·0s·(gc)39 ·--·used·0.000204704s·(cpu);·9.127e-06s·(thread);·0s·(gc)
  
40 o5·=·Partition{9,·2}40 o5·=·Partition{9,·2}
  
41 o5·:·Partition41 o5·:·Partition
42 The·Greene-Kleitman·partition·of·the·$n$·_\x8c_\x8h_\x8a_\x8i_\x8n·is·the·partition·of·$n$·with·$1$42 The·Greene-Kleitman·partition·of·the·$n$·_\x8c_\x8h_\x8a_\x8i_\x8n·is·the·partition·of·$n$·with·$1$
43 part.43 part.
44 i6·:·greeneKleitmanPartition·chain·1044 i6·:·greeneKleitmanPartition·chain·10
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758 B
./usr/share/doc/Macaulay2/PositivityToricBundles/dump/rawdocumentation.dump
    
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9 #:len=26269 #:len=2626
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu
11 dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp11 dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp
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./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_kernel__Of__Localization.out
    
Offset 24, 35 lines modifiedOffset 24, 35 lines modified
24 ··············|·0············0·············0············x_1^3-x_0x_2^2·0··············|24 ··············|·0············0·············0············x_1^3-x_0x_2^2·0··············|
25 ··············|·0············0·············0············0··············x_1^5-x_0x_2^4·|25 ··············|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
26 ····························326 ····························3
27 o3·:·R-module,·quotient·of·R27 o3·:·R-module,·quotient·of·R
  
28 i4·:·elapsedTime·kernelOfLocalization(M,·I1)28 i4·:·elapsedTime·kernelOfLocalization(M,·I1)
29 ·--·.146576s·elapsed29 ·--·.14969s·elapsed
  
30 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)30 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
31 ··················|·1·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|31 ··················|·1·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
32 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|32 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
33 ·······························333 ·······························3
34 o4·:·R-module,·subquotient·of·R34 o4·:·R-module,·subquotient·of·R
  
35 i5·:·elapsedTime·kernelOfLocalization(M,·I2)35 i5·:·elapsedTime·kernelOfLocalization(M,·I2)
36 ·--·.0226296s·elapsed36 ·--·.0210733s·elapsed
  
37 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)37 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
38 ··················|·0·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|38 ··················|·0·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
39 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|39 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
40 ·······························340 ·······························3
41 o5·:·R-module,·subquotient·of·R41 o5·:·R-module,·subquotient·of·R
  
42 i6·:·elapsedTime·kernelOfLocalization(M,·I3)42 i6·:·elapsedTime·kernelOfLocalization(M,·I3)
43 ·--·.0185809s·elapsed43 ·--·.0166003s·elapsed
  
44 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)44 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
45 ··················|·0·1·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|45 ··················|·0·1·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
46 ··················|·0·0·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|46 ··················|·0·0·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
47 ·······························347 ·······························3
48 o6·:·R-module,·subquotient·of·R48 o6·:·R-module,·subquotient·of·R
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./usr/share/doc/Macaulay2/PrimaryDecomposition/example-output/_reg__Seq__In__Ideal.out
    
Offset 13, 15 lines modifiedOffset 13, 15 lines modified
13 ·····------------------------------------------------------------------------13 ·····------------------------------------------------------------------------
14 ·····x·x·)14 ·····x·x·)
15 ······0·415 ······0·4
  
16 o2·:·Ideal·of·R16 o2·:·Ideal·of·R
  
17 i3·:·elapsedTime·regSeqInIdeal·I17 i3·:·elapsedTime·regSeqInIdeal·I
18 ·--·.0621889s·elapsed18 ·--·.0842477s·elapsed
  
19 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)19 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)
20 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·720 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·7
  
21 o3·:·Ideal·of·R21 o3·:·Ideal·of·R
  
22 i4·:·R·=·QQ[h,l,s,x,y,z]22 i4·:·R·=·QQ[h,l,s,x,y,z]
Offset 41, 15 lines modifiedOffset 41, 15 lines modified
41 o5·:·Ideal·of·R41 o5·:·Ideal·of·R
  
42 i6·:·isSubset(I,·ideal(s,l,h))·--·implies·codim·I·==·342 i6·:·isSubset(I,·ideal(s,l,h))·--·implies·codim·I·==·3
  
43 o6·=·true43 o6·=·true
  
44 i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)44 i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)
45 ·--·.00637862s·elapsed45 ·--·.00887527s·elapsed
  
46 ···················2················3····2·····2····8····3····2·····246 ···················2················3····2·····2····8····3····2·····2
47 o7·=·ideal·(h*l·-·l··-·4l*s·+·h*y,·h··+·l·s·-·h·x,·s··+·h··+·l·s·-·h·x)47 o7·=·ideal·(h*l·-·l··-·4l*s·+·h*y,·h··+·l·s·-·h·x,·s··+·h··+·l·s·-·h·x)
  
48 o7·:·Ideal·of·R48 o7·:·Ideal·of·R
  
49 i8·:·49 i8·:·
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./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_kernel__Of__Localization.html
    
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101 ··············|·0············0·············0············0··············x_1^5-x_0x_2^4·|101 ··············|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
102 ····························3102 ····························3
103 o3·:·R-module,·quotient·of·R</code></pre>103 o3·:·R-module,·quotient·of·R</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·kernelOfLocalization(M,·I1)106 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·kernelOfLocalization(M,·I1)
107 ·--·.146576s·elapsed107 ·--·.14969s·elapsed
  
108 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)108 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
109 ··················|·1·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|109 ··················|·1·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
110 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|110 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
111 ·······························3111 ·······························3
112 o4·:·R-module,·subquotient·of·R</code></pre>112 o4·:·R-module,·subquotient·of·R</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·kernelOfLocalization(M,·I2)115 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·kernelOfLocalization(M,·I2)
116 ·--·.0226296s·elapsed116 ·--·.0210733s·elapsed
  
117 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)117 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
118 ··················|·0·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|118 ··················|·0·0·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
119 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|119 ··················|·0·1·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
120 ·······························3120 ·······························3
121 o5·:·R-module,·subquotient·of·R</code></pre>121 o5·:·R-module,·subquotient·of·R</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·kernelOfLocalization(M,·I3)124 <td>··············<pre><code·class="language-macaulay2">i6·:·elapsedTime·kernelOfLocalization(M,·I3)
125 ·--·.0185809s·elapsed125 ·--·.0166003s·elapsed
  
126 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)126 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0··············0··············|)
127 ··················|·0·1·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|127 ··················|·0·1·|··|·0············0·············0············x_1^3-x_0x_2^2·0··············|
128 ··················|·0·0·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|128 ··················|·0·0·|··|·0············0·············0············0··············x_1^5-x_0x_2^4·|
  
129 ·······························3129 ·······························3
130 o6·:·R-module,·subquotient·of·R</code></pre>130 o6·:·R-module,·subquotient·of·R</code></pre>
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42 |42 |
43 ··············|·0············0·············0············0··············x_1^5-43 ··············|·0············0·············0············0··············x_1^5-
44 x_0x_2^4·|44 x_0x_2^4·|
  
45 ····························345 ····························3
46 o3·:·R-module,·quotient·of·R46 o3·:·R-module,·quotient·of·R
47 i4·:·elapsedTime·kernelOfLocalization(M,·I1)47 i4·:·elapsedTime·kernelOfLocalization(M,·I1)
48 ·--·.146576s·elapsed48 ·--·.14969s·elapsed
  
49 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·049 o4·=·subquotient·(|·0·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0
50 0··············|)50 0··············|)
51 ··················|·1·0·|··|·0············0·············0············x_1^3-51 ··················|·1·0·|··|·0············0·············0············x_1^3-
52 x_0x_2^2·0··············|52 x_0x_2^2·0··············|
53 ··················|·0·1·|··|·0············0·············0············053 ··················|·0·1·|··|·0············0·············0············0
54 x_1^5-x_0x_2^4·|54 x_1^5-x_0x_2^4·|
  
55 ·······························355 ·······························3
56 o4·:·R-module,·subquotient·of·R56 o4·:·R-module,·subquotient·of·R
57 i5·:·elapsedTime·kernelOfLocalization(M,·I2)57 i5·:·elapsedTime·kernelOfLocalization(M,·I2)
58 ·--·.0226296s·elapsed58 ·--·.0210733s·elapsed
  
59 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·059 o5·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0
60 0··············|)60 0··············|)
61 ··················|·0·0·|··|·0············0·············0············x_1^3-61 ··················|·0·0·|··|·0············0·············0············x_1^3-
62 x_0x_2^2·0··············|62 x_0x_2^2·0··············|
63 ··················|·0·1·|··|·0············0·············0············063 ··················|·0·1·|··|·0············0·············0············0
64 x_1^5-x_0x_2^4·|64 x_1^5-x_0x_2^4·|
  
65 ·······························365 ·······························3
66 o5·:·R-module,·subquotient·of·R66 o5·:·R-module,·subquotient·of·R
67 i6·:·elapsedTime·kernelOfLocalization(M,·I3)67 i6·:·elapsedTime·kernelOfLocalization(M,·I3)
68 ·--·.0185809s·elapsed68 ·--·.0166003s·elapsed
  
69 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·069 o6·=·subquotient·(|·1·0·|,·|·x_2^2-x_1x_3·x_1x_2-x_0x_3·x_1^2-x_0x_2·0
70 0··············|)70 0··············|)
71 ··················|·0·1·|··|·0············0·············0············x_1^3-71 ··················|·0·1·|··|·0············0·············0············x_1^3-
72 x_0x_2^2·0··············|72 x_0x_2^2·0··············|
73 ··················|·0·0·|··|·0············0·············0············073 ··················|·0·0·|··|·0············0·············0············0
74 x_1^5-x_0x_2^4·|74 x_1^5-x_0x_2^4·|
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./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html
    
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102 ·····x·x·)102 ·····x·x·)
103 ······0·4103 ······0·4
  
104 o2·:·Ideal·of·R</code></pre>104 o2·:·Ideal·of·R</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·regSeqInIdeal·I107 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·regSeqInIdeal·I
108 ·--·.0621889s·elapsed108 ·--·.0842477s·elapsed
  
109 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)109 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)
110 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·7110 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·7
  
111 o3·:·Ideal·of·R</code></pre>111 o3·:·Ideal·of·R</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
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140 ··········<tr>140 ··········<tr>
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142 o6·=·true</code></pre>142 o6·=·true</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)145 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)
146 ·--·.00637862s·elapsed146 ·--·.00887527s·elapsed
  
147 ···················2················3····2·····2····8····3····2·····2147 ···················2················3····2·····2····8····3····2·····2
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149 o7·:·Ideal·of·R</code></pre>149 o7·:·Ideal·of·R</code></pre>
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41 ·············2·7···0·7···3·6···2·6···1·6···0·6···2·5···0·5···3·4···2·4···1·441 ·············2·7···0·7···3·6···2·6···1·6···0·6···2·5···0·5···3·4···2·4···1·4
42 ·····------------------------------------------------------------------------42 ·····------------------------------------------------------------------------
43 ·····x·x·)43 ·····x·x·)
44 ······0·444 ······0·4
  
45 o2·:·Ideal·of·R45 o2·:·Ideal·of·R
46 i3·:·elapsedTime·regSeqInIdeal·I46 i3·:·elapsedTime·regSeqInIdeal·I
47 ·--·.0621889s·elapsed47 ·--·.0842477s·elapsed
  
48 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)48 o3·=·ideal·(x·x·,·x·x··+·x·x·,·x·x··+·x·x·,·x·x··+·x·x·)
49 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·749 ·············2·7···3·6····0·7···2·5····0·7···1·4····0·7
  
50 o3·:·Ideal·of·R50 o3·:·Ideal·of·R
51 If·I·is·the·unit·ideal,·then·an·ideal·of·variables·of·the·ring·is·returned.51 If·I·is·the·unit·ideal,·then·an·ideal·of·variables·of·the·ring·is·returned.
52 If·the·codimension·of·I·is·already·known,·then·one·can·specify·this,·along·with52 If·the·codimension·of·I·is·already·known,·then·one·can·specify·this,·along·with
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71 ·····l·,·s·)71 ·····l·,·s·)
  
72 o5·:·Ideal·of·R72 o5·:·Ideal·of·R
73 i6·:·isSubset(I,·ideal(s,l,h))·--·implies·codim·I·==·373 i6·:·isSubset(I,·ideal(s,l,h))·--·implies·codim·I·==·3
  
74 o6·=·true74 o6·=·true
75 i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)75 i7·:·elapsedTime·regSeqInIdeal(I,·3,·3,·1)
76 ·--·.00637862s·elapsed76 ·--·.00887527s·elapsed
  
77 ···················2················3····2·····2····8····3····2·····277 ···················2················3····2·····2····8····3····2·····2
78 o7·=·ideal·(h*l·-·l··-·4l*s·+·h*y,·h··+·l·s·-·h·x,·s··+·h··+·l·s·-·h·x)78 o7·=·ideal·(h*l·-·l··-·4l*s·+·h*y,·h··+·l·s·-·h·x,·s··+·h··+·l·s·-·h·x)
  
79 o7·:·Ideal·of·R79 o7·:·Ideal·of·R
80 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*80 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
81 ····*·_\x8r_\x8a_\x8d_\x8i_\x8c_\x8a_\x8l·--·the·radical·of·an·ideal81 ····*·_\x8r_\x8a_\x8d_\x8i_\x8c_\x8a_\x8l·--·the·radical·of·an·ideal
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25 ····*·_\x8n_\x8e_\x8x_\x8t_\x8(_\x8P_\x8y_\x8t_\x8h_\x8o_\x8n_\x8O_\x8b_\x8j_\x8e_\x8c_\x8t_\x8)·--·retrieve·the·next·item·from·a·python·iterator25 ····*·_\x8n_\x8e_\x8x_\x8t_\x8(_\x8P_\x8y_\x8t_\x8h_\x8o_\x8n_\x8O_\x8b_\x8j_\x8e_\x8c_\x8t_\x8)·--·retrieve·the·next·item·from·a·python·iterator
26 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*26 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
27 ····*·_\x8i_\x8t_\x8e_\x8r_\x8a_\x8t_\x8o_\x8r_\x8(_\x8P_\x8y_\x8t_\x8h_\x8o_\x8n_\x8O_\x8b_\x8j_\x8e_\x8c_\x8t_\x8)·--·get·iterator·of·iterable·python·object27 ····*·_\x8i_\x8t_\x8e_\x8r_\x8a_\x8t_\x8o_\x8r_\x8(_\x8P_\x8y_\x8t_\x8h_\x8o_\x8n_\x8O_\x8b_\x8j_\x8e_\x8c_\x8t_\x8)·--·get·iterator·of·iterable·python·object
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975 B
./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html
    
Offset 73, 15 lines modifiedOffset 73, 15 lines modified
73 o1·=·range(0,·3)73 o1·=·range(0,·3)
  
74 o1·:·PythonObject·of·class·range</code></pre>74 o1·:·PythonObject·of·class·range</code></pre>
75 </td>··········</tr>75 </td>··········</tr>
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i2·:·i·=·iterator·x77 <td>··············<pre><code·class="language-macaulay2">i2·:·i·=·iterator·x
  
78 o2·=·&lt;range_iterator·object·at·0x7f4259ec76f0>78 o2·=·&lt;range_iterator·object·at·0x7f3521267750>
  
79 o2·:·PythonObject·of·class·range_iterator</code></pre>79 o2·:·PythonObject·of·class·range_iterator</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·next·i82 <td>··············<pre><code·class="language-macaulay2">i3·:·next·i
  
83 o3·=·083 o3·=·0
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354 B
html2text {}
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 i1·:·x·=·pythonValue·"range(3)"17 i1·:·x·=·pythonValue·"range(3)"
  
18 o1·=·range(0,·3)18 o1·=·range(0,·3)
  
19 o1·:·PythonObject·of·class·range19 o1·:·PythonObject·of·class·range
20 i2·:·i·=·iterator·x20 i2·:·i·=·iterator·x
  
21 o2·=·<range_iterator·object·at·0x7f4259ec76f0>21 o2·=·<range_iterator·object·at·0x7f3521267750>
  
22 o2·:·PythonObject·of·class·range_iterator22 o2·:·PythonObject·of·class·range_iterator
23 i3·:·next·i23 i3·:·next·i
  
24 o3·=·024 o3·=·0
  
25 o3·:·PythonObject·of·class·int25 o3·:·PythonObject·of·class·int
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1.07 KB
./usr/share/doc/Macaulay2/Python/html/_to__Python.html
    
Offset 156, 15 lines modifiedOffset 156, 15 lines modified
156 o12·=·m2sqrt156 o12·=·m2sqrt
  
157 o12·:·FunctionClosure</code></pre>157 o12·:·FunctionClosure</code></pre>
158 </td>··········</tr>158 </td>··········</tr>
159 ··········<tr>159 ··········<tr>
160 <td>··············<pre><code·class="language-macaulay2">i13·:·pysqrt·=·toPython·m2sqrt160 <td>··············<pre><code·class="language-macaulay2">i13·:·pysqrt·=·toPython·m2sqrt
  
161 o13·=·&lt;built-in·method·m2sqrt·of·PyCapsule·object·at·0x7f2e9d2aa8e0>161 o13·=·&lt;built-in·method·m2sqrt·of·PyCapsule·object·at·0x7f2826fce8e0>
  
162 o13·:·PythonObject·of·class·builtin_function_or_method</code></pre>162 o13·:·PythonObject·of·class·builtin_function_or_method</code></pre>
163 </td>··········</tr>163 </td>··········</tr>
164 ··········<tr>164 ··········<tr>
165 <td>··············<pre><code·class="language-macaulay2">i14·:·pysqrt·2165 <td>··············<pre><code·class="language-macaulay2">i14·:·pysqrt·2
166 calling·Macaulay2·code·from·Python!166 calling·Macaulay2·code·from·Python!
  
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html2text {}
    
Offset 73, 15 lines modifiedOffset 73, 15 lines modified
73 ··········sqrt·x)73 ··········sqrt·x)
  
74 o12·=·m2sqrt74 o12·=·m2sqrt
  
75 o12·:·FunctionClosure75 o12·:·FunctionClosure
76 i13·:·pysqrt·=·toPython·m2sqrt76 i13·:·pysqrt·=·toPython·m2sqrt
  
77 o13·=·<built-in·method·m2sqrt·of·PyCapsule·object·at·0x7f2e9d2aa8e0>77 o13·=·<built-in·method·m2sqrt·of·PyCapsule·object·at·0x7f2826fce8e0>
  
78 o13·:·PythonObject·of·class·builtin_function_or_method78 o13·:·PythonObject·of·class·builtin_function_or_method
79 i14·:·pysqrt·279 i14·:·pysqrt·2
80 calling·Macaulay2·code·from·Python!80 calling·Macaulay2·code·from·Python!
  
81 o14·=·1.414213562373095181 o14·=·1.4142135623730951
  
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718 B
./usr/share/doc/Macaulay2/QthPower/dump/rawdocumentation.dump
    
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765 B
./usr/share/doc/Macaulay2/QuadraticIdealExamplesByRoos/dump/rawdocumentation.dump
    
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11 LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl11 LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl
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746 B
./usr/share/doc/Macaulay2/Quasidegrees/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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8 cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d58 cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d5
9 #:len=38819 #:len=3881
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg
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728 B
./usr/share/doc/Macaulay2/QuaternaryQuartics/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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8 W1FRXQ==8 W1FRXQ==
9 #:len=5449 #:len=544
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z
11 IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl11 IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl
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26.2 KB
./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
    
Offset 179, 15 lines modifiedOffset 179, 15 lines modified
179 i21·:·L·=·trim·groebnerStratum·F;179 i21·:·L·=·trim·groebnerStratum·F;
  
180 o21·:·Ideal·of·T180 o21·:·Ideal·of·T
  
181 i22·:·assert(dim·L·==·18)181 i22·:·assert(dim·L·==·18)
  
182 i23·:·elapsedTime·isPrime·L182 i23·:·elapsedTime·isPrime·L
183 ·--·2.25923s·elapsed183 ·--·4.27494s·elapsed
  
184 o23·=·true184 o23·=·true
  
185 i24·:·I·=·pointsIdeal·randomPoints(S,·6)185 i24·:·I·=·pointsIdeal·randomPoints(S,·6)
  
186 ·····························2······························2···2··········186 ·····························2······························2···2··········
187 o24·=·ideal·(a*c·-·7b*c·-·49c··+·40a*d·-·42b*d·+·12c*d·+·28d·,·b··-·36b*c·-187 o24·=·ideal·(a*c·-·7b*c·-·49c··+·40a*d·-·42b*d·+·12c*d·+·28d·,·b··-·36b*c·-
Offset 301, 15 lines modifiedOffset 301, 15 lines modified
301 o38·=·true301 o38·=·true
  
302 i39·:·L441·=·trim(L·+·ideal·M1);302 i39·:·L441·=·trim(L·+·ideal·M1);
  
303 o39·:·Ideal·of·T303 o39·:·Ideal·of·T
  
304 i40·:·elapsedTime·compsL441·=·decompose·L441;304 i40·:·elapsedTime·compsL441·=·decompose·L441;
305 ·--·2.10589s·elapsed305 ·--·3.10075s·elapsed
  
306 i41·:·#compsL441306 i41·:·#compsL441
  
307 o41·=·2307 o41·=·2
  
308 i42·:·compsL441/dim·--·two·components,·of·dimensions·14·and·16.308 i42·:·compsL441/dim·--·two·components,·of·dimensions·14·and·16.
  
Offset 319, 37 lines modifiedOffset 319, 37 lines modified
  
319 i43·:·compsL441/dim·==·{16,·14}319 i43·:·compsL441/dim·==·{16,·14}
  
320 o43·=·true320 o43·=·true
  
321 i44·:·pta·=·randomPointOnRationalVariety·compsL441_0321 i44·:·pta·=·randomPointOnRationalVariety·compsL441_0
  
322 o44·=·|·-49·11·-25·17·44·-16·-27·40·-34·20·-19·29·41·19·-40·42·-13·5·17·39·31322 o44·=·|·-40·-4·-40·44·-22·-50·-30·13·-1·-16·45·-23·29·19·14·23·-21·19·14·29·6
323 ······-----------------------------------------------------------------------323 ······-----------------------------------------------------------------------
324 ······10·45·26·43·49·-32·-29·19·-50·15·18·37·-10·34·-28·|324 ······10·-32·18·-22·37·34·15·5·-10·-29·26·49·-50·45·-28·|
  
325 ···············1·······36325 ···············1·······36
326 o44·:·Matrix·kk··<--·kk326 o44·:·Matrix·kk··<--·kk
  
327 i45·:·Fa·=·sub(F,·(vars·S)·|·pta)327 i45·:·Fa·=·sub(F,·(vars·S)·|·pta)
  
328 ··············2··············2······························2···············328 ··············2··············2······························2···············
329 o45·=·ideal·(a··-·40b*c·+·20c··+·29a*d·-·16b*d·-·25c*d·-·49d·,·a*b·+·43b*c·+329 o45·=·ideal·(a··+·14b*c·-·16c··-·23a*d·-·50b*d·-·40c*d·-·40d·,·a*b·-·22b*c·+
330 ······-----------------------------------------------------------------------330 ······-----------------------------------------------------------------------
331 ·········2······························2···2··············2················331 ·········2··························2···2··············2··················
332 ······17c··+·31a*d·+·41b*d·-·34c*d·+·11d·,·b··+·34b*c·-·29c··+·37a*d·+·10b*d332 ······14c··+·6a*d·+·29b*d·-·c*d·-·4d·,·b··+·45b*c·+·15c··+·49a*d·+·10b*d·+
333 ······-----------------------------------------------------------------------333 ······-----------------------------------------------------------------------
334 ···················2···················2······························2·····2334 ·················2···················2·····························2·····2··
335 ······+·42c*d·-·27d·,·a*c·+·18b*c·+·49c··+·19a*d·+·39b*d·+·19c*d·+·44d·,·b*c·335 ······23c*d·-·30d·,·a*c·+·26b*c·+·37c··+·5a*d·+·29b*d·+·19c*d·-·22d·,·b*c··-
336 ······-----------------------------------------------------------------------336 ······-----------------------------------------------------------------------
337 ·····················2·········2········2········2······3···3············337 ···················2·········2········2········2······3···3················2·
338 ······-·50b*c*d·+·45c·d·-·32a*d··-·13b*d··-·19c*d··+·17d·,·c··-·28b*c*d·+338 ······10b*c*d·-·32c·d·+·34a*d··-·21b*d··+·45c*d··+·44d·,·c··-·28b*c*d·-·29c·d
339 ······-----------------------------------------------------------------------339 ······-----------------------------------------------------------------------
340 ·········2·········2········2·······2······3340 ·············2········2········2······3
341 ······15c·d·-·10a*d··+·26b*d··+·5c*d··+·40d·)341 ······-·50a*d··+·18b*d··+·19c*d··+·13d·)
  
342 o45·:·Ideal·of·S342 o45·:·Ideal·of·S
  
343 i46·:·betti·res·Fa343 i46·:·betti·res·Fa
  
344 ·············0·1·2·3344 ·············0·1·2·3
345 o46·=·total:·1·6·8·3345 o46·=·total:·1·6·8·3
Offset 357, 81 lines modifiedOffset 357, 84 lines modified
357 ··········1:·.·4·4·1357 ··········1:·.·4·4·1
358 ··········2:·.·2·4·2358 ··········2:·.·2·4·2
  
359 o46·:·BettiTally359 o46·:·BettiTally
  
360 i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another·point360 i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another·point
  
361 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+361 ······+------------------------------------------------------------------------------------------------------------+
362 o47·=·|ideal·(c·+·19d,·b·-·10d,·a·+·6d)···································································································································|362 o47·=·|ideal·(c·+·5d,·b·-·33d,·a·-·21d)············································································|
363 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+363 ······+------------------------------------------------------------------------------------------------------------+
364 ······|····························2··············2······················2···3················2········2········2······3·····2················2·········2········2······3·| 
365 ······|ideal·(a·+·18b·+·49c·-·3d,·b··+·34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c··-·28b*c*d·+·15c·d·+·4b*d··-·10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+·34c*d··+·22d·)|364 ······|······································2··············2······················································|
 365 ······|ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,·c··+·49c*d·+·42d·)·····················································|
366 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+366 ······+------------------------------------------------------------------------------------------------------------+
 367 ······|·····························2······················2···························2···2·····················2·|
 368 ······|ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+·27d·,·b*c·-·30b*d·+·16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)|
 369 ······+------------------------------------------------------------------------------------------------------------+
  
367 i48·:·CFa·=·minimalPrimes·Fa370 i48·:·CFa·=·minimalPrimes·Fa
  
368 ·····································································2··371 ·············································································
369 o48·=·{ideal·(c·+·19d,·b·-·10d,·a·+·6d),·ideal·(a·+·18b·+·49c·-·3d,·b··+372 o48·=·{ideal·(c·+·5d,·b·-·33d,·a·-·21d),·ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,
370 ······-----------------------------------------------------------------------373 ······-----------------------------------------------------------------------
371 ·················2······················2···3················2········2··374 ·······2··············2································2··················
372 ······34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c··-·28b*c*d·+·15c·d·+·4b*d··-375 ······c··+·49c*d·+·42d·),·ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+
373 ······-----------------------------------------------------------------------376 ······-----------------------------------------------------------------------
374 ···········2······3·····2················2·········2········2······3 
375 ······10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+·34c*d··+·22d·)}377 ·········2···························2···2·····················2
 378 ······27d·,·b*c·-·30b*d·+·16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)}
  
376 o48·:·List379 o48·:·List
  
377 i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.380 i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.
  
378 o49·=·a·+·18b·+·49c·-·3d381 o49·=·b·+·45c·+·49d
  
379 o49·:·S382 o49·:·S
  
380 i50·:·CFa/degree383 i50·:·CFa/degree
  
381 o50·=·{1,·5}384 o50·=·{1,·2,·3}
  
382 o50·:·List385 o50·:·List
  
383 i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.386 i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.
  
384 o51·=·{false,·true}387 o51·=·{false,·true,·false}
  
385 o51·:·List388 o51·:·List
  
386 i52·:·degree(Fa·:·(Fa·:·lin))··--·somewhat·simpler(?)·way·to·see·the·ideal·of·the·5·points389 i52·:·degree(Fa·:·(Fa·:·lin))··--·somewhat·simpler(?)·way·to·see·the·ideal·of·the·5·points
  
387 o52·=·5390 o52·=·2
  
388 i53·:·ptb·=·randomPointOnRationalVariety·compsL441_1391 i53·:·ptb·=·randomPointOnRationalVariety·compsL441_1
  
389 o53·=·|·27·12·-34·9·-19·-43·-32·27·40·45·-13·29·-41·-13·22·-49·-4·-4·9·-23·43392 o53·=·|·31·42·28·25·19·3·43·-7·-3·-42·-29·-29·14·2·50·5·36·-13·-42·47·13·31
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Offset 291, 15 lines modifiedOffset 291, 15 lines modified
291 o21·:·Ideal·of·T</code></pre>291 o21·:·Ideal·of·T</code></pre>
292 </td>··········</tr>292 </td>··········</tr>
293 ··········<tr>293 ··········<tr>
294 <td>··············<pre><code·class="language-macaulay2">i22·:·assert(dim·L·==·18)</code></pre>294 <td>··············<pre><code·class="language-macaulay2">i22·:·assert(dim·L·==·18)</code></pre>
295 </td>··········</tr>295 </td>··········</tr>
296 ··········<tr>296 ··········<tr>
297 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·isPrime·L297 <td>··············<pre><code·class="language-macaulay2">i23·:·elapsedTime·isPrime·L
298 ·--·2.25923s·elapsed298 ·--·4.27494s·elapsed
  
299 o23·=·true</code></pre>299 o23·=·true</code></pre>
300 </td>··········</tr>300 </td>··········</tr>
301 ········</table>301 ········</table>
302 ········<div>302 ········<div>
303 ··········<h2>The·Schreyer·resolution·and·minimal·Betti·numbers</h2>303 ··········<h2>The·Schreyer·resolution·and·minimal·Betti·numbers</h2>
304 ········</div>304 ········</div>
Offset 469, 15 lines modifiedOffset 469, 15 lines modified
469 ··········<tr>469 ··········<tr>
470 <td>··············<pre><code·class="language-macaulay2">i39·:·L441·=·trim(L·+·ideal·M1);470 <td>··············<pre><code·class="language-macaulay2">i39·:·L441·=·trim(L·+·ideal·M1);
  
471 o39·:·Ideal·of·T</code></pre>471 o39·:·Ideal·of·T</code></pre>
472 </td>··········</tr>472 </td>··········</tr>
473 ··········<tr>473 ··········<tr>
474 <td>··············<pre><code·class="language-macaulay2">i40·:·elapsedTime·compsL441·=·decompose·L441;474 <td>··············<pre><code·class="language-macaulay2">i40·:·elapsedTime·compsL441·=·decompose·L441;
475 ·--·2.10589s·elapsed</code></pre>475 ·--·3.10075s·elapsed</code></pre>
476 </td>··········</tr>476 </td>··········</tr>
477 ··········<tr>477 ··········<tr>
478 <td>··············<pre><code·class="language-macaulay2">i41·:·#compsL441478 <td>··············<pre><code·class="language-macaulay2">i41·:·#compsL441
  
479 o41·=·2</code></pre>479 o41·=·2</code></pre>
480 </td>··········</tr>480 </td>··········</tr>
481 ··········<tr>481 ··········<tr>
Offset 496, 38 lines modifiedOffset 496, 38 lines modified
496 ········<div>496 ········<div>
497 ··········<p>Both·components·are·rational,·and·here·are·random·points,·one·on·each·component:</p>497 ··········<p>Both·components·are·rational,·and·here·are·random·points,·one·on·each·component:</p>
498 ········</div>498 ········</div>
499 ········<table·class="examples">499 ········<table·class="examples">
500 ··········<tr>500 ··········<tr>
501 <td>··············<pre><code·class="language-macaulay2">i44·:·pta·=·randomPointOnRationalVariety·compsL441_0501 <td>··············<pre><code·class="language-macaulay2">i44·:·pta·=·randomPointOnRationalVariety·compsL441_0
  
502 o44·=·|·-49·11·-25·17·44·-16·-27·40·-34·20·-19·29·41·19·-40·42·-13·5·17·39·31502 o44·=·|·-40·-4·-40·44·-22·-50·-30·13·-1·-16·45·-23·29·19·14·23·-21·19·14·29·6
503 ······-----------------------------------------------------------------------503 ······-----------------------------------------------------------------------
504 ······10·45·26·43·49·-32·-29·19·-50·15·18·37·-10·34·-28·|504 ······10·-32·18·-22·37·34·15·5·-10·-29·26·49·-50·45·-28·|
  
505 ···············1·······36505 ···············1·······36
506 o44·:·Matrix·kk··&lt;--·kk</code></pre>506 o44·:·Matrix·kk··&lt;--·kk</code></pre>
507 </td>··········</tr>507 </td>··········</tr>
508 ··········<tr>508 ··········<tr>
509 <td>··············<pre><code·class="language-macaulay2">i45·:·Fa·=·sub(F,·(vars·S)·|·pta)509 <td>··············<pre><code·class="language-macaulay2">i45·:·Fa·=·sub(F,·(vars·S)·|·pta)
  
510 ··············2··············2······························2···············510 ··············2··············2······························2···············
511 o45·=·ideal·(a··-·40b*c·+·20c··+·29a*d·-·16b*d·-·25c*d·-·49d·,·a*b·+·43b*c·+511 o45·=·ideal·(a··+·14b*c·-·16c··-·23a*d·-·50b*d·-·40c*d·-·40d·,·a*b·-·22b*c·+
512 ······-----------------------------------------------------------------------512 ······-----------------------------------------------------------------------
513 ·········2······························2···2··············2················513 ·········2··························2···2··············2··················
514 ······17c··+·31a*d·+·41b*d·-·34c*d·+·11d·,·b··+·34b*c·-·29c··+·37a*d·+·10b*d514 ······14c··+·6a*d·+·29b*d·-·c*d·-·4d·,·b··+·45b*c·+·15c··+·49a*d·+·10b*d·+
515 ······-----------------------------------------------------------------------515 ······-----------------------------------------------------------------------
516 ···················2···················2······························2·····2516 ·················2···················2·····························2·····2··
517 ······+·42c*d·-·27d·,·a*c·+·18b*c·+·49c··+·19a*d·+·39b*d·+·19c*d·+·44d·,·b*c·517 ······23c*d·-·30d·,·a*c·+·26b*c·+·37c··+·5a*d·+·29b*d·+·19c*d·-·22d·,·b*c··-
518 ······-----------------------------------------------------------------------518 ······-----------------------------------------------------------------------
519 ·····················2·········2········2········2······3···3············519 ···················2·········2········2········2······3···3················2·
520 ······-·50b*c*d·+·45c·d·-·32a*d··-·13b*d··-·19c*d··+·17d·,·c··-·28b*c*d·+520 ······10b*c*d·-·32c·d·+·34a*d··-·21b*d··+·45c*d··+·44d·,·c··-·28b*c*d·-·29c·d
521 ······-----------------------------------------------------------------------521 ······-----------------------------------------------------------------------
522 ·········2·········2········2·······2······3522 ·············2········2········2······3
523 ······15c·d·-·10a*d··+·26b*d··+·5c*d··+·40d·)523 ······-·50a*d··+·18b*d··+·19c*d··+·13d·)
  
524 o45·:·Ideal·of·S</code></pre>524 o45·:·Ideal·of·S</code></pre>
525 </td>··········</tr>525 </td>··········</tr>
526 ··········<tr>526 ··········<tr>
527 <td>··············<pre><code·class="language-macaulay2">i46·:·betti·res·Fa527 <td>··············<pre><code·class="language-macaulay2">i46·:·betti·res·Fa
  
528 ·············0·1·2·3528 ·············0·1·2·3
Offset 537, 90 lines modifiedOffset 537, 93 lines modified
537 ··········2:·.·2·4·2537 ··········2:·.·2·4·2
  
538 o46·:·BettiTally</code></pre>538 o46·:·BettiTally</code></pre>
539 </td>··········</tr>539 </td>··········</tr>
540 ··········<tr>540 ··········<tr>
541 <td>··············<pre><code·class="language-macaulay2">i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another·point541 <td>··············<pre><code·class="language-macaulay2">i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another·point
  
542 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+542 ······+------------------------------------------------------------------------------------------------------------+
543 o47·=·|ideal·(c·+·19d,·b·-·10d,·a·+·6d)···································································································································|543 o47·=·|ideal·(c·+·5d,·b·-·33d,·a·-·21d)············································································|
544 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+544 ······+------------------------------------------------------------------------------------------------------------+
545 ······|····························2··············2······················2···3················2········2········2······3·····2················2·········2········2······3·| 
546 ······|ideal·(a·+·18b·+·49c·-·3d,·b··+·34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c··-·28b*c*d·+·15c·d·+·4b*d··-·10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+·34c*d··+·22d·)|545 ······|······································2··············2······················································|
 546 ······|ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,·c··+·49c*d·+·42d·)·····················································|
 547 ······+------------------------------------------------------------------------------------------------------------+
 548 ······|·····························2······················2···························2···2·····················2·|
 549 ······|ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+·27d·,·b*c·-·30b*d·+·16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)|
547 ······+-------------------------------------------------------------------------------------------------------------------------------------------------------------------+</code></pre>550 ······+------------------------------------------------------------------------------------------------------------+</code></pre>
548 </td>··········</tr>551 </td>··········</tr>
549 ··········<tr>552 ··········<tr>
550 <td>··············<pre><code·class="language-macaulay2">i48·:·CFa·=·minimalPrimes·Fa553 <td>··············<pre><code·class="language-macaulay2">i48·:·CFa·=·minimalPrimes·Fa
  
551 ·····································································2··554 ·············································································
552 o48·=·{ideal·(c·+·19d,·b·-·10d,·a·+·6d),·ideal·(a·+·18b·+·49c·-·3d,·b··+555 o48·=·{ideal·(c·+·5d,·b·-·33d,·a·-·21d),·ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,
553 ······-----------------------------------------------------------------------556 ······-----------------------------------------------------------------------
554 ·················2······················2···3················2········2··557 ·······2··············2································2··················
555 ······34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c··-·28b*c*d·+·15c·d·+·4b*d··-558 ······c··+·49c*d·+·42d·),·ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+
556 ······-----------------------------------------------------------------------559 ······-----------------------------------------------------------------------
557 ···········2······3·····2················2·········2········2······3 
558 ······10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+·34c*d··+·22d·)}560 ·········2···························2···2·····················2
 561 ······27d·,·b*c·-·30b*d·+·16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)}
  
559 o48·:·List</code></pre>562 o48·:·List</code></pre>
560 </td>··········</tr>563 </td>··········</tr>
561 ··········<tr>564 ··········<tr>
562 <td>··············<pre><code·class="language-macaulay2">i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.565 <td>··············<pre><code·class="language-macaulay2">i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.
  
563 o49·=·a·+·18b·+·49c·-·3d566 o49·=·b·+·45c·+·49d
  
564 o49·:·S</code></pre>567 o49·:·S</code></pre>
565 </td>··········</tr>568 </td>··········</tr>
566 ··········<tr>569 ··········<tr>
567 <td>··············<pre><code·class="language-macaulay2">i50·:·CFa/degree570 <td>··············<pre><code·class="language-macaulay2">i50·:·CFa/degree
  
568 o50·=·{1,·5}571 o50·=·{1,·2,·3}
  
569 o50·:·List</code></pre>572 o50·:·List</code></pre>
570 </td>··········</tr>573 </td>··········</tr>
571 ··········<tr>574 ··········<tr>
572 <td>··············<pre><code·class="language-macaulay2">i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.575 <td>··············<pre><code·class="language-macaulay2">i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.
  
573 o51·=·{false,·true}576 o51·=·{false,·true,·false}
  
574 o51·:·List</code></pre>577 o51·:·List</code></pre>
575 </td>··········</tr>578 </td>··········</tr>
576 ··········<tr>579 ··········<tr>
577 <td>··············<pre><code·class="language-macaulay2">i52·:·degree(Fa·:·(Fa·:·lin))··--·somewhat·simpler(?)·way·to·see·the·ideal·of·the·5·points580 <td>··············<pre><code·class="language-macaulay2">i52·:·degree(Fa·:·(Fa·:·lin))··--·somewhat·simpler(?)·way·to·see·the·ideal·of·the·5·points
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250 ······|······31·········33········32·······34········35········36····|250 ······|······31·········33········32·······34········35········36····|
251 ······+--------------------------------------------------------------+251 ······+--------------------------------------------------------------+
252 i21·:·L·=·trim·groebnerStratum·F;252 i21·:·L·=·trim·groebnerStratum·F;
  
253 o21·:·Ideal·of·T253 o21·:·Ideal·of·T
254 i22·:·assert(dim·L·==·18)254 i22·:·assert(dim·L·==·18)
255 i23·:·elapsedTime·isPrime·L255 i23·:·elapsedTime·isPrime·L
256 ·--·2.25923s·elapsed256 ·--·4.27494s·elapsed
  
257 o23·=·true257 o23·=·true
258 *\x8**\x8**\x8**\x8**\x8*·T\x8Th\x8he\x8e·S\x8Sc\x8ch\x8hr\x8re\x8ey\x8ye\x8er\x8r·r\x8re\x8es\x8so\x8ol\x8lu\x8ut\x8ti\x8io\x8on\x8n·a\x8an\x8nd\x8d·m\x8mi\x8in\x8ni\x8im\x8ma\x8al\x8l·B\x8Be\x8et\x8tt\x8ti\x8i·n\x8nu\x8um\x8mb\x8be\x8er\x8rs\x8s·*\x8**\x8**\x8**\x8**\x8*258 *\x8**\x8**\x8**\x8**\x8*·T\x8Th\x8he\x8e·S\x8Sc\x8ch\x8hr\x8re\x8ey\x8ye\x8er\x8r·r\x8re\x8es\x8so\x8ol\x8lu\x8ut\x8ti\x8io\x8on\x8n·a\x8an\x8nd\x8d·m\x8mi\x8in\x8ni\x8im\x8ma\x8al\x8l·B\x8Be\x8et\x8tt\x8ti\x8i·n\x8nu\x8um\x8mb\x8be\x8er\x8rs\x8s·*\x8**\x8**\x8**\x8**\x8*
259 Schreyer's·construction·of·a·nonminimal·free·resolution·starts·with·a·Groebner259 Schreyer's·construction·of·a·nonminimal·free·resolution·starts·with·a·Groebner
260 basis.·First,·one·constructs·the·S\x8Sc\x8ch\x8hr\x8re\x8ey\x8ye\x8er\x8r·f\x8fr\x8ra\x8am\x8me\x8e·(see·La·Scala,·Stillman).·This260 basis.·First,·one·constructs·the·S\x8Sc\x8ch\x8hr\x8re\x8ey\x8ye\x8er\x8r·f\x8fr\x8ra\x8am\x8me\x8e·(see·La·Scala,·Stillman).·This
261 is·determined·solely·from·the·initial·ideal·$J$·and·its·minimal·generators·(but261 is·determined·solely·from·the·initial·ideal·$J$·and·its·minimal·generators·(but
262 depends·on·some·choices·of·ordering,·but·otherwise·is·combinatorial).·This262 depends·on·some·choices·of·ordering,·but·otherwise·is·combinatorial).·This
Offset 414, 15 lines modifiedOffset 414, 15 lines modified
414 We·now·compute·the·locus·in·$V(L)$·where·the·Betti·diagram·has·no·cancellation.414 We·now·compute·the·locus·in·$V(L)$·where·the·Betti·diagram·has·no·cancellation.
415 This·is·a·closed·subscheme·of·$V(L)$,·which·is·a·closed·subscheme·of·the415 This·is·a·closed·subscheme·of·$V(L)$,·which·is·a·closed·subscheme·of·the
416 Hilbert·scheme.·Notice·that·there·are·two·components.416 Hilbert·scheme.·Notice·that·there·are·two·components.
417 i39·:·L441·=·trim(L·+·ideal·M1);417 i39·:·L441·=·trim(L·+·ideal·M1);
  
418 o39·:·Ideal·of·T418 o39·:·Ideal·of·T
419 i40·:·elapsedTime·compsL441·=·decompose·L441;419 i40·:·elapsedTime·compsL441·=·decompose·L441;
420 ·--·2.10589s·elapsed420 ·--·3.10075s·elapsed
421 i41·:·#compsL441421 i41·:·#compsL441
  
422 o41·=·2422 o41·=·2
423 i42·:·compsL441/dim·--·two·components,·of·dimensions·14·and·16.423 i42·:·compsL441/dim·--·two·components,·of·dimensions·14·and·16.
  
424 o42·=·{16,·14}424 o42·=·{16,·14}
  
Offset 430, 36 lines modifiedOffset 430, 36 lines modified
430 i43·:·compsL441/dim·==·{16,·14}430 i43·:·compsL441/dim·==·{16,·14}
  
431 o43·=·true431 o43·=·true
432 Both·components·are·rational,·and·here·are·random·points,·one·on·each432 Both·components·are·rational,·and·here·are·random·points,·one·on·each
433 component:433 component:
434 i44·:·pta·=·randomPointOnRationalVariety·compsL441_0434 i44·:·pta·=·randomPointOnRationalVariety·compsL441_0
  
435 o44·=·|·-49·11·-25·17·44·-16·-27·40·-34·20·-19·29·41·19·-40·42·-13·5·17·39·31435 o44·=·|·-40·-4·-40·44·-22·-50·-30·13·-1·-16·45·-23·29·19·14·23·-21·19·14·29·6
436 ······-----------------------------------------------------------------------436 ······-----------------------------------------------------------------------
437 ······10·45·26·43·49·-32·-29·19·-50·15·18·37·-10·34·-28·|437 ······10·-32·18·-22·37·34·15·5·-10·-29·26·49·-50·45·-28·|
  
438 ···············1·······36438 ···············1·······36
439 o44·:·Matrix·kk··<--·kk439 o44·:·Matrix·kk··<--·kk
440 i45·:·Fa·=·sub(F,·(vars·S)·|·pta)440 i45·:·Fa·=·sub(F,·(vars·S)·|·pta)
  
441 ··············2··············2······························2441 ··············2··············2······························2
442 o45·=·ideal·(a··-·40b*c·+·20c··+·29a*d·-·16b*d·-·25c*d·-·49d·,·a*b·+·43b*c·+442 o45·=·ideal·(a··+·14b*c·-·16c··-·23a*d·-·50b*d·-·40c*d·-·40d·,·a*b·-·22b*c·+
443 ······-----------------------------------------------------------------------443 ······-----------------------------------------------------------------------
444 ·········2······························2···2··············2444 ·········2··························2···2··············2
445 ······17c··+·31a*d·+·41b*d·-·34c*d·+·11d·,·b··+·34b*c·-·29c··+·37a*d·+·10b*d445 ······14c··+·6a*d·+·29b*d·-·c*d·-·4d·,·b··+·45b*c·+·15c··+·49a*d·+·10b*d·+
446 ······-----------------------------------------------------------------------446 ······-----------------------------------------------------------------------
447 ···················2···················2······························2·····2447 ·················2···················2·····························2·····2
448 ······+·42c*d·-·27d·,·a*c·+·18b*c·+·49c··+·19a*d·+·39b*d·+·19c*d·+·44d·,·b*c448 ······23c*d·-·30d·,·a*c·+·26b*c·+·37c··+·5a*d·+·29b*d·+·19c*d·-·22d·,·b*c··-
449 ······-----------------------------------------------------------------------449 ······-----------------------------------------------------------------------
450 ·····················2·········2········2········2······3···3450 ···················2·········2········2········2······3···3················2
451 ······-·50b*c*d·+·45c·d·-·32a*d··-·13b*d··-·19c*d··+·17d·,·c··-·28b*c*d·+451 ······10b*c*d·-·32c·d·+·34a*d··-·21b*d··+·45c*d··+·44d·,·c··-·28b*c*d·-·29c·d
452 ······-----------------------------------------------------------------------452 ······-----------------------------------------------------------------------
453 ·········2·········2········2·······2······3453 ·············2········2········2······3
454 ······15c·d·-·10a*d··+·26b*d··+·5c*d··+·40d·)454 ······-·50a*d··+·18b*d··+·19c*d··+·13d·)
  
455 o45·:·Ideal·of·S455 o45·:·Ideal·of·S
456 i46·:·betti·res·Fa456 i46·:·betti·res·Fa
  
457 ·············0·1·2·3457 ·············0·1·2·3
458 o46·=·total:·1·6·8·3458 o46·=·total:·1·6·8·3
459 ··········0:·1·.·.·.459 ··········0:·1·.·.·.
Offset 467, 172 lines modifiedOffset 467, 256 lines modified
467 ··········2:·.·2·4·2467 ··········2:·.·2·4·2
  
468 o46·:·BettiTally468 o46·:·BettiTally
469 i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another469 i47·:·netList·decompose·Fa·--·this·one·is·5·points·on·a·plane,·and·another
470 point470 point
  
471 ······+------------------------------------------------------------------------471 ······+------------------------------------------------------------------------
 472 ------------------------------------+
472 ------------------------------------------------------------------------------- 
473 ------------+ 
474 o47·=·|ideal·(c·+·19d,·b·-·10d,·a·+·6d)473 o47·=·|ideal·(c·+·5d,·b·-·33d,·a·-·21d)
475 |474 |
476 ······+------------------------------------------------------------------------475 ······+------------------------------------------------------------------------
 476 ------------------------------------+
477 ------------------------------------------------------------------------------- 
478 ------------+ 
479 ······|····························2··············2······················2···3 
480 2········2········2······3·····2················2·········2········2······3·|477 ······|······································2··············2
481 ······|ideal·(a·+·18b·+·49c·-·3d,·b··+·34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c 
482 -·28b*c*d·+·15c·d·+·4b*d··-·10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+ 
483 34c*d··+·22d·)|478 |
 479 ······|ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,·c··+·49c*d·+·42d·)
 480 |
484 ······+------------------------------------------------------------------------481 ······+------------------------------------------------------------------------
 482 ------------------------------------+
 483 ······|·····························2······················2
 484 2···2·····················2·|
 485 ······|ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+·27d·,·b*c·-·30b*d·+
 486 16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)|
485 -------------------------------------------------------------------------------487 ······+------------------------------------------------------------------------
486 ------------+488 ------------------------------------+
487 i48·:·CFa·=·minimalPrimes·Fa489 i48·:·CFa·=·minimalPrimes·Fa
  
488 ·····································································2 
489 o48·=·{ideal·(c·+·19d,·b·-·10d,·a·+·6d),·ideal·(a·+·18b·+·49c·-·3d,·b··+ 
 490 o48·=·{ideal·(c·+·5d,·b·-·33d,·a·-·21d),·ideal·(b·+·45c·+·49d,·a·-·22c·-·26d,
490 ······-----------------------------------------------------------------------491 ······-----------------------------------------------------------------------
491 ·················2······················2···3················2········2492 ·······2··············2································2
492 ······34b*c·-·29c··-·50b*d·+·47c*d·-·17d·,·c··-·28b*c*d·+·15c·d·+·4b*d··-493 ······c··+·49c*d·+·42d·),·ideal·(a·+·26b·+·37c·+·36d,·c··-·21b*d·+·43c*d·+
493 ······-----------------------------------------------------------------------494 ······-----------------------------------------------------------------------
494 ···········2······3·····2················2·········2········2······3 
495 ······10c*d··+·10d·,·b*c··-·50b*c*d·+·45c·d·-·43b*d··+·34c*d··+·22d·)}495 ·········2···························2···2·····················2
 496 ······27d·,·b*c·-·30b*d·+·16c*d·+·26d·,·b··-·3b*d·-·24c*d·-·36d·)}
  
496 o48·:·List497 o48·:·List
497 i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.498 i49·:·lin·=·CFa_1_0·--·a·linear·form,·defining·a·plane.
  
498 o49·=·a·+·18b·+·49c·-·3d499 o49·=·b·+·45c·+·49d
  
499 o49·:·S500 o49·:·S
500 i50·:·CFa/degree501 i50·:·CFa/degree
  
501 o50·=·{1,·5}502 o50·=·{1,·2,·3}
  
502 o50·:·List503 o50·:·List
503 i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.504 i51·:·CFa/(I·->·lin·%·I·==·0)·--·so·5·points·on·the·plane.
  
504 o51·=·{false,·true}505 o51·=·{false,·true,·false}
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./usr/share/doc/Macaulay2/QuillenSuslin/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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./usr/share/doc/Macaulay2/RInterface/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFJPYmplY3QsQ0MpLCJuZXcg11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFJPYmplY3QsQ0MpLCJuZXcg
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./usr/share/doc/Macaulay2/RandomCanonicalCurves/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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9 #:len=2189 #:len=218
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11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InJhbmRvbUNh11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InJhbmRvbUNh
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495 B
./usr/share/doc/Macaulay2/RandomCanonicalCurves/example-output/_canonical__Curve.out
    
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5 i2·:·g=14;5 i2·:·g=14;
  
6 i3·:·FF=ZZ/10007;6 i3·:·FF=ZZ/10007;
  
7 i4·:·R=FF[x_0..x_(g-1)];7 i4·:·R=FF[x_0..x_(g-1)];
  
8 i5·:·time·betti(I=(random·canonicalCurve)(g,R))8 i5·:·time·betti(I=(random·canonicalCurve)(g,R))
9 ·--·used·5.36236s·(cpu);·4.354s·(thread);·0s·(gc)9 ·--·used·5.75307s·(cpu);·4.43895s·(thread);·0s·(gc)
  
10 ············0··110 ············0··1
11 o5·=·total:·1·6611 o5·=·total:·1·66
12 ·········0:·1··.12 ·········0:·1··.
13 ·········1:·.·6613 ·········1:·.·66
  
14 o5·:·BettiTally14 o5·:·BettiTally
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./usr/share/doc/Macaulay2/RandomCanonicalCurves/html/_canonical__Curve.html
    
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83 <td>··············<pre><code·class="language-macaulay2">i3·:·FF=ZZ/10007;</code></pre>83 <td>··············<pre><code·class="language-macaulay2">i3·:·FF=ZZ/10007;</code></pre>
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i4·:·R=FF[x_0..x_(g-1)];</code></pre>86 <td>··············<pre><code·class="language-macaulay2">i4·:·R=FF[x_0..x_(g-1)];</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·betti(I=(random·canonicalCurve)(g,R))89 <td>··············<pre><code·class="language-macaulay2">i5·:·time·betti(I=(random·canonicalCurve)(g,R))
90 ·--·used·5.36236s·(cpu);·4.354s·(thread);·0s·(gc)90 ·--·used·5.75307s·(cpu);·4.43895s·(thread);·0s·(gc)
  
91 ············0··191 ············0··1
92 o5·=·total:·1·6692 o5·=·total:·1·66
93 ·········0:·1··.93 ·········0:·1··.
94 ·········1:·.·6694 ·········1:·.·66
  
95 o5·:·BettiTally</code></pre>95 o5·:·BettiTally</code></pre>
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html2text {}
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 Compute·a·random·canonical·curve·of·genus·$g·\le{}·14$,·based·on·the·proofs·of17 Compute·a·random·canonical·curve·of·genus·$g·\le{}·14$,·based·on·the·proofs·of
18 unirationality·of·$M_g$·by·Severi,·Sernesi,·Chang-Ran·and·Verra.18 unirationality·of·$M_g$·by·Severi,·Sernesi,·Chang-Ran·and·Verra.
19 i1·:·setRandomSeed·"alpha";19 i1·:·setRandomSeed·"alpha";
20 i2·:·g=14;20 i2·:·g=14;
21 i3·:·FF=ZZ/10007;21 i3·:·FF=ZZ/10007;
22 i4·:·R=FF[x_0..x_(g-1)];22 i4·:·R=FF[x_0..x_(g-1)];
23 i5·:·time·betti(I=(random·canonicalCurve)(g,R))23 i5·:·time·betti(I=(random·canonicalCurve)(g,R))
24 ·--·used·5.36236s·(cpu);·4.354s·(thread);·0s·(gc)24 ·--·used·5.75307s·(cpu);·4.43895s·(thread);·0s·(gc)
  
25 ············0··125 ············0··1
26 o5·=·total:·1·6626 o5·=·total:·1·66
27 ·········0:·1··.27 ·········0:·1··.
28 ·········1:·.·6628 ·········1:·.·66
  
29 o5·:·BettiTally29 o5·:·BettiTally
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./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump
    
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1.6 KB
./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out
    
Offset 6, 42 lines modifiedOffset 6, 42 lines modified
  
6 o2·=·(10,·20)6 o2·=·(10,·20)
  
7 o2·:·Sequence7 o2·:·Sequence
  
8 i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)8 i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
  
9 o3·=·({5,·2.91596e52,·9},·0,·.00401094)9 o3·=·({5,·2.91596e52,·9},·0,·0)
  
10 o3·:·Sequence10 o3·:·Sequence
  
11 i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)11 i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
  
12 o4·=·({50,·2.30853e454,·98},·.00395691,·.0342736)12 o4·=·({50,·2.30853e454,·98},·.00838679,·.0327403)
  
13 o4·:·Sequence13 o4·:·Sequence
  
14 i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})14 i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
  
15 o5·=·{{.00399997,·.0119955},·{.00400264,·0},·{.00360428,·.0080176},15 o5·=·{{.00685534,·.00799669},·{.00425286,·.0036916},·{.00572619,·.00400115},
16 ·····------------------------------------------------------------------------16 ·····------------------------------------------------------------------------
17 ·····{.00385678,·.00799236},·{.0568536,·.0130524},·{.00777651,·.0119991},17 ·····{.00790714,·.00797849},·{.0579225,·.011997},·{.00400143,·.0119194},
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ·····{.00802694,·.00790754},·{.00706792,·.00667325},·{.00395036,·.00400722},19 ·····{.00400046,·.00799564},·{.00400709,·.00401843},·{.00400051,·.00398355},
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ·····{.00403522,·.0072114}}21 ·····{.00590819,·.00806039}}
  
22 o5·:·List22 o5·:·List
  
23 i6·:·1/10*sum(L,t->t_0)23 i6·:·1/10*sum(L,t->t_0)
  
24 o6·=·.010317421500000124 o6·=·.01045816749999999
  
25 o6·:·RR·(of·precision·53)25 o6·:·RR·(of·precision·53)
  
26 i7·:·1/10*sum(L,t->t_1)26 i7·:·1/10*sum(L,t->t_1)
  
27 o7·=·.00788563099999999427 o7·=·.007164236799999957
  
28 o7·:·RR·(of·precision·53)28 o7·:·RR·(of·precision·53)
  
29 i8·:·29 i8·:·
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./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
    
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92 o2·=·(10,·20)92 o2·=·(10,·20)
  
93 o2·:·Sequence</code></pre>93 o2·:·Sequence</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)96 <td>··············<pre><code·class="language-macaulay2">i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
  
97 o3·=·({5,·2.91596e52,·9},·0,·.00401094)97 o3·=·({5,·2.91596e52,·9},·0,·0)
  
98 o3·:·Sequence</code></pre>98 o3·:·Sequence</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)101 <td>··············<pre><code·class="language-macaulay2">i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
  
102 o4·=·({50,·2.30853e454,·98},·.00395691,·.0342736)102 o4·=·({50,·2.30853e454,·98},·.00838679,·.0327403)
  
103 o4·:·Sequence</code></pre>103 o4·:·Sequence</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})106 <td>··············<pre><code·class="language-macaulay2">i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
  
107 o5·=·{{.00399997,·.0119955},·{.00400264,·0},·{.00360428,·.0080176},107 o5·=·{{.00685534,·.00799669},·{.00425286,·.0036916},·{.00572619,·.00400115},
108 ·····------------------------------------------------------------------------108 ·····------------------------------------------------------------------------
109 ·····{.00385678,·.00799236},·{.0568536,·.0130524},·{.00777651,·.0119991},109 ·····{.00790714,·.00797849},·{.0579225,·.011997},·{.00400143,·.0119194},
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
111 ·····{.00802694,·.00790754},·{.00706792,·.00667325},·{.00395036,·.00400722},111 ·····{.00400046,·.00799564},·{.00400709,·.00401843},·{.00400051,·.00398355},
112 ·····------------------------------------------------------------------------112 ·····------------------------------------------------------------------------
113 ·····{.00403522,·.0072114}}113 ·····{.00590819,·.00806039}}
  
114 o5·:·List</code></pre>114 o5·:·List</code></pre>
115 </td>··········</tr>115 </td>··········</tr>
116 ··········<tr>116 ··········<tr>
117 <td>··············<pre><code·class="language-macaulay2">i6·:·1/10*sum(L,t->t_0)117 <td>··············<pre><code·class="language-macaulay2">i6·:·1/10*sum(L,t->t_0)
  
118 o6·=·.0103174215000001118 o6·=·.01045816749999999
  
119 o6·:·RR·(of·precision·53)</code></pre>119 o6·:·RR·(of·precision·53)</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i7·:·1/10*sum(L,t->t_1)122 <td>··············<pre><code·class="language-macaulay2">i7·:·1/10*sum(L,t->t_1)
  
123 o7·=·.007885630999999994123 o7·=·.007164236799999957
  
124 o7·:·RR·(of·precision·53)</code></pre>124 o7·:·RR·(of·precision·53)</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ········</table>126 ········</table>
127 ······</div>127 ······</div>
128 ······<div·class="waystouse">128 ······<div·class="waystouse">
129 ········<h2>Ways·to·use·<span·class="tt">testTimeForLLLonSyzygies</span>:</h2>129 ········<h2>Ways·to·use·<span·class="tt">testTimeForLLLonSyzygies</span>:</h2>
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25 i2·:·r=10,n=2025 i2·:·r=10,n=20
  
26 o2·=·(10,·20)26 o2·=·(10,·20)
  
27 o2·:·Sequence27 o2·:·Sequence
28 i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)28 i3·:·(m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
  
29 o3·=·({5,·2.91596e52,·9},·0,·.00401094)29 o3·=·({5,·2.91596e52,·9},·0,·0)
  
30 o3·:·Sequence30 o3·:·Sequence
31 i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)31 i4·:·(m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
  
32 o4·=·({50,·2.30853e454,·98},·.00395691,·.0342736)32 o4·=·({50,·2.30853e454,·98},·.00838679,·.0327403)
  
33 o4·:·Sequence33 o4·:·Sequence
34 i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})34 i5·:·L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
  
35 o5·=·{{.00399997,·.0119955},·{.00400264,·0},·{.00360428,·.0080176},35 o5·=·{{.00685534,·.00799669},·{.00425286,·.0036916},·{.00572619,·.00400115},
36 ·····------------------------------------------------------------------------36 ·····------------------------------------------------------------------------
37 ·····{.00385678,·.00799236},·{.0568536,·.0130524},·{.00777651,·.0119991},37 ·····{.00790714,·.00797849},·{.0579225,·.011997},·{.00400143,·.0119194},
38 ·····------------------------------------------------------------------------38 ·····------------------------------------------------------------------------
39 ·····{.00802694,·.00790754},·{.00706792,·.00667325},·{.00395036,·.00400722},39 ·····{.00400046,·.00799564},·{.00400709,·.00401843},·{.00400051,·.00398355},
40 ·····------------------------------------------------------------------------40 ·····------------------------------------------------------------------------
41 ·····{.00403522,·.0072114}}41 ·····{.00590819,·.00806039}}
  
42 o5·:·List42 o5·:·List
43 i6·:·1/10*sum(L,t->t_0)43 i6·:·1/10*sum(L,t->t_0)
  
44 o6·=·.010317421500000144 o6·=·.01045816749999999
  
45 o6·:·RR·(of·precision·53)45 o6·:·RR·(of·precision·53)
46 i7·:·1/10*sum(L,t->t_1)46 i7·:·1/10*sum(L,t->t_1)
  
47 o7·=·.00788563099999999447 o7·=·.007164236799999957
  
48 o7·:·RR·(of·precision·53)48 o7·:·RR·(of·precision·53)
49 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8te\x8es\x8st\x8tT\x8Ti\x8im\x8me\x8eF\x8Fo\x8or\x8rL\x8LL\x8LL\x8Lo\x8on\x8nS\x8Sy\x8yz\x8zy\x8yg\x8gi\x8ie\x8es\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*49 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8te\x8es\x8st\x8tT\x8Ti\x8im\x8me\x8eF\x8Fo\x8or\x8rL\x8LL\x8LL\x8Lo\x8on\x8nS\x8Sy\x8yz\x8zy\x8yg\x8gi\x8ie\x8es\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
50 ····*·testTimeForLLLonSyzygies(ZZ,ZZ)50 ····*·testTimeForLLLonSyzygies(ZZ,ZZ)
51 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*51 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
52 The·object·_\x8t_\x8e_\x8s_\x8t_\x8T_\x8i_\x8m_\x8e_\x8F_\x8o_\x8r_\x8L_\x8L_\x8L_\x8o_\x8n_\x8S_\x8y_\x8z_\x8y_\x8g_\x8i_\x8e_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.52 The·object·_\x8t_\x8e_\x8s_\x8t_\x8T_\x8i_\x8m_\x8e_\x8F_\x8o_\x8r_\x8L_\x8L_\x8L_\x8o_\x8n_\x8S_\x8y_\x8z_\x8y_\x8g_\x8i_\x8e_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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8 UmFuZG9tQ3VydmVz8 UmFuZG9tQ3VydmVz
9 #:len=7759 #:len=775
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmFuZG9tIGN1cnZlcyIsIERlc2NyaXB010 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmFuZG9tIGN1cnZlcyIsIERlc2NyaXB0
11 aW9uID0+IDE6KERJVntQQVJBe1RFWHsiVGhpcyBwYWNrYWdlIGxvYWRzIHRoZSAifSxUT3tuZXcg11 aW9uID0+IDE6KERJVntQQVJBe1RFWHsiVGhpcyBwYWNrYWdlIGxvYWRzIHRoZSAifSxUT3tuZXcg
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./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:38·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy
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./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/example-output/_smooth__Canonical__Curve.out
    
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1 --·-*-·M2-comint·-*-·hash:·115495276897903451521 --·-*-·M2-comint·-*-·hash:·11549527689790345152
  
2 i1·:·time·ICan·=·smoothCanonicalCurve(11,5);2 i1·:·time·ICan·=·smoothCanonicalCurve(11,5);
3 ·--·used·1.28953s·(cpu);·0.924076s·(thread);·0s·(gc)3 ·--·used·1.40591s·(cpu);·1.12683s·(thread);·0s·(gc)
  
4 ··············ZZ4 ··············ZZ
5 o1·:·Ideal·of·--[t·..t··]5 o1·:·Ideal·of·--[t·..t··]
6 ···············5··0···106 ···············5··0···10
  
7 i2·:·(dim·ICan,·genus·ICan,·degree·ICan)7 i2·:·(dim·ICan,·genus·ICan,·degree·ICan)
  
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./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
    
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85 ··········<p>For·g=15·the·curves·are·constructed·via·matrix·factorizations.</p>85 ··········<p>For·g=15·the·curves·are·constructed·via·matrix·factorizations.</p>
86 ··········<p>If·the·option·<span·class="tt">Printing</span>·is·set·to·<span·class="tt">true</span>·then·printings·about·the·current·step·in·the·construction·are·displayed.</p>86 ··········<p>If·the·option·<span·class="tt">Printing</span>·is·set·to·<span·class="tt">true</span>·then·printings·about·the·current·step·in·the·construction·are·displayed.</p>
87 ··········<p></p>87 ··········<p></p>
88 ········</div>88 ········</div>
89 ········<table·class="examples">89 ········<table·class="examples">
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i1·:·time·ICan·=·smoothCanonicalCurve(11,5);91 <td>··············<pre><code·class="language-macaulay2">i1·:·time·ICan·=·smoothCanonicalCurve(11,5);
92 ·--·used·1.28953s·(cpu);·0.924076s·(thread);·0s·(gc)92 ·--·used·1.40591s·(cpu);·1.12683s·(thread);·0s·(gc)
  
93 ··············ZZ93 ··············ZZ
94 o1·:·Ideal·of·--[t·..t··]94 o1·:·Ideal·of·--[t·..t··]
95 ···············5··0···10</code></pre>95 ···············5··0···10</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i2·:·(dim·ICan,·genus·ICan,·degree·ICan)98 <td>··············<pre><code·class="language-macaulay2">i2·:·(dim·ICan,·genus·ICan,·degree·ICan)
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30 For·g<=10·the·curves·are·constructed·via·plane·models.30 For·g<=10·the·curves·are·constructed·via·plane·models.
31 For·g<=13·the·curves·are·constructed·via·space·models.31 For·g<=13·the·curves·are·constructed·via·space·models.
32 For·g=14·the·curves·are·constructed·by·Verra's·method.32 For·g=14·the·curves·are·constructed·by·Verra's·method.
33 For·g=15·the·curves·are·constructed·via·matrix·factorizations.33 For·g=15·the·curves·are·constructed·via·matrix·factorizations.
34 If·the·option·Printing·is·set·to·true·then·printings·about·the·current·step·in34 If·the·option·Printing·is·set·to·true·then·printings·about·the·current·step·in
35 the·construction·are·displayed.35 the·construction·are·displayed.
36 i1·:·time·ICan·=·smoothCanonicalCurve(11,5);36 i1·:·time·ICan·=·smoothCanonicalCurve(11,5);
37 ·--·used·1.28953s·(cpu);·0.924076s·(thread);·0s·(gc)37 ·--·used·1.40591s·(cpu);·1.12683s·(thread);·0s·(gc)
  
38 ··············ZZ38 ··············ZZ
39 o1·:·Ideal·of·--[t·..t··]39 o1·:·Ideal·of·--[t·..t··]
40 ···············5··0···1040 ···············5··0···10
41 i2·:·(dim·ICan,·genus·ICan,·degree·ICan)41 i2·:·(dim·ICan,·genus·ICan,·degree·ICan)
  
42 o2·=·(2,·11,·20)42 o2·=·(2,·11,·20)
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./usr/share/doc/Macaulay2/RandomGenus14Curves/example-output/_random__Curve__Genus14__Degree18in__P6.out
    
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3 i1·:·setRandomSeed("alpha");3 i1·:·setRandomSeed("alpha");
  
4 i2·:·FF=ZZ/10007;4 i2·:·FF=ZZ/10007;
  
5 i3·:·S=FF[x_0..x_6];5 i3·:·S=FF[x_0..x_6];
  
6 i4·:·time·I=randomCurveGenus14Degree18inP6(S);6 i4·:·time·I=randomCurveGenus14Degree18inP6(S);
7 ·--·used·1.18226s·(cpu);·1.05174s·(thread);·0s·(gc)7 ·--·used·1.31011s·(cpu);·1.15799s·(thread);·0s·(gc)
  
8 o4·:·Ideal·of·S8 o4·:·Ideal·of·S
  
9 i5·:·betti·res·I9 i5·:·betti·res·I
  
10 ············0··1··2··3··4·510 ············0··1··2··3··4·5
11 o5·=·total:·1·13·45·56·25·211 o5·=·total:·1·13·45·56·25·2
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./usr/share/doc/Macaulay2/RandomGenus14Curves/html/_random__Curve__Genus14__Degree18in__P6.html
    
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87 <td>··············<pre><code·class="language-macaulay2">i2·:·FF=ZZ/10007;</code></pre>87 <td>··············<pre><code·class="language-macaulay2">i2·:·FF=ZZ/10007;</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i3·:·S=FF[x_0..x_6];</code></pre>90 <td>··············<pre><code·class="language-macaulay2">i3·:·S=FF[x_0..x_6];</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·I=randomCurveGenus14Degree18inP6(S);93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·I=randomCurveGenus14Degree18inP6(S);
94 ·--·used·1.18226s·(cpu);·1.05174s·(thread);·0s·(gc)94 ·--·used·1.31011s·(cpu);·1.15799s·(thread);·0s·(gc)
  
95 o4·:·Ideal·of·S</code></pre>95 o4·:·Ideal·of·S</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i5·:·betti·res·I98 <td>··············<pre><code·class="language-macaulay2">i5·:·betti·res·I
  
99 ············0··1··2··3··4·599 ············0··1··2··3··4·5
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28 the·unirationality·of·$M_{14}$.·It·proves·the·unirationality·of·$M_{14}$·for28 the·unirationality·of·$M_{14}$.·It·proves·the·unirationality·of·$M_{14}$·for
29 fields·of·the·chosen·finite·characteristic·10007,·for·fields·of·characteristic29 fields·of·the·chosen·finite·characteristic·10007,·for·fields·of·characteristic
30 0·by·semi-continuity,·and,·hence,·for·all·but·finitely·many·primes·$p$.30 0·by·semi-continuity,·and,·hence,·for·all·but·finitely·many·primes·$p$.
31 i1·:·setRandomSeed("alpha");31 i1·:·setRandomSeed("alpha");
32 i2·:·FF=ZZ/10007;32 i2·:·FF=ZZ/10007;
33 i3·:·S=FF[x_0..x_6];33 i3·:·S=FF[x_0..x_6];
34 i4·:·time·I=randomCurveGenus14Degree18inP6(S);34 i4·:·time·I=randomCurveGenus14Degree18inP6(S);
35 ·--·used·1.18226s·(cpu);·1.05174s·(thread);·0s·(gc)35 ·--·used·1.31011s·(cpu);·1.15799s·(thread);·0s·(gc)
  
36 o4·:·Ideal·of·S36 o4·:·Ideal·of·S
37 i5·:·betti·res·I37 i5·:·betti·res·I
  
38 ············0··1··2··3··4·538 ············0··1··2··3··4·5
39 o5·=·total:·1·13·45·56·25·239 o5·=·total:·1·13·45·56·25·2
40 ·········0:·1··.··.··.··.·.40 ·········0:·1··.··.··.··.·.
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./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=257 #:len=25
8 cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==8 cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==
9 #:len=18079 #:len=1807
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh
11 bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjksIElucHV0cyA9PiB711 bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjksIElucHV0cyA9PiB7
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./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out
    
Offset 1, 23 lines modifiedOffset 1, 23 lines modified
1 --·-*-·M2-comint·-*-·hash:·95428017424294951611 --·-*-·M2-comint·-*-·hash:·9542801742429495161
  
2 i1·:·setRandomSeed(currentTime())2 i1·:·setRandomSeed(currentTime())
  
3 o1·=·17489624713 o1·=·1783380567
  
4 i2·:·kk=ZZ/101;4 i2·:·kk=ZZ/101;
  
5 i3·:·S=kk[vars(0..5)];5 i3·:·S=kk[vars(0..5)];
  
6 i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)6 i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)
7 ·--·used·1.32939s·(cpu);·1.05608s·(thread);·0s·(gc)7 ·--·used·1.55s·(cpu);·1.17772s·(thread);·0s·(gc)
  
8 o4·=·Tally{4·=>·43·}8 o4·=·Tally{4·=>·50·}
9 ···········5·=>·1959 ···········5·=>·214
10 ···········6·=>·18710 ···········6·=>·159
11 ···········7·=>·6411 ···········7·=>·59
12 ···········8·=>·1012 ···········8·=>·17
13 ···········9·=>·113 ···········9·=>·1
  
14 o4·:·Tally14 o4·:·Tally
  
15 i5·:·15 i5·:·
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./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out
    
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1 --·-*-·M2-comint·-*-·hash:·59594655671978210461 --·-*-·M2-comint·-*-·hash:·5959465567197821046
  
2 i1·:·setRandomSeed(currentTime())2 i1·:·setRandomSeed(currentTime())
  
3 o1·=·17489625093 o1·=·1783380611
  
4 i2·:·kk=ZZ/1014 i2·:·kk=ZZ/101
  
5 o2·=·kk5 o2·=·kk
  
6 o2·:·QuotientRing6 o2·:·QuotientRing
  
Offset 14, 13 lines modifiedOffset 14, 13 lines modified
  
14 o3·=·S14 o3·=·S
  
15 o3·:·PolynomialRing15 o3·:·PolynomialRing
  
16 i4·:·randomMonomial(3,S)16 i4·:·randomMonomial(3,S)
  
17 ······317 ······2
18 o4·=·a18 o4·=·b·c
  
19 o4·:·S19 o4·:·S
  
20 i5·:·20 i5·:·
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./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out
    
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1 --·-*-·M2-comint·-*-·hash:·88763405620218654471 --·-*-·M2-comint·-*-·hash:·8876340562021865447
  
2 i1·:·setRandomSeed(currentTime())2 i1·:·setRandomSeed(currentTime())
  
3 o1·=·17489625593 o1·=·1783380671
  
4 i2·:·kk=ZZ/1014 i2·:·kk=ZZ/101
  
5 o2·=·kk5 o2·=·kk
  
6 o2·:·QuotientRing6 o2·:·QuotientRing
  
Offset 21, 18 lines modifiedOffset 21, 18 lines modified
21 o4·=·{3,·5,·7}21 o4·=·{3,·5,·7}
  
22 o4·:·List22 o4·:·List
  
23 i5·:·randomSquareFreeMonomialIdeal(L,·S)23 i5·:·randomSquareFreeMonomialIdeal(L,·S)
24 low·degree·gens·generated·everything24 low·degree·gens·generated·everything
  
25 o5·=·ideal(c*d*e)25 o5·=·ideal(a*b*d)
  
26 o5·:·Ideal·of·S26 o5·:·Ideal·of·S
  
27 i6·:·randomSquareFreeMonomialIdeal(5:2,·S)27 i6·:·randomSquareFreeMonomialIdeal(5:2,·S)
  
28 o6·=·ideal·(b*e,·d*e,·a*b,·a*e,·b*d)28 o6·=·ideal·(b*e,·a*e,·a*d,·b*c,·c*d)
  
29 o6·:·Ideal·of·S29 o6·:·Ideal·of·S
  
30 i7·:·30 i7·:·
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./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
    
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1 --·-*-·M2-comint·-*-·hash:·105049112135082813151 --·-*-·M2-comint·-*-·hash:·10504911213508281315
  
2 i1·:·setRandomSeed(currentTime())2 i1·:·setRandomSeed(currentTime())
  
3 o1·=·17489625463 o1·=·1783380653
  
4 i2·:·S=ZZ/2[vars(0..3)]4 i2·:·S=ZZ/2[vars(0..3)]
  
5 o2·=·S5 o2·=·S
  
6 o2·:·PolynomialRing6 o2·:·PolynomialRing
  
Offset 14, 15 lines modifiedOffset 14, 17 lines modified
  
14 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)14 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)
  
15 o3·:·MonomialIdeal·of·S15 o3·:·MonomialIdeal·of·S
  
16 i4·:·randomSquareFreeStep·J16 i4·:·randomSquareFreeStep·J
  
17 o4·=·{monomialIdeal·(a*b,·a*d,·b*d),·{a*b,·a*d,·b*d},·{c*d,·b*c,·a*c}}17 o4·=·{monomialIdeal·(a*b*c,·a*d,·b*c*d),·{a*b*c,·a*d,·b*c*d},·{c*d,·b*d,·b*c,
 18 ·····------------------------------------------------------------------------
 19 ·····a*c,·a*b}}
  
18 o4·:·List20 o4·:·List
  
19 i5·:·setRandomSeed(1)21 i5·:·setRandomSeed(1)
  
20 o5·=·122 o5·=·1
  
Offset 35, 15 lines modifiedOffset 37, 15 lines modified
35 i7·:·J·=·monomialIdeal·0_S37 i7·:·J·=·monomialIdeal·0_S
  
36 o7·=·monomialIdeal·()38 o7·=·monomialIdeal·()
  
37 o7·:·MonomialIdeal·of·S39 o7·:·MonomialIdeal·of·S
  
38 i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs(J,AlexanderProbability·=>·.01));40 i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs(J,AlexanderProbability·=>·.01));
39 ·--·used·3.16184s·(cpu);·2.23249s·(thread);·0s·(gc)41 ·--·used·4.45924s·(cpu);·3.07563s·(thread);·0s·(gc)
  
40 i9·:·#T42 i9·:·#T
  
41 o9·=·16843 o9·=·168
  
42 i10·:·T44 i10·:·T
  
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72 ········<div>72 ········<div>
73 ··········<p>Chooses·a·random·monomial.</p>73 ··········<p>Chooses·a·random·monomial.</p>
74 ········</div>74 ········</div>
75 ········<table·class="examples">75 ········<table·class="examples">
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())77 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())
  
78 o1·=·1748962509</code></pre>78 o1·=·1783380611</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/10181 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/101
  
82 o2·=·kk82 o2·=·kk
  
83 o2·:·QuotientRing</code></pre>83 o2·:·QuotientRing</code></pre>
Offset 91, 16 lines modifiedOffset 91, 16 lines modified
91 o3·=·S91 o3·=·S
  
92 o3·:·PolynomialRing</code></pre>92 o3·:·PolynomialRing</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i4·:·randomMonomial(3,S)95 <td>··············<pre><code·class="language-macaulay2">i4·:·randomMonomial(3,S)
  
96 ······396 ······2
97 o4·=·a97 o4·=·b·c
  
98 o4·:·S</code></pre>98 o4·:·S</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ········</table>100 ········</table>
101 ······</div>101 ······</div>
102 ······<div>102 ······<div>
103 ········<h2>See·also</h2>103 ········<h2>See·also</h2>
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Offset 13, 29 lines modifiedOffset 13, 29 lines modified
13 ··········o·S,·a·_\x8r_\x8i_\x8n_\x8g,·polynomial·ring13 ··········o·S,·a·_\x8r_\x8i_\x8n_\x8g,·polynomial·ring
14 ····*·Outputs:14 ····*·Outputs:
15 ··········o·m,·a·_\x8r_\x8i_\x8n_\x8g_\x8·_\x8e_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t,·monomial·of·S15 ··········o·m,·a·_\x8r_\x8i_\x8n_\x8g_\x8·_\x8e_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t,·monomial·of·S
16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
17 Chooses·a·random·monomial.17 Chooses·a·random·monomial.
18 i1·:·setRandomSeed(currentTime())18 i1·:·setRandomSeed(currentTime())
  
19 o1·=·174896250919 o1·=·1783380611
20 i2·:·kk=ZZ/10120 i2·:·kk=ZZ/101
  
21 o2·=·kk21 o2·=·kk
  
22 o2·:·QuotientRing22 o2·:·QuotientRing
23 i3·:·S=kk[a,b,c]23 i3·:·S=kk[a,b,c]
  
24 o3·=·S24 o3·=·S
  
25 o3·:·PolynomialRing25 o3·:·PolynomialRing
26 i4·:·randomMonomial(3,S)26 i4·:·randomMonomial(3,S)
  
27 ······327 ······2
28 o4·=·a28 o4·=·b·c
  
29 o4·:·S29 o4·:·S
30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
31 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8I_\x8d_\x8e_\x8a_\x8l·--·random·monomial·ideal·with·given·degree·generators31 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8I_\x8d_\x8e_\x8a_\x8l·--·random·monomial·ideal·with·given·degree·generators
32 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8S_\x8q_\x8u_\x8a_\x8r_\x8e_\x8F_\x8r_\x8e_\x8e_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8I_\x8d_\x8e_\x8a_\x8l·--·random·square-free·monomial·ideal·with32 ····*·_\x8r_\x8a_\x8n_\x8d_\x8o_\x8m_\x8S_\x8q_\x8u_\x8a_\x8r_\x8e_\x8F_\x8r_\x8e_\x8e_\x8M_\x8o_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8I_\x8d_\x8e_\x8a_\x8l·--·random·square-free·monomial·ideal·with
33 ······given·degree·generators33 ······given·degree·generators
34 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mM\x8Mo\x8on\x8no\x8om\x8mi\x8ia\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*34 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mM\x8Mo\x8on\x8no\x8om\x8mi\x8ia\x8al\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Monomial__Ideal.html
    
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72 ········<div>72 ········<div>
73 ··········<p>Choose·a·random·square-free·monomial·ideal·whose·generators·have·the·degrees·specified·by·the·list·or·sequence·L.</p>73 ··········<p>Choose·a·random·square-free·monomial·ideal·whose·generators·have·the·degrees·specified·by·the·list·or·sequence·L.</p>
74 ········</div>74 ········</div>
75 ········<table·class="examples">75 ········<table·class="examples">
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())77 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())
  
78 o1·=·1748962559</code></pre>78 o1·=·1783380671</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/10181 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/101
  
82 o2·=·kk82 o2·=·kk
  
83 o2·:·QuotientRing</code></pre>83 o2·:·QuotientRing</code></pre>
Offset 99, 22 lines modifiedOffset 99, 22 lines modified
  
99 o4·:·List</code></pre>99 o4·:·List</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i5·:·randomSquareFreeMonomialIdeal(L,·S)102 <td>··············<pre><code·class="language-macaulay2">i5·:·randomSquareFreeMonomialIdeal(L,·S)
103 low·degree·gens·generated·everything103 low·degree·gens·generated·everything
  
104 o5·=·ideal(c*d*e)104 o5·=·ideal(a*b*d)
  
105 o5·:·Ideal·of·S</code></pre>105 o5·:·Ideal·of·S</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i6·:·randomSquareFreeMonomialIdeal(5:2,·S)108 <td>··············<pre><code·class="language-macaulay2">i6·:·randomSquareFreeMonomialIdeal(5:2,·S)
  
109 o6·=·ideal·(b*e,·d*e,·a*b,·a*e,·b*d)109 o6·=·ideal·(b*e,·a*e,·a*d,·b*c,·c*d)
  
110 o6·:·Ideal·of·S</code></pre>110 o6·:·Ideal·of·S</code></pre>
111 </td>··········</tr>111 </td>··········</tr>
112 ········</table>112 ········</table>
113 ······</div>113 ······</div>
114 ······<div>114 ······<div>
115 ········<h2>Caveat</h2>115 ········<h2>Caveat</h2>
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15 ··········o·I,·an·_\x8i_\x8d_\x8e_\x8a_\x8l,·square-free·monomial·ideal·with·generators·of15 ··········o·I,·an·_\x8i_\x8d_\x8e_\x8a_\x8l,·square-free·monomial·ideal·with·generators·of
16 ············specified·degrees16 ············specified·degrees
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 Choose·a·random·square-free·monomial·ideal·whose·generators·have·the·degrees18 Choose·a·random·square-free·monomial·ideal·whose·generators·have·the·degrees
19 specified·by·the·list·or·sequence·L.19 specified·by·the·list·or·sequence·L.
20 i1·:·setRandomSeed(currentTime())20 i1·:·setRandomSeed(currentTime())
  
21 o1·=·174896255921 o1·=·1783380671
22 i2·:·kk=ZZ/10122 i2·:·kk=ZZ/101
  
23 o2·=·kk23 o2·=·kk
  
24 o2·:·QuotientRing24 o2·:·QuotientRing
25 i3·:·S=kk[a..e]25 i3·:·S=kk[a..e]
  
Offset 34, 20 lines modifiedOffset 34, 20 lines modified
  
34 o4·=·{3,·5,·7}34 o4·=·{3,·5,·7}
  
35 o4·:·List35 o4·:·List
36 i5·:·randomSquareFreeMonomialIdeal(L,·S)36 i5·:·randomSquareFreeMonomialIdeal(L,·S)
37 low·degree·gens·generated·everything37 low·degree·gens·generated·everything
  
38 o5·=·ideal(c*d*e)38 o5·=·ideal(a*b*d)
  
39 o5·:·Ideal·of·S39 o5·:·Ideal·of·S
40 i6·:·randomSquareFreeMonomialIdeal(5:2,·S)40 i6·:·randomSquareFreeMonomialIdeal(5:2,·S)
  
41 o6·=·ideal·(b*e,·d*e,·a*b,·a*e,·b*d)41 o6·=·ideal·(b*e,·a*e,·a*d,·b*c,·c*d)
  
42 o6·:·Ideal·of·S42 o6·:·Ideal·of·S
43 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*43 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
44 The·ideal·is·constructed·degree·by·degree,·starting·from·the·lowest·degree44 The·ideal·is·constructed·degree·by·degree,·starting·from·the·lowest·degree
45 specified.·If·there·are·not·enough·monomials·of·the·next·specified·degree·that45 specified.·If·there·are·not·enough·monomials·of·the·next·specified·degree·that
46 are·not·already·in·the·ideal,·the·function·prints·a·warning·and·returns·an46 are·not·already·in·the·ideal,·the·function·prints·a·warning·and·returns·an
47 ideal·containing·all·the·generators·of·that·degree.47 ideal·containing·all·the·generators·of·that·degree.
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82 ··········<p>With·probability·p·the·routine·takes·the·Alexander·dual·of·I;·the·default·value·of·p·is·.05,·and·it·can·be·set·with·the·option·AlexanderProbility.</p>82 ··········<p>With·probability·p·the·routine·takes·the·Alexander·dual·of·I;·the·default·value·of·p·is·.05,·and·it·can·be·set·with·the·option·AlexanderProbility.</p>
83 ··········<p>Otherwise·uses·the·Metropolis·algorithm·to·produce·a·random·walk·on·the·space·of·square-free·ideals.·Note·that·there·are·a·LOT·of·square-free·ideals;·these·are·the·Dedekind·numbers,·and·the·sequence·(with·1,2,3,4,5,6,7,8·variables)·begins·3,6,20,168,7581,·7828354,·2414682040998,·56130437228687557907788.·(see·the·Online·Encyclopedia·of·Integer·Sequences·for·more·information).·Given·I·in·a·polynomial·ring·S,·we·make·a·list·ISocgens·of·the·square-free·minimal·monomial·generators·of·the·socle·of·S/(squares+I)·and·a·list·of·minimal·generators·Igens·of·I.·A·candidate·&quot;next&quot;·ideal·J·is·formed·as·follows:·We·choose·randomly·from·the·union·of·these·lists;·if·a·socle·element·is·chosen,·it's·added·to·I;·if·a·minimal·generator·is·chosen,·it's·replaced·by·the·square-free·part·of·the·maximal·ideal·times·it.·the·chance·of·making·the·given·move·is·then·1/(#ISocgens+#Igens),·and·the·chance·of·making·the·move·back·would·be·the·similar·quantity·for·J,·so·we·make·the·move·or·not·depending·on·whether·random·RR·&lt;·(nJ+nSocJ)/(nI+nSocI)·or·not;·here·random·RR·is·a·random·number·in·[0,1].</p>83 ··········<p>Otherwise·uses·the·Metropolis·algorithm·to·produce·a·random·walk·on·the·space·of·square-free·ideals.·Note·that·there·are·a·LOT·of·square-free·ideals;·these·are·the·Dedekind·numbers,·and·the·sequence·(with·1,2,3,4,5,6,7,8·variables)·begins·3,6,20,168,7581,·7828354,·2414682040998,·56130437228687557907788.·(see·the·Online·Encyclopedia·of·Integer·Sequences·for·more·information).·Given·I·in·a·polynomial·ring·S,·we·make·a·list·ISocgens·of·the·square-free·minimal·monomial·generators·of·the·socle·of·S/(squares+I)·and·a·list·of·minimal·generators·Igens·of·I.·A·candidate·&quot;next&quot;·ideal·J·is·formed·as·follows:·We·choose·randomly·from·the·union·of·these·lists;·if·a·socle·element·is·chosen,·it's·added·to·I;·if·a·minimal·generator·is·chosen,·it's·replaced·by·the·square-free·part·of·the·maximal·ideal·times·it.·the·chance·of·making·the·given·move·is·then·1/(#ISocgens+#Igens),·and·the·chance·of·making·the·move·back·would·be·the·similar·quantity·for·J,·so·we·make·the·move·or·not·depending·on·whether·random·RR·&lt;·(nJ+nSocJ)/(nI+nSocI)·or·not;·here·random·RR·is·a·random·number·in·[0,1].</p>
84 ········</div>84 ········</div>
85 ········<table·class="examples">85 ········<table·class="examples">
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())87 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())
  
88 o1·=·1748962546</code></pre>88 o1·=·1783380653</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i2·:·S=ZZ/2[vars(0..3)]91 <td>··············<pre><code·class="language-macaulay2">i2·:·S=ZZ/2[vars(0..3)]
  
92 o2·=·S92 o2·=·S
  
93 o2·:·PolynomialRing</code></pre>93 o2·:·PolynomialRing</code></pre>
Offset 101, 15 lines modifiedOffset 101, 17 lines modified
101 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)101 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)
  
102 o3·:·MonomialIdeal·of·S</code></pre>102 o3·:·MonomialIdeal·of·S</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·randomSquareFreeStep·J105 <td>··············<pre><code·class="language-macaulay2">i4·:·randomSquareFreeStep·J
  
106 o4·=·{monomialIdeal·(a*b,·a*d,·b*d),·{a*b,·a*d,·b*d},·{c*d,·b*c,·a*c}}106 o4·=·{monomialIdeal·(a*b*c,·a*d,·b*c*d),·{a*b*c,·a*d,·b*c*d},·{c*d,·b*d,·b*c,
 107 ·····------------------------------------------------------------------------
 108 ·····a*c,·a*b}}
  
107 o4·:·List</code></pre>109 o4·:·List</code></pre>
108 </td>··········</tr>110 </td>··········</tr>
109 ········</table>111 ········</table>
110 ········<div>112 ········<div>
111 ··········<p>With·4·variables·and·168·possible·monomial·ideals,·a·run·of·5000·takes·less·than·6·seconds·on·a·reasonably·fast·machine.·With·10·variables·a·run·of·1000·takes·about·2·seconds.</p>113 ··········<p>With·4·variables·and·168·possible·monomial·ideals,·a·run·of·5000·takes·less·than·6·seconds·on·a·reasonably·fast·machine.·With·10·variables·a·run·of·1000·takes·about·2·seconds.</p>
112 ········</div>114 ········</div>
Offset 131, 15 lines modifiedOffset 133, 15 lines modified
  
131 o7·=·monomialIdeal·()133 o7·=·monomialIdeal·()
  
132 o7·:·MonomialIdeal·of·S</code></pre>134 o7·:·MonomialIdeal·of·S</code></pre>
133 </td>··········</tr>135 </td>··········</tr>
134 ··········<tr>136 ··········<tr>
135 <td>··············<pre><code·class="language-macaulay2">i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs(J,AlexanderProbability·=>·.01));137 <td>··············<pre><code·class="language-macaulay2">i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs(J,AlexanderProbability·=>·.01));
136 ·--·used·3.16184s·(cpu);·2.23249s·(thread);·0s·(gc)</code></pre>138 ·--·used·4.45924s·(cpu);·3.07563s·(thread);·0s·(gc)</code></pre>
137 </td>··········</tr>139 </td>··········</tr>
138 ··········<tr>140 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i9·:·#T141 <td>··············<pre><code·class="language-macaulay2">i9·:·#T
  
140 o9·=·168</code></pre>142 o9·=·168</code></pre>
141 </td>··········</tr>143 </td>··········</tr>
142 ··········<tr>144 ··········<tr>
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37 it's·added·to·I;·if·a·minimal·generator·is·chosen,·it's·replaced·by·the·square-37 it's·added·to·I;·if·a·minimal·generator·is·chosen,·it's·replaced·by·the·square-
38 free·part·of·the·maximal·ideal·times·it.·the·chance·of·making·the·given·move·is38 free·part·of·the·maximal·ideal·times·it.·the·chance·of·making·the·given·move·is
39 then·1/(#ISocgens+#Igens),·and·the·chance·of·making·the·move·back·would·be·the39 then·1/(#ISocgens+#Igens),·and·the·chance·of·making·the·move·back·would·be·the
40 similar·quantity·for·J,·so·we·make·the·move·or·not·depending·on·whether·random40 similar·quantity·for·J,·so·we·make·the·move·or·not·depending·on·whether·random
41 RR·<·(nJ+nSocJ)/(nI+nSocI)·or·not;·here·random·RR·is·a·random·number·in·[0,1].41 RR·<·(nJ+nSocJ)/(nI+nSocI)·or·not;·here·random·RR·is·a·random·number·in·[0,1].
42 i1·:·setRandomSeed(currentTime())42 i1·:·setRandomSeed(currentTime())
  
43 o1·=·174896254643 o1·=·1783380653
44 i2·:·S=ZZ/2[vars(0..3)]44 i2·:·S=ZZ/2[vars(0..3)]
  
45 o2·=·S45 o2·=·S
  
46 o2·:·PolynomialRing46 o2·:·PolynomialRing
47 i3·:·J·=·monomialIdeal"ab,ad,·bcd"47 i3·:·J·=·monomialIdeal"ab,ad,·bcd"
  
48 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)48 o3·=·monomialIdeal·(a*b,·a*d,·b*c*d)
  
49 o3·:·MonomialIdeal·of·S49 o3·:·MonomialIdeal·of·S
50 i4·:·randomSquareFreeStep·J50 i4·:·randomSquareFreeStep·J
  
51 o4·=·{monomialIdeal·(a*b,·a*d,·b*d),·{a*b,·a*d,·b*d},·{c*d,·b*c,·a*c}}51 o4·=·{monomialIdeal·(a*b*c,·a*d,·b*c*d),·{a*b*c,·a*d,·b*c*d},·{c*d,·b*d,·b*c,
 52 ·····------------------------------------------------------------------------
 53 ·····a*c,·a*b}}
  
52 o4·:·List54 o4·:·List
53 With·4·variables·and·168·possible·monomial·ideals,·a·run·of·5000·takes·less55 With·4·variables·and·168·possible·monomial·ideals,·a·run·of·5000·takes·less
54 than·6·seconds·on·a·reasonably·fast·machine.·With·10·variables·a·run·of·100056 than·6·seconds·on·a·reasonably·fast·machine.·With·10·variables·a·run·of·1000
55 takes·about·2·seconds.57 takes·about·2·seconds.
56 i5·:·setRandomSeed(1)58 i5·:·setRandomSeed(1)
  
Offset 71, 15 lines modifiedOffset 73, 15 lines modified
71 i7·:·J·=·monomialIdeal·0_S73 i7·:·J·=·monomialIdeal·0_S
  
72 o7·=·monomialIdeal·()74 o7·=·monomialIdeal·()
  
73 o7·:·MonomialIdeal·of·S75 o7·:·MonomialIdeal·of·S
74 i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs76 i8·:·time·T=tally·for·t·from·1·to·5000·list·first·(J=rsfs
75 (J,AlexanderProbability·=>·.01));77 (J,AlexanderProbability·=>·.01));
76 ·--·used·3.16184s·(cpu);·2.23249s·(thread);·0s·(gc)78 ·--·used·4.45924s·(cpu);·3.07563s·(thread);·0s·(gc)
77 i9·:·#T79 i9·:·#T
  
78 o9·=·16880 o9·=·168
79 i10·:·T81 i10·:·T
  
80 o10·=·Tally{monomialIdeal·()·=>·45····························}82 o10·=·Tally{monomialIdeal·()·=>·45····························}
81 ············monomialIdeal·(a*b*c,·a*b*d)·=>·3383 ············monomialIdeal·(a*b*c,·a*b*d)·=>·33
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47 ········<div>47 ········<div>
48 ··········<p>This·package·can·be·used·to·make·experiments,·trying·many·ideals,·perhaps·over·small·fields.·For·example...what·would·you·expect·the·regularities·of·&quot;typical&quot;·monomial·ideals·with·10·generators·of·degree·3·in·6·variables·to·be?·Try·a·bunch·of·examples·--·it's·fast.·Here·we·do·only·500·--·this·takes·about·a·second·on·a·fast·machine·--·but·with·a·little·patience,·thousands·can·be·done·conveniently.</p>48 ··········<p>This·package·can·be·used·to·make·experiments,·trying·many·ideals,·perhaps·over·small·fields.·For·example...what·would·you·expect·the·regularities·of·&quot;typical&quot;·monomial·ideals·with·10·generators·of·degree·3·in·6·variables·to·be?·Try·a·bunch·of·examples·--·it's·fast.·Here·we·do·only·500·--·this·takes·about·a·second·on·a·fast·machine·--·but·with·a·little·patience,·thousands·can·be·done·conveniently.</p>
49 ········</div>49 ········</div>
50 ········<table·class="examples">50 ········<table·class="examples">
51 ··········<tr>51 ··········<tr>
52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())52 <td>··············<pre><code·class="language-macaulay2">i1·:·setRandomSeed(currentTime())
  
53 o1·=·1748962471</code></pre>53 o1·=·1783380567</code></pre>
54 </td>··········</tr>54 </td>··········</tr>
55 ··········<tr>55 ··········<tr>
56 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/101;</code></pre>56 <td>··············<pre><code·class="language-macaulay2">i2·:·kk=ZZ/101;</code></pre>
57 </td>··········</tr>57 </td>··········</tr>
58 ··········<tr>58 ··········<tr>
59 <td>··············<pre><code·class="language-macaulay2">i3·:·S=kk[vars(0..5)];</code></pre>59 <td>··············<pre><code·class="language-macaulay2">i3·:·S=kk[vars(0..5)];</code></pre>
60 </td>··········</tr>60 </td>··········</tr>
61 ··········<tr>61 ··········<tr>
62 <td>··············<pre><code·class="language-macaulay2">i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)62 <td>··············<pre><code·class="language-macaulay2">i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)
63 ·--·used·1.32939s·(cpu);·1.05608s·(thread);·0s·(gc)63 ·--·used·1.55s·(cpu);·1.17772s·(thread);·0s·(gc)
  
64 o4·=·Tally{4·=>·43·}64 o4·=·Tally{4·=>·50·}
65 ···········5·=>·19565 ···········5·=>·214
66 ···········6·=>·18766 ···········6·=>·159
67 ···········7·=>·6467 ···········7·=>·59
68 ···········8·=>·1068 ···········8·=>·17
69 ···········9·=>·169 ···········9·=>·1
  
70 o4·:·Tally</code></pre>70 o4·:·Tally</code></pre>
71 </td>··········</tr>71 </td>··········</tr>
72 ········</table>72 ········</table>
73 ········<div>73 ········<div>
74 ··········<p>How·does·this·compare·with·the·case·of·binomial·ideals?·or·pure·binomial·ideals?·We·invite·the·reader·to·experiment,·replacing·&quot;randomMonomialIdeal&quot;·above·with·&quot;randomBinomialIdeal&quot;·or·&quot;randomPureBinomialIdeal&quot;,·or·taking·larger·numbers·of·examples.·Click·the·link·&quot;Finding·Extreme·Examples&quot;·below·to·see·some·other,·more·elaborate·ways·to·search.</p>74 ··········<p>How·does·this·compare·with·the·case·of·binomial·ideals?·or·pure·binomial·ideals?·We·invite·the·reader·to·experiment,·replacing·&quot;randomMonomialIdeal&quot;·above·with·&quot;randomBinomialIdeal&quot;·or·&quot;randomPureBinomialIdeal&quot;,·or·taking·larger·numbers·of·examples.·Click·the·link·&quot;Finding·Extreme·Examples&quot;·below·to·see·some·other,·more·elaborate·ways·to·search.</p>
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10 small·fields.·For·example...what·would·you·expect·the·regularities·of·"typical"10 small·fields.·For·example...what·would·you·expect·the·regularities·of·"typical"
11 monomial·ideals·with·10·generators·of·degree·3·in·6·variables·to·be?·Try·a11 monomial·ideals·with·10·generators·of·degree·3·in·6·variables·to·be?·Try·a
12 bunch·of·examples·--·it's·fast.·Here·we·do·only·500·--·this·takes·about·a12 bunch·of·examples·--·it's·fast.·Here·we·do·only·500·--·this·takes·about·a
13 second·on·a·fast·machine·--·but·with·a·little·patience,·thousands·can·be·done13 second·on·a·fast·machine·--·but·with·a·little·patience,·thousands·can·be·done
14 conveniently.14 conveniently.
15 i1·:·setRandomSeed(currentTime())15 i1·:·setRandomSeed(currentTime())
  
16 o1·=·174896247116 o1·=·1783380567
17 i2·:·kk=ZZ/101;17 i2·:·kk=ZZ/101;
18 i3·:·S=kk[vars(0..5)];18 i3·:·S=kk[vars(0..5)];
19 i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)19 i4·:·time·tally·for·n·from·1·to·500·list·regularity·randomMonomialIdeal(10:3,S)
20 ·--·used·1.32939s·(cpu);·1.05608s·(thread);·0s·(gc)20 ·--·used·1.55s·(cpu);·1.17772s·(thread);·0s·(gc)
  
21 o4·=·Tally{4·=>·43·}21 o4·=·Tally{4·=>·50·}
22 ···········5·=>·19522 ···········5·=>·214
23 ···········6·=>·18723 ···········6·=>·159
24 ···········7·=>·6424 ···········7·=>·59
25 ···········8·=>·1025 ···········8·=>·17
26 ···········9·=>·126 ···········9·=>·1
  
27 o4·:·Tally27 o4·:·Tally
28 How·does·this·compare·with·the·case·of·binomial·ideals?·or·pure·binomial28 How·does·this·compare·with·the·case·of·binomial·ideals?·or·pure·binomial
29 ideals?·We·invite·the·reader·to·experiment,·replacing·"randomMonomialIdeal"29 ideals?·We·invite·the·reader·to·experiment,·replacing·"randomMonomialIdeal"
30 above·with·"randomBinomialIdeal"·or·"randomPureBinomialIdeal",·or·taking·larger30 above·with·"randomBinomialIdeal"·or·"randomPureBinomialIdeal",·or·taking·larger
31 numbers·of·examples.·Click·the·link·"Finding·Extreme·Examples"·below·to·see31 numbers·of·examples.·Click·the·link·"Finding·Extreme·Examples"·below·to·see
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./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=127 #:len=12
8 VmFyaWFibGVOYW1l8 VmFyaWFibGVOYW1l
9 #:len=12419 #:len=1241
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl
11 IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu11 IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu
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./usr/share/doc/Macaulay2/RandomObjects/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 QXR0ZW1wdHM=8 QXR0ZW1wdHM=
9 #:len=5029 #:len=502
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo
11 ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp11 ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp
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./usr/share/doc/Macaulay2/RandomPlaneCurves/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=497 #:len=49
8 Y29tcGxldGVMaW5lYXJTeXN0ZW1Pbk5vZGFsUGxhbmVDdXJ2ZShJZGVhbCxMaXN0KQ==8 Y29tcGxldGVMaW5lYXJTeXN0ZW1Pbk5vZGFsUGxhbmVDdXJ2ZShJZGVhbCxMaXN0KQ==
9 #:len=3829 #:len=382
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc2LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV0ZUxpbmVhclN5c3RlbU9uTm9kYWxQbGFu11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV0ZUxpbmVhclN5c3RlbU9uTm9kYWxQbGFu
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./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=467 #:len=46
8 ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ==8 ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ==
9 #:len=3159 #:len=315
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg5MSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg5MSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F011 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F0
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./usr/share/doc/Macaulay2/RandomPoints/example-output/_dim__Via__Bezout.out
    
Offset 5, 17 lines modifiedOffset 5, 17 lines modified
5 i2·:·S=kk[y_0..y_8];5 i2·:·S=kk[y_0..y_8];
  
6 i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;6 i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;
  
7 o3·:·Ideal·of·S7 o3·:·Ideal·of·S
  
8 i4·:·elapsedTime·dimViaBezout(I)8 i4·:·elapsedTime·dimViaBezout(I)
9 ·--·1.96318s·elapsed9 ·--·1.69929s·elapsed
  
10 o4·=·410 o4·=·4
  
11 i5·:·elapsedTime·dim·I11 i5·:·elapsedTime·dim·I
12 ·--·4.75969s·elapsed12 ·--·4.26792s·elapsed
  
13 o5·=·413 o5·=·4
  
14 i6·:·14 i6·:·
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./usr/share/doc/Macaulay2/RandomPoints/example-output/_extend__Ideal__By__Non__Zero__Minor.out
    
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35 i8·:·i·=·0;35 i8·:·i·=·0;
  
36 i9·:·J·=·I;36 i9·:·J·=·I;
  
37 o9·:·Ideal·of·T37 o9·:·Ideal·of·T
  
38 i10·:·elapsedTime(while·(i·<·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=·extendIdealByNonZeroMinor(4,·M,·J))·);38 i10·:·elapsedTime(while·(i·<·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=·extendIdealByNonZeroMinor(4,·M,·J))·);
39 ·--·2.63409s·elapsed39 ·--·1.79424s·elapsed
  
40 i11·:·dim·J40 i11·:·dim·J
  
41 o11·=·141 o11·=·1
  
42 i12·:·i42 i12·:·i
  
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./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out
    
Offset 27, 24 lines modifiedOffset 27, 24 lines modified
27 i6·:·S=ZZ/103[y_0..y_30];27 i6·:·S=ZZ/103[y_0..y_30];
  
28 i7·:·I=minors(2,random(S^3,S^{3:-1}));28 i7·:·I=minors(2,random(S^3,S^{3:-1}));
  
29 o7·:·Ideal·of·S29 o7·:·Ideal·of·S
  
30 i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>MultiplicationTable)30 i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>MultiplicationTable)
31 ·--·3.9442s·elapsed31 ·--·3.74561s·elapsed
  
32 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,32 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,
33 ·····------------------------------------------------------------------------33 ·····------------------------------------------------------------------------
34 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}34 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}
  
35 o8·:·List35 o8·:·List
  
36 i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>Decompose)36 i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>Decompose)
37 ·--·3.99026s·elapsed37 ·--·2.33444s·elapsed
  
38 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,38 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}40 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}
  
41 o9·:·List41 o9·:·List
  
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./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html
    
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94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;95 <td>··············<pre><code·class="language-macaulay2">i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;
  
96 o3·:·Ideal·of·S</code></pre>96 o3·:·Ideal·of·S</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·dimViaBezout(I)99 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·dimViaBezout(I)
100 ·--·1.96318s·elapsed100 ·--·1.69929s·elapsed
  
101 o4·=·4</code></pre>101 o4·=·4</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·dim·I104 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·dim·I
105 ·--·4.75969s·elapsed105 ·--·4.26792s·elapsed
  
106 o5·=·4</code></pre>106 o5·=·4</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ········</table>108 ········</table>
109 ········<div>109 ········<div>
110 ··········<p>The·user·may·set·the·<span·class="tt">MinimumFieldSize</span>·to·ensure·that·the·field·being·worked·over·is·big·enough.··For·instance,·there·are·relatively·few·linear·spaces·over·a·field·of·characteristic·2,·and·this·can·cause·incorrect·results·to·be·provided.··If·no·size·is·provided,·the·function·tries·to·guess·a·good·size·based·on·ambient·ring.</p>110 ··········<p>The·user·may·set·the·<span·class="tt">MinimumFieldSize</span>·to·ensure·that·the·field·being·worked·over·is·big·enough.··For·instance,·there·are·relatively·few·linear·spaces·over·a·field·of·characteristic·2,·and·this·can·cause·incorrect·results·to·be·provided.··If·no·size·is·provided,·the·function·tries·to·guess·a·good·size·based·on·ambient·ring.</p>
111 ········</div>111 ········</div>
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33 examples,·the·built·in·dim·function·is·much·faster.33 examples,·the·built·in·dim·function·is·much·faster.
34 i1·:·kk=ZZ/101;34 i1·:·kk=ZZ/101;
35 i2·:·S=kk[y_0..y_8];35 i2·:·S=kk[y_0..y_8];
36 i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;36 i3·:·I=ideal·random(S^1,S^{-2,-2,-2,-2})+(ideal·random(2,S))^2;
  
37 o3·:·Ideal·of·S37 o3·:·Ideal·of·S
38 i4·:·elapsedTime·dimViaBezout(I)38 i4·:·elapsedTime·dimViaBezout(I)
39 ·--·1.96318s·elapsed39 ·--·1.69929s·elapsed
  
40 o4·=·440 o4·=·4
41 i5·:·elapsedTime·dim·I41 i5·:·elapsedTime·dim·I
42 ·--·4.75969s·elapsed42 ·--·4.26792s·elapsed
  
43 o5·=·443 o5·=·4
44 The·user·may·set·the·MinimumFieldSize·to·ensure·that·the·field·being·worked44 The·user·may·set·the·MinimumFieldSize·to·ensure·that·the·field·being·worked
45 over·is·big·enough.·For·instance,·there·are·relatively·few·linear·spaces·over·a45 over·is·big·enough.·For·instance,·there·are·relatively·few·linear·spaces·over·a
46 field·of·characteristic·2,·and·this·can·cause·incorrect·results·to·be·provided.46 field·of·characteristic·2,·and·this·can·cause·incorrect·results·to·be·provided.
47 If·no·size·is·provided,·the·function·tries·to·guess·a·good·size·based·on47 If·no·size·is·provided,·the·function·tries·to·guess·a·good·size·based·on
48 ambient·ring.48 ambient·ring.
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149 ··········<tr>149 ··········<tr>
150 <td>··············<pre><code·class="language-macaulay2">i9·:·J·=·I;150 <td>··············<pre><code·class="language-macaulay2">i9·:·J·=·I;
  
151 o9·:·Ideal·of·T</code></pre>151 o9·:·Ideal·of·T</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime(while·(i·&lt;·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=·extendIdealByNonZeroMinor(4,·M,·J))·);154 <td>··············<pre><code·class="language-macaulay2">i10·:·elapsedTime(while·(i·&lt;·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=·extendIdealByNonZeroMinor(4,·M,·J))·);
155 ·--·2.63409s·elapsed</code></pre>155 ·--·1.79424s·elapsed</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i11·:·dim·J158 <td>··············<pre><code·class="language-macaulay2">i11·:·dim·J
  
159 o11·=·1</code></pre>159 o11·=·1</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
161 ··········<tr>161 ··········<tr>
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80 o7·:·Matrix·T··<--·T80 o7·:·Matrix·T··<--·T
81 i8·:·i·=·0;81 i8·:·i·=·0;
82 i9·:·J·=·I;82 i9·:·J·=·I;
  
83 o9·:·Ideal·of·T83 o9·:·Ideal·of·T
84 i10·:·elapsedTime(while·(i·<·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=84 i10·:·elapsedTime(while·(i·<·10)·and·dim·J·>·1·do·(i·=·i+1;·J·=
85 extendIdealByNonZeroMinor(4,·M,·J))·);85 extendIdealByNonZeroMinor(4,·M,·J))·);
86 ·--·2.63409s·elapsed86 ·--·1.79424s·elapsed
87 i11·:·dim·J87 i11·:·dim·J
  
88 o11·=·188 o11·=·1
89 i12·:·i89 i12·:·i
  
90 o12·=·490 o12·=·4
91 In·this·particular·example,·there·tend·to·be·about·5·associated·primes·when91 In·this·particular·example,·there·tend·to·be·about·5·associated·primes·when
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./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html
    
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141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i7·:·I=minors(2,random(S^3,S^{3:-1}));142 <td>··············<pre><code·class="language-macaulay2">i7·:·I=minors(2,random(S^3,S^{3:-1}));
  
143 o7·:·Ideal·of·S</code></pre>143 o7·:·Ideal·of·S</code></pre>
144 </td>··········</tr>144 </td>··········</tr>
145 ··········<tr>145 ··········<tr>
146 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>MultiplicationTable)146 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>MultiplicationTable)
147 ·--·3.9442s·elapsed147 ·--·3.74561s·elapsed
  
148 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,148 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,
149 ·····------------------------------------------------------------------------149 ·····------------------------------------------------------------------------
150 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}150 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}
  
151 o8·:·List</code></pre>151 o8·:·List</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>Decompose)154 <td>··············<pre><code·class="language-macaulay2">i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,·DecompositionStrategy=>Decompose)
155 ·--·3.99026s·elapsed155 ·--·2.33444s·elapsed
  
156 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,156 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,
157 ·····------------------------------------------------------------------------157 ·····------------------------------------------------------------------------
158 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}158 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}
  
159 o9·:·List</code></pre>159 o9·:·List</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
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Offset 67, 24 lines modifiedOffset 67, 24 lines modified
67 first·in·rings·with·more·variables.67 first·in·rings·with·more·variables.
68 i6·:·S=ZZ/103[y_0..y_30];68 i6·:·S=ZZ/103[y_0..y_30];
69 i7·:·I=minors(2,random(S^3,S^{3:-1}));69 i7·:·I=minors(2,random(S^3,S^{3:-1}));
  
70 o7·:·Ideal·of·S70 o7·:·Ideal·of·S
71 i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,71 i8·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,
72 DecompositionStrategy=>MultiplicationTable)72 DecompositionStrategy=>MultiplicationTable)
73 ·--·3.9442s·elapsed73 ·--·3.74561s·elapsed
  
74 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,74 o8·=·{{-4,·-35,·-7,·0,·0,·1,·5,·-13,·0,·-47,·0,·41,·0,·-51,·-46,·35,·0,·0,
75 ·····------------------------------------------------------------------------75 ·····------------------------------------------------------------------------
76 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}76 ·····-47,·14,·-30,·42,·30,·4,·-41,·24,·0,·0,·15,·20,·1}}
  
77 o8·:·List77 o8·:·List
78 i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,78 i9·:·elapsedTime·randomPoints(I,·Strategy=>LinearIntersection,
79 DecompositionStrategy=>Decompose)79 DecompositionStrategy=>Decompose)
80 ·--·3.99026s·elapsed80 ·--·2.33444s·elapsed
  
81 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,81 o9·=·{{11,·9,·-9,·-15,·-7,·27,·19,·-36,·48,·26,·-4,·3,·29,·-8,·7,·-32,·16,
82 ·····------------------------------------------------------------------------82 ·····------------------------------------------------------------------------
83 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}83 ·····11,·7,·7,·25,·-14,·-39,·17,·-16,·4,·-50,·-12,·21,·-50,·51}}
  
84 o9·:·List84 o9·:·List
85 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mP\x8Po\x8oi\x8in\x8nt\x8ts\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*85 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8an\x8nd\x8do\x8om\x8mP\x8Po\x8oi\x8in\x8nt\x8ts\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
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./usr/share/doc/Macaulay2/RationalMaps/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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./usr/share/doc/Macaulay2/RationalMaps/example-output/_inverse__Of__Map.out
    
Offset 49, 15 lines modifiedOffset 49, 15 lines modified
49 i12·:·Q=QQ[x,y,z,t,u];49 i12·:·Q=QQ[x,y,z,t,u];
  
50 i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});50 i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
  
51 o13·:·RingMap·Q·<--·Q51 o13·:·RingMap·Q·<--·Q
  
52 i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)52 i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)
53 ·--·used·2.29236s·(cpu);·0.512094s·(thread);·0s·(gc)53 ·--·used·2.35606s·(cpu);·0.61485s·(thread);·0s·(gc)
  
54 ································125···124······120·5····124····100·25·····104·20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120········8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6··········28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70·11···········48·65·12···········52·60·13···········56·55·14···········60·50·15···········64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25·20·········84·20·21········88·15·22·······92·10·23······96·5·24····100·25·····24·100········28·95··········32·90·2·········36·85·3··········40·80·4··········44·75·5···········48·70·6···········52·65·7···········56·60·8···········60·55·9···········64·50·10···········68·45·11···········72·40·12···········76·35·13···········80·30·14··········84·25·15··········88·20·16·········92·15·17········96·10·18········100·5·19······104·20·······48·75·2·······52·70···2········56·65·2·2········60·60·3·2·········64·55·4·2·········68·50·5·2·········72·45·6·2·········76·40·7·2·········80·35·8·2·········84·30·9·2·········88·25·10·2·········92·20·11·2········96·15·12·2········100·10·13·2·······104·5·14·2······108·15·2······72·50·3·······76·45···3·······80·40·2·3········84·35·3·3········88·30·4·3········92·25·5·3········96·20·6·3········100·15·7·3·······104·10·8·3·······108·5·9·3······112·10·3·····96·25·4······100·20···4······104·15·2·4······108·10·3·4······112·5·4·4·····116·5·4····120·5····12454 ································125···124······120·5····124····100·25·····104·20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120········8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6··········28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70·11···········48·65·12···········52·60·13···········56·55·14···········60·50·15···········64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25·20·········84·20·21········88·15·22·······92·10·23······96·5·24····100·25·····24·100········28·95··········32·90·2·········36·85·3··········40·80·4··········44·75·5···········48·70·6···········52·65·7···········56·60·8···········60·55·9···········64·50·10···········68·45·11···········72·40·12···········76·35·13···········80·30·14··········84·25·15··········88·20·16·········92·15·17········96·10·18········100·5·19······104·20·······48·75·2·······52·70···2········56·65·2·2········60·60·3·2·········64·55·4·2·········68·50·5·2·········72·45·6·2·········76·40·7·2·········80·35·8·2·········84·30·9·2·········88·25·10·2·········92·20·11·2········96·15·12·2········100·10·13·2·······104·5·14·2······108·15·2······72·50·3·······76·45···3·······80·40·2·3········84·35·3·3········88·30·4·3········92·25·5·3········96·20·6·3········100·15·7·3·······104·10·8·3·······108·5·9·3······112·10·3·····96·25·4······100·20···4······104·15·2·4······108·10·3·4······112·5·4·4·····116·5·4····120·5····124
55 o14·=·Proj·Q·-·-·-·>·Proj·Q···{x···,·x···y,·-·x···y··+·x···z,·x···y···-·5x···y··z·+·10x···y··z··-·10x···y··z··+·5x···y·z··-·x···z··+·x···t,·-·y····+·25x·y···z·-·300x·y···z··+·2300x··y···z··-·12650x··y···z··+·53130x··y···z··-·177100x··y··z··+·480700x··y··z··-·1081575x··y··z··+·2042975x··y··z··-·3268760x··y··z···+·4457400x··y··z···-·5200300x··y··z···+·5200300x··y··z···-·4457400x··y··z···+·3268760x··y··z···-·2042975x··y··z···+·1081575x··y··z···-·480700x··y··z···+·177100x··y··z···-·53130x··y··z···+·12650x··y··z···-·2300x··y··z···+·300x··y··z···-·25x··y·z···+·x···z···-·5x··y···t·+·100x··y··z*t·-·950x··y··z·t·+·5700x··y··z·t·-·24225x··y··z·t·+·77520x··y··z·t·-·193800x··y··z·t·+·387600x··y··z·t·-·629850x··y··z·t·+·839800x··y··z·t·-·923780x··y··z··t·+·839800x··y··z··t·-·629850x··y··z··t·+·387600x··y··z··t·-·193800x··y··z··t·+·77520x··y··z··t·-·24225x··y··z··t·+·5700x··y··z··t·-·950x··y··z··t·+·100x···y·z··t·-·5x···z··t·-·10x··y··t··+·150x··y··z*t··-·1050x··y··z·t··+·4550x··y··z·t··-·13650x··y··z·t··+·30030x··y··z·t··-·50050x··y··z·t··+·64350x··y··z·t··-·64350x··y··z·t··+·50050x··y··z·t··-·30030x··y··z··t··+·13650x··y··z··t··-·4550x··y··z··t··+·1050x···y··z··t··-·150x···y·z··t··+·10x···z··t··-·10x··y··t··+·100x··y··z*t··-·450x··y··z·t··+·1200x··y··z·t··-·2100x··y··z·t··+·2520x··y··z·t··-·2100x··y··z·t··+·1200x···y··z·t··-·450x···y··z·t··+·100x···y·z·t··-·10x···z··t··-·5x··y··t··+·25x···y··z*t··-·50x···y··z·t··+·50x···y··z·t··-·25x···y·z·t··+·5x···z·t··-·x···t··+·x···u}55 o14·=·Proj·Q·-·-·-·>·Proj·Q···{x···,·x···y,·-·x···y··+·x···z,·x···y···-·5x···y··z·+·10x···y··z··-·10x···y··z··+·5x···y·z··-·x···z··+·x···t,·-·y····+·25x·y···z·-·300x·y···z··+·2300x··y···z··-·12650x··y···z··+·53130x··y···z··-·177100x··y··z··+·480700x··y··z··-·1081575x··y··z··+·2042975x··y··z··-·3268760x··y··z···+·4457400x··y··z···-·5200300x··y··z···+·5200300x··y··z···-·4457400x··y··z···+·3268760x··y··z···-·2042975x··y··z···+·1081575x··y··z···-·480700x··y··z···+·177100x··y··z···-·53130x··y··z···+·12650x··y··z···-·2300x··y··z···+·300x··y··z···-·25x··y·z···+·x···z···-·5x··y···t·+·100x··y··z*t·-·950x··y··z·t·+·5700x··y··z·t·-·24225x··y··z·t·+·77520x··y··z·t·-·193800x··y··z·t·+·387600x··y··z·t·-·629850x··y··z·t·+·839800x··y··z·t·-·923780x··y··z··t·+·839800x··y··z··t·-·629850x··y··z··t·+·387600x··y··z··t·-·193800x··y··z··t·+·77520x··y··z··t·-·24225x··y··z··t·+·5700x··y··z··t·-·950x··y··z··t·+·100x···y·z··t·-·5x···z··t·-·10x··y··t··+·150x··y··z*t··-·1050x··y··z·t··+·4550x··y··z·t··-·13650x··y··z·t··+·30030x··y··z·t··-·50050x··y··z·t··+·64350x··y··z·t··-·64350x··y··z·t··+·50050x··y··z·t··-·30030x··y··z··t··+·13650x··y··z··t··-·4550x··y··z··t··+·1050x···y··z··t··-·150x···y·z··t··+·10x···z··t··-·10x··y··t··+·100x··y··z*t··-·450x··y··z·t··+·1200x··y··z·t··-·2100x··y··z·t··+·2520x··y··z·t··-·2100x··y··z·t··+·1200x···y··z·t··-·450x···y··z·t··+·100x···y·z·t··-·10x···z··t··-·5x··y··t··+·25x···y··z*t··-·50x···y··z·t··+·50x···y··z·t··-·25x···y·z·t··+·5x···z·t··-·x···t··+·x···u}
  
56 o14·:·RationalMapping56 o14·:·RationalMapping
  
57 i15·:·R=QQ[x,y,z,t]/(z-2*t);57 i15·:·R=QQ[x,y,z,t]/(z-2*t);
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./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html
    
Offset 171, 15 lines modifiedOffset 171, 15 lines modified
171 ··········<tr>171 ··········<tr>
172 <td>··············<pre><code·class="language-macaulay2">i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});172 <td>··············<pre><code·class="language-macaulay2">i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
  
173 o13·:·RingMap·Q·&lt;--·Q</code></pre>173 o13·:·RingMap·Q·&lt;--·Q</code></pre>
174 </td>··········</tr>174 </td>··········</tr>
175 ··········<tr>175 ··········<tr>
176 <td>··············<pre><code·class="language-macaulay2">i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)176 <td>··············<pre><code·class="language-macaulay2">i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)
177 ·--·used·2.29236s·(cpu);·0.512094s·(thread);·0s·(gc)177 ·--·used·2.35606s·(cpu);·0.61485s·(thread);·0s·(gc)
  
178 ································125···124······120·5····124····100·25·····104·20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120········8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6··········28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70·11···········48·65·12···········52·60·13···········56·55·14···········60·50·15···········64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25·20·········84·20·21········88·15·22·······92·10·23······96·5·24····100·25·····24·100········28·95··········32·90·2·········36·85·3··········40·80·4··········44·75·5···········48·70·6···········52·65·7···········56·60·8···········60·55·9···········64·50·10···········68·45·11···········72·40·12···········76·35·13···········80·30·14··········84·25·15··········88·20·16·········92·15·17········96·10·18········100·5·19······104·20·······48·75·2·······52·70···2········56·65·2·2········60·60·3·2·········64·55·4·2·········68·50·5·2·········72·45·6·2·········76·40·7·2·········80·35·8·2·········84·30·9·2·········88·25·10·2·········92·20·11·2········96·15·12·2········100·10·13·2·······104·5·14·2······108·15·2······72·50·3·······76·45···3·······80·40·2·3········84·35·3·3········88·30·4·3········92·25·5·3········96·20·6·3········100·15·7·3·······104·10·8·3·······108·5·9·3······112·10·3·····96·25·4······100·20···4······104·15·2·4······108·10·3·4······112·5·4·4·····116·5·4····120·5····124178 ································125···124······120·5····124····100·25·····104·20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120········8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6··········28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70·11···········48·65·12···········52·60·13···········56·55·14···········60·50·15···········64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25·20·········84·20·21········88·15·22·······92·10·23······96·5·24····100·25·····24·100········28·95··········32·90·2·········36·85·3··········40·80·4··········44·75·5···········48·70·6···········52·65·7···········56·60·8···········60·55·9···········64·50·10···········68·45·11···········72·40·12···········76·35·13···········80·30·14··········84·25·15··········88·20·16·········92·15·17········96·10·18········100·5·19······104·20·······48·75·2·······52·70···2········56·65·2·2········60·60·3·2·········64·55·4·2·········68·50·5·2·········72·45·6·2·········76·40·7·2·········80·35·8·2·········84·30·9·2·········88·25·10·2·········92·20·11·2········96·15·12·2········100·10·13·2·······104·5·14·2······108·15·2······72·50·3·······76·45···3·······80·40·2·3········84·35·3·3········88·30·4·3········92·25·5·3········96·20·6·3········100·15·7·3·······104·10·8·3·······108·5·9·3······112·10·3·····96·25·4······100·20···4······104·15·2·4······108·10·3·4······112·5·4·4·····116·5·4····120·5····124
179 o14·=·Proj·Q·-·-·-·>·Proj·Q···{x···,·x···y,·-·x···y··+·x···z,·x···y···-·5x···y··z·+·10x···y··z··-·10x···y··z··+·5x···y·z··-·x···z··+·x···t,·-·y····+·25x·y···z·-·300x·y···z··+·2300x··y···z··-·12650x··y···z··+·53130x··y···z··-·177100x··y··z··+·480700x··y··z··-·1081575x··y··z··+·2042975x··y··z··-·3268760x··y··z···+·4457400x··y··z···-·5200300x··y··z···+·5200300x··y··z···-·4457400x··y··z···+·3268760x··y··z···-·2042975x··y··z···+·1081575x··y··z···-·480700x··y··z···+·177100x··y··z···-·53130x··y··z···+·12650x··y··z···-·2300x··y··z···+·300x··y··z···-·25x··y·z···+·x···z···-·5x··y···t·+·100x··y··z*t·-·950x··y··z·t·+·5700x··y··z·t·-·24225x··y··z·t·+·77520x··y··z·t·-·193800x··y··z·t·+·387600x··y··z·t·-·629850x··y··z·t·+·839800x··y··z·t·-·923780x··y··z··t·+·839800x··y··z··t·-·629850x··y··z··t·+·387600x··y··z··t·-·193800x··y··z··t·+·77520x··y··z··t·-·24225x··y··z··t·+·5700x··y··z··t·-·950x··y··z··t·+·100x···y·z··t·-·5x···z··t·-·10x··y··t··+·150x··y··z*t··-·1050x··y··z·t··+·4550x··y··z·t··-·13650x··y··z·t··+·30030x··y··z·t··-·50050x··y··z·t··+·64350x··y··z·t··-·64350x··y··z·t··+·50050x··y··z·t··-·30030x··y··z··t··+·13650x··y··z··t··-·4550x··y··z··t··+·1050x···y··z··t··-·150x···y·z··t··+·10x···z··t··-·10x··y··t··+·100x··y··z*t··-·450x··y··z·t··+·1200x··y··z·t··-·2100x··y··z·t··+·2520x··y··z·t··-·2100x··y··z·t··+·1200x···y··z·t··-·450x···y··z·t··+·100x···y·z·t··-·10x···z··t··-·5x··y··t··+·25x···y··z*t··-·50x···y··z·t··+·50x···y··z·t··-·25x···y·z·t··+·5x···z·t··-·x···t··+·x···u}179 o14·=·Proj·Q·-·-·-·>·Proj·Q···{x···,·x···y,·-·x···y··+·x···z,·x···y···-·5x···y··z·+·10x···y··z··-·10x···y··z··+·5x···y·z··-·x···z··+·x···t,·-·y····+·25x·y···z·-·300x·y···z··+·2300x··y···z··-·12650x··y···z··+·53130x··y···z··-·177100x··y··z··+·480700x··y··z··-·1081575x··y··z··+·2042975x··y··z··-·3268760x··y··z···+·4457400x··y··z···-·5200300x··y··z···+·5200300x··y··z···-·4457400x··y··z···+·3268760x··y··z···-·2042975x··y··z···+·1081575x··y··z···-·480700x··y··z···+·177100x··y··z···-·53130x··y··z···+·12650x··y··z···-·2300x··y··z···+·300x··y··z···-·25x··y·z···+·x···z···-·5x··y···t·+·100x··y··z*t·-·950x··y··z·t·+·5700x··y··z·t·-·24225x··y··z·t·+·77520x··y··z·t·-·193800x··y··z·t·+·387600x··y··z·t·-·629850x··y··z·t·+·839800x··y··z·t·-·923780x··y··z··t·+·839800x··y··z··t·-·629850x··y··z··t·+·387600x··y··z··t·-·193800x··y··z··t·+·77520x··y··z··t·-·24225x··y··z··t·+·5700x··y··z··t·-·950x··y··z··t·+·100x···y·z··t·-·5x···z··t·-·10x··y··t··+·150x··y··z*t··-·1050x··y··z·t··+·4550x··y··z·t··-·13650x··y··z·t··+·30030x··y··z·t··-·50050x··y··z·t··+·64350x··y··z·t··-·64350x··y··z·t··+·50050x··y··z·t··-·30030x··y··z··t··+·13650x··y··z··t··-·4550x··y··z··t··+·1050x···y··z··t··-·150x···y·z··t··+·10x···z··t··-·10x··y··t··+·100x··y··z*t··-·450x··y··z·t··+·1200x··y··z·t··-·2100x··y··z·t··+·2520x··y··z·t··-·2100x··y··z·t··+·1200x···y··z·t··-·450x···y··z·t··+·100x···y·z·t··-·10x···z··t··-·5x··y··t··+·25x···y··z*t··-·50x···y··z·t··+·50x···y··z·t··-·25x···y·z·t··+·5x···z·t··-·x···t··+·x···u}
  
180 o14·:·RationalMapping</code></pre>180 o14·:·RationalMapping</code></pre>
181 </td>··········</tr>181 </td>··········</tr>
182 ········</table>182 ········</table>
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Offset 95, 15 lines modifiedOffset 95, 15 lines modified
95 o11·:·Ideal·of·blowUpSubvar95 o11·:·Ideal·of·blowUpSubvar
96 The·next·example·is·a·birational·map·on·$\mathbb{P}^4$.96 The·next·example·is·a·birational·map·on·$\mathbb{P}^4$.
97 i12·:·Q=QQ[x,y,z,t,u];97 i12·:·Q=QQ[x,y,z,t,u];
98 i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});98 i13·:·phi=map(Q,Q,matrix{{x^5,y*x^4,z*x^4+y^5,t*x^4+z^5,u*x^4+t^5}});
  
99 o13·:·RingMap·Q·<--·Q99 o13·:·RingMap·Q·<--·Q
100 i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)100 i14·:·time·inverseOfMap(phi,CheckBirational=>false,·Verbosity=>0)
101 ·--·used·2.29236s·(cpu);·0.512094s·(thread);·0s·(gc)101 ·--·used·2.35606s·(cpu);·0.61485s·(thread);·0s·(gc)
  
102 ································125···124······120·5····124····100·25·····104102 ································125···124······120·5····124····100·25·····104
103 20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120103 20·······108·15·2······112·10·3·····116·5·4····120·5····124······125······4·120
104 8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6104 8·115·2········12·110·3·········16·105·4·········20·100·5··········24·95·6
105 28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70105 28·90·7···········32·85·8···········36·80·9···········40·75·10···········44·70
106 11···········48·65·12···········52·60·13···········56·55·14···········60·50·15106 11···········48·65·12···········52·60·13···········56·55·14···········60·50·15
107 64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25107 64·45·16···········68·40·17··········72·35·18··········76·30·19·········80·25
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./usr/share/doc/Macaulay2/RationalPoints/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 U29ydEdlbnM=8 U29ydEdlbnM=
9 #:len=2469 #:len=246
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ
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./usr/share/doc/Macaulay2/RationalPoints2/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=207 #:len=20
8 bmV0KFByb2plY3RpdmVQb2ludCk=8 bmV0KFByb2plY3RpdmVQb2ludCk=
9 #:len=2059 #:len=205
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy
11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q11 dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q
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1.27 KB
./usr/share/doc/Macaulay2/RationalPoints2/example-output/_rational__Points.out
    
Offset 48, 15 lines modifiedOffset 48, 15 lines modified
48 ·············0····1····2····3····4····5····6····7····8····9····1048 ·············0····1····2····3····4····5····6····7····8····9····10
  
49 ················ZZ49 ················ZZ
50 o13·:·Ideal·of·---[u·..u··]50 o13·:·Ideal·of·---[u·..u··]
51 ···············101··0···1051 ···············101··0···10
  
52 i14·:·time·rationalPoints(I,·Amount·=>·true)52 i14·:·time·rationalPoints(I,·Amount·=>·true)
53 ·--·used·0.00291924s·(cpu);·0.00333793s·(thread);·0s·(gc)53 ·--·used·0.00399481s·(cpu);·0.00378789s·(thread);·0s·(gc)
  
54 o14·=·11046221254112045100154 o14·=·110462212541120451001
  
55 i15·:·QQ[x,y,z];·I·=·homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)),·z);55 i15·:·QQ[x,y,z];·I·=·homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)),·z);
  
56 o16·:·Ideal·of·QQ[x..z]56 o16·:·Ideal·of·QQ[x..z]
  
Offset 141, 24 lines modifiedOffset 141, 24 lines modified
141 o30·:·Ideal·of·R141 o30·:·Ideal·of·R
  
142 i31·:·nodes·=·I·+·ideal·jacobian·I;142 i31·:·nodes·=·I·+·ideal·jacobian·I;
  
143 o31·:·Ideal·of·R143 o31·:·Ideal·of·R
  
144 i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);144 i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);
145 ·--·used·0.745649s·(cpu);·0.627411s·(thread);·0s·(gc)145 ·--·used·0.915937s·(cpu);·0.744727s·(thread);·0s·(gc)
146 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)146 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
  
147 i33·:·#oo147 i33·:·#oo
  
148 o33·=·31148 o33·=·31
  
149 i34·:·nodes'·=·baseChange_32003·nodes;149 i34·:·nodes'·=·baseChange_32003·nodes;
  
150 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]150 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]
  
151 i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)151 i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)
152 ·--·used·0.211654s·(cpu);·0.170283s·(thread);·0s·(gc)152 ·--·used·0.229179s·(cpu);·0.170806s·(thread);·0s·(gc)
  
153 o35·=·31153 o35·=·31
  
154 i36·:·154 i36·:·
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./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html
    
Offset 167, 15 lines modifiedOffset 167, 15 lines modified
  
167 ················ZZ167 ················ZZ
168 o13·:·Ideal·of·---[u·..u··]168 o13·:·Ideal·of·---[u·..u··]
169 ···············101··0···10</code></pre>169 ···············101··0···10</code></pre>
170 </td>··········</tr>170 </td>··········</tr>
171 ··········<tr>171 ··········<tr>
172 <td>··············<pre><code·class="language-macaulay2">i14·:·time·rationalPoints(I,·Amount·=>·true)172 <td>··············<pre><code·class="language-macaulay2">i14·:·time·rationalPoints(I,·Amount·=>·true)
173 ·--·used·0.00291924s·(cpu);·0.00333793s·(thread);·0s·(gc)173 ·--·used·0.00399481s·(cpu);·0.00378789s·(thread);·0s·(gc)
  
174 o14·=·110462212541120451001</code></pre>174 o14·=·110462212541120451001</code></pre>
175 </td>··········</tr>175 </td>··········</tr>
176 ········</table>176 ········</table>
177 ········<div>177 ········<div>
178 ··········<h3>Over·number·fields</h3>178 ··········<h3>Over·number·fields</h3>
179 ········</div>179 ········</div>
Offset 308, 15 lines modifiedOffset 308, 15 lines modified
308 ··········<tr>308 ··········<tr>
309 <td>··············<pre><code·class="language-macaulay2">i31·:·nodes·=·I·+·ideal·jacobian·I;309 <td>··············<pre><code·class="language-macaulay2">i31·:·nodes·=·I·+·ideal·jacobian·I;
  
310 o31·:·Ideal·of·R</code></pre>310 o31·:·Ideal·of·R</code></pre>
311 </td>··········</tr>311 </td>··········</tr>
312 ··········<tr>312 ··········<tr>
313 <td>··············<pre><code·class="language-macaulay2">i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);313 <td>··············<pre><code·class="language-macaulay2">i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);
314 ·--·used·0.745649s·(cpu);·0.627411s·(thread);·0s·(gc)314 ·--·used·0.915937s·(cpu);·0.744727s·(thread);·0s·(gc)
315 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)</code></pre>315 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)</code></pre>
316 </td>··········</tr>316 </td>··········</tr>
317 ··········<tr>317 ··········<tr>
318 <td>··············<pre><code·class="language-macaulay2">i33·:·#oo318 <td>··············<pre><code·class="language-macaulay2">i33·:·#oo
  
319 o33·=·31</code></pre>319 o33·=·31</code></pre>
320 </td>··········</tr>320 </td>··········</tr>
Offset 328, 15 lines modifiedOffset 328, 15 lines modified
328 ··········<tr>328 ··········<tr>
329 <td>··············<pre><code·class="language-macaulay2">i34·:·nodes'·=·baseChange_32003·nodes;329 <td>··············<pre><code·class="language-macaulay2">i34·:·nodes'·=·baseChange_32003·nodes;
  
330 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]</code></pre>330 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]</code></pre>
331 </td>··········</tr>331 </td>··········</tr>
332 ··········<tr>332 ··········<tr>
333 <td>··············<pre><code·class="language-macaulay2">i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)333 <td>··············<pre><code·class="language-macaulay2">i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)
334 ·--·used·0.211654s·(cpu);·0.170283s·(thread);·0s·(gc)334 ·--·used·0.229179s·(cpu);·0.170806s·(thread);·0s·(gc)
  
335 o35·=·31</code></pre>335 o35·=·31</code></pre>
336 </td>··········</tr>336 </td>··········</tr>
337 ········</table>337 ········</table>
338 ······</div>338 ······</div>
339 ······<div>339 ······<div>
340 ········<h2>Caveat</h2>340 ········<h2>Caveat</h2>
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Offset 90, 15 lines modifiedOffset 90, 15 lines modified
90 o13·=·ideal(u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··)90 o13·=·ideal(u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··+·u··)
91 ·············0····1····2····3····4····5····6····7····8····9····1091 ·············0····1····2····3····4····5····6····7····8····9····10
  
92 ················ZZ92 ················ZZ
93 o13·:·Ideal·of·---[u·..u··]93 o13·:·Ideal·of·---[u·..u··]
94 ···············101··0···1094 ···············101··0···10
95 i14·:·time·rationalPoints(I,·Amount·=>·true)95 i14·:·time·rationalPoints(I,·Amount·=>·true)
96 ·--·used·0.00291924s·(cpu);·0.00333793s·(thread);·0s·(gc)96 ·--·used·0.00399481s·(cpu);·0.00378789s·(thread);·0s·(gc)
  
97 o14·=·11046221254112045100197 o14·=·110462212541120451001
98 *\x8**\x8**\x8**\x8*·O\x8Ov\x8ve\x8er\x8r·n\x8nu\x8um\x8mb\x8be\x8er\x8r·f\x8fi\x8ie\x8el\x8ld\x8ds\x8s·*\x8**\x8**\x8**\x8*98 *\x8**\x8**\x8**\x8*·O\x8Ov\x8ve\x8er\x8r·n\x8nu\x8um\x8mb\x8be\x8er\x8r·f\x8fi\x8ie\x8el\x8ld\x8ds\x8s·*\x8**\x8**\x8**\x8*
99 Over·a·number·field·one·can·use·the·option·Bound·to·specify·a·maximal99 Over·a·number·field·one·can·use·the·option·Bound·to·specify·a·maximal
100 multiplicative·height·given·by·$(x_0:\dots:x_n)\mapsto·\prod_{v}\max_i|x_i|_v·^100 multiplicative·height·given·by·$(x_0:\dots:x_n)\mapsto·\prod_{v}\max_i|x_i|_v·^
101 {d_v/d}$·(this·is·also·available·as·a·method·_\x8g_\x8l_\x8o_\x8b_\x8a_\x8l_\x8H_\x8e_\x8i_\x8g_\x8h_\x8t).101 {d_v/d}$·(this·is·also·available·as·a·method·_\x8g_\x8l_\x8o_\x8b_\x8a_\x8l_\x8H_\x8e_\x8i_\x8g_\x8h_\x8t).
102 i15·:·QQ[x,y,z];·I·=·homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)),·z);102 i15·:·QQ[x,y,z];·I·=·homogenize(ideal(y^2-x*(x-1)*(x-2)*(x-5)*(x-6)),·z);
Offset 197, 25 lines modifiedOffset 197, 25 lines modified
197 (2z-qw)(4(x2+y2-z2)+(1+3(5-q2))w2)2";197 (2z-qw)(4(x2+y2-z2)+(1+3(5-q2))w2)2";
  
198 o30·:·Ideal·of·R198 o30·:·Ideal·of·R
199 i31·:·nodes·=·I·+·ideal·jacobian·I;199 i31·:·nodes·=·I·+·ideal·jacobian·I;
  
200 o31·:·Ideal·of·R200 o31·:·Ideal·of·R
201 i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);201 i32·:·time·rationalPoints(variety·nodes,·Split=>true,·Verbose=>true);
202 ·--·used·0.745649s·(cpu);·0.627411s·(thread);·0s·(gc)202 ·--·used·0.915937s·(cpu);·0.744727s·(thread);·0s·(gc)
203 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)203 --·base·change·to·the·field·QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
204 i33·:·#oo204 i33·:·#oo
  
205 o33·=·31205 o33·=·31
206 Still·it·runs·a·lot·faster·when·reduced·to·a·positive·characteristic.206 Still·it·runs·a·lot·faster·when·reduced·to·a·positive·characteristic.
207 i34·:·nodes'·=·baseChange_32003·nodes;207 i34·:·nodes'·=·baseChange_32003·nodes;
  
208 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]208 o34·:·Ideal·of·GF·1048969271299456081[x..z,·w]
209 i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)209 i35·:·time·#rationalPoints(variety·nodes',·Split=>true,·Verbose=>true)
210 ·--·used·0.211654s·(cpu);·0.170283s·(thread);·0s·(gc)210 ·--·used·0.229179s·(cpu);·0.170806s·(thread);·0s·(gc)
  
211 o35·=·31211 o35·=·31
212 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*212 *\x8**\x8**\x8**\x8**\x8*·C\x8Ca\x8av\x8ve\x8ea\x8at\x8t·*\x8**\x8**\x8**\x8**\x8*
213 For·a·number·field·other·than·QQ,·the·enumeration·of·elements·with·bounded213 For·a·number·field·other·than·QQ,·the·enumeration·of·elements·with·bounded
214 height·depends·on·an·algorithm·by·Doyle–Krumm,·which·is·currently·only214 height·depends·on·an·algorithm·by·Doyle–Krumm,·which·is·currently·only
215 implemented·in·Sage.215 implemented·in·Sage.
216 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lP\x8Po\x8oi\x8in\x8nt\x8ts\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*216 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·r\x8ra\x8at\x8ti\x8io\x8on\x8na\x8al\x8lP\x8Po\x8oi\x8in\x8nt\x8ts\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
7 #:len=307 #:len=30
8 bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp8 bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp
9 #:len=3449 #:len=344
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp
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./usr/share/doc/Macaulay2/RealRoots/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=147 #:len=14
8 U3lsdmVzdGVyQ291bnQ=8 U3lsdmVzdGVyQ291bnQ=
9 #:len=21459 #:len=2145
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGRpZmZlcmVuY2UgaW4gdmFyaWF010 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGRpZmZlcmVuY2UgaW4gdmFyaWF0
11 aW9ucyBvZiB0aGUgU3lsdmVzdGVyIHNlcXVlbmNlIG9mIHR3byByYXRpb25hbCB1bml2YXJpYXRl11 aW9ucyBvZiB0aGUgU3lsdmVzdGVyIHNlcXVlbmNlIG9mIHR3byByYXRpb25hbCB1bml2YXJpYXRl
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./usr/share/doc/Macaulay2/ReesAlgebra/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=207 #:len=20
8 aXNMaW5lYXJUeXBlKE1vZHVsZSk=8 aXNMaW5lYXJUeXBlKE1vZHVsZSk=
9 #:len=2579 #:len=257
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIxMCwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIxMCwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy
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./usr/share/doc/Macaulay2/ReesAlgebra/example-output/___Plane__Curve__Singularities.out
    
Offset 331, 15 lines modifiedOffset 331, 15 lines modified
331 ···············2·2····2···2·············2·2····2331 ···············2·2····2···2·············2·2····2
332 ······-·p·w·,·p·y··-·p·,·p·w·y·-·p·p·,·p·w··-·p·)332 ······-·p·w·,·p·y··-·p·,·p·w·y·-·p·p·,·p·w··-·p·)
333 ·········2·1···0······1···0·0·····1·2···0·0····2333 ·········2·1···0······1···0·0·····1·2···0·0····2
  
334 o47·:·Ideal·of·B2334 o47·:·Ideal·of·B2
  
335 i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;335 i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;
336 ·--·used·0.711338s·(cpu);·0.617792s·(thread);·0s·(gc)336 ·--·used·0.906964s·(cpu);·0.795793s·(thread);·0s·(gc)
  
337 ·················ZZ337 ·················ZZ
338 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]338 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]
339 ···············32003··0···2···0···1339 ···············32003··0···2···0···1
  
340 i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))340 i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))
  
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./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_expected__Rees__Ideal.out
    
Offset 57, 15 lines modifiedOffset 57, 15 lines modified
57 o5·:·Matrix·S··<--·S57 o5·:·Matrix·S··<--·S
  
58 i6·:·I·=·minors(n-1,·M);58 i6·:·I·=·minors(n-1,·M);
  
59 o6·:·Ideal·of·S59 o6·:·Ideal·of·S
  
60 i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·<·1·sec.60 i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·<·1·sec.
61 ·--·used·1.35211s·(cpu);·0.594392s·(thread);·0s·(gc)61 ·--·used·1.53546s·(cpu);·0.870243s·(thread);·0s·(gc)
  
62 o7·:·Ideal·of·S[w·..w·]62 o7·:·Ideal·of·S[w·..w·]
63 ·················0···463 ·················0···4
  
64 i8·:·kk·=·ZZ/101;64 i8·:·kk·=·ZZ/101;
  
65 i9·:·S·=·kk[x,y,z];65 i9·:·S·=·kk[x,y,z];
Offset 76, 19 lines modifiedOffset 76, 19 lines modified
76 o10·:·Matrix·S··<--·S76 o10·:·Matrix·S··<--·S
  
77 i11·:·I·=·minors(3,m);77 i11·:·I·=·minors(3,m);
  
78 o11·:·Ideal·of·S78 o11·:·Ideal·of·S
  
79 i12·:·time·reesIdeal·(I,·I_0);79 i12·:·time·reesIdeal·(I,·I_0);
80 ·--·used·1.32402s·(cpu);·0.958128s·(thread);·0s·(gc)80 ·--·used·1.68004s·(cpu);·1.20197s·(thread);·0s·(gc)
  
81 o12·:·Ideal·of·S[w·..w·]81 o12·:·Ideal·of·S[w·..w·]
82 ··················0···382 ··················0···3
  
83 i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);83 i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);
84 ·--·used·1.14388s·(cpu);·0.99447s·(thread);·0s·(gc)84 ·--·used·1.31339s·(cpu);·1.08531s·(thread);·0s·(gc)
  
85 o13·:·Ideal·of·S[w·..w·]85 o13·:·Ideal·of·S[w·..w·]
86 ··················0···386 ··················0···3
  
87 i14·:·87 i14·:·
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./usr/share/doc/Macaulay2/ReesAlgebra/example-output/_rees__Ideal.out
    
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13 ················3····213 ················3····2
14 ·····-·x·x·x·,·x··-·x·x·)14 ·····-·x·x·x·,·x··-·x·x·)
15 ········0·2·4···1····0·415 ········0·2·4···1····0·4
  
16 o3·:·Ideal·of·S16 o3·:·Ideal·of·S
  
17 i4·:·time·V1·=·reesIdeal·i;17 i4·:·time·V1·=·reesIdeal·i;
18 ·--·used·1.81112s·(cpu);·0.207173s·(thread);·0s·(gc)18 ·--·used·1.44928s·(cpu);·0.222208s·(thread);·0s·(gc)
  
19 o4·:·Ideal·of·S[w·..w·]19 o4·:·Ideal·of·S[w·..w·]
20 ·················0···620 ·················0···6
  
21 i5·:·time·V2·=·reesIdeal(i,i_0);21 i5·:·time·V2·=·reesIdeal(i,i_0);
22 ·--·used·0.803722s·(cpu);·0.194966s·(thread);·0s·(gc)22 ·--·used·0.701859s·(cpu);·0.200337s·(thread);·0s·(gc)
  
23 o5·:·Ideal·of·S[w·..w·]23 o5·:·Ideal·of·S[w·..w·]
24 ·················0···624 ·················0···6
  
25 i6·:·S=kk[a,b,c]25 i6·:·S=kk[a,b,c]
  
26 o6·=·S26 o6·=·S
Offset 47, 21 lines modifiedOffset 47, 21 lines modified
  
47 ·············2········247 ·············2········2
48 o8·=·ideal·(a·,·a*b,·b·)48 o8·=·ideal·(a·,·a*b,·b·)
  
49 o8·:·Ideal·of·S49 o8·:·Ideal·of·S
  
50 i9·:·time·I1·=·reesIdeal·i;50 i9·:·time·I1·=·reesIdeal·i;
51 ·--·used·0.855879s·(cpu);·0.102717s·(thread);·0s·(gc)51 ·--·used·0.785306s·(cpu);·0.116432s·(thread);·0s·(gc)
  
52 o9·:·Ideal·of·S[w·..w·]52 o9·:·Ideal·of·S[w·..w·]
53 ·················0···253 ·················0···2
  
54 i10·:·time·I2·=·reesIdeal(i,i_0);54 i10·:·time·I2·=·reesIdeal(i,i_0);
55 ·--·used·0.257167s·(cpu);·0.0465803s·(thread);·0s·(gc)55 ·--·used·0.136396s·(cpu);·0.0182049s·(thread);·0s·(gc)
  
56 o10·:·Ideal·of·S[w·..w·]56 o10·:·Ideal·of·S[w·..w·]
57 ··················0···257 ··················0···2
  
58 i11·:·transpose·gens·I158 i11·:·transpose·gens·I1
  
59 o11·=·{-1,·-3}·|·aw_1-bw_2····|59 o11·=·{-1,·-3}·|·aw_1-bw_2····|
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./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html
    
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485 ········</table>485 ········</table>
486 ········<div>486 ········<div>
487 ··········<p>We·compute·the·singular·locus·once·again:</p>487 ··········<p>We·compute·the·singular·locus·once·again:</p>
488 ········</div>488 ········</div>
489 ········<table·class="examples">489 ········<table·class="examples">
490 ··········<tr>490 ··········<tr>
491 <td>··············<pre><code·class="language-macaulay2">i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;491 <td>··············<pre><code·class="language-macaulay2">i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;
492 ·--·used·0.711338s·(cpu);·0.617792s·(thread);·0s·(gc)492 ·--·used·0.906964s·(cpu);·0.795793s·(thread);·0s·(gc)
  
493 ·················ZZ493 ·················ZZ
494 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]494 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]
495 ···············32003··0···2···0···1</code></pre>495 ···············32003··0···2···0···1</code></pre>
496 </td>··········</tr>496 </td>··········</tr>
497 ··········<tr>497 ··········<tr>
498 <td>··············<pre><code·class="language-macaulay2">i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))498 <td>··············<pre><code·class="language-macaulay2">i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))
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325 ···············2·2····2···2·············2·2····2325 ···············2·2····2···2·············2·2····2
326 ······-·p·w·,·p·y··-·p·,·p·w·y·-·p·p·,·p·w··-·p·)326 ······-·p·w·,·p·y··-·p·,·p·w·y·-·p·p·,·p·w··-·p·)
327 ·········2·1···0······1···0·0·····1·2···0·0····2327 ·········2·1···0······1···0·0·····1·2···0·0····2
  
328 o47·:·Ideal·of·B2328 o47·:·Ideal·of·B2
329 We·compute·the·singular·locus·once·again:329 We·compute·the·singular·locus·once·again:
330 i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;330 i48·:·time·sing2·=·ideal·singularLocus·strictTransform2;
331 ·--·used·0.711338s·(cpu);·0.617792s·(thread);·0s·(gc)331 ·--·used·0.906964s·(cpu);·0.795793s·(thread);·0s·(gc)
  
332 ·················ZZ332 ·················ZZ
333 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]333 o48·:·Ideal·of·-----[p·..p·,·w·..w·,·x..y]
334 ···············32003··0···2···0···1334 ···············32003··0···2···0···1
335 i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))335 i49·:·saturate(sing2,·sub(irrelTot,·ring·sing2))
  
336 o49·=·ideal·1336 o49·=·ideal·1
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Offset 138, 15 lines modifiedOffset 138, 15 lines modified
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·minors(n-1,·M);139 <td>··············<pre><code·class="language-macaulay2">i6·:·I·=·minors(n-1,·M);
  
140 o6·:·Ideal·of·S</code></pre>140 o6·:·Ideal·of·S</code></pre>
141 </td>··········</tr>141 </td>··········</tr>
142 ··········<tr>142 ··········<tr>
143 <td>··············<pre><code·class="language-macaulay2">i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·&lt;·1·sec.143 <td>··············<pre><code·class="language-macaulay2">i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·&lt;·1·sec.
144 ·--·used·1.35211s·(cpu);·0.594392s·(thread);·0s·(gc)144 ·--·used·1.53546s·(cpu);·0.870243s·(thread);·0s·(gc)
  
145 o7·:·Ideal·of·S[w·..w·]145 o7·:·Ideal·of·S[w·..w·]
146 ·················0···4</code></pre>146 ·················0···4</code></pre>
147 </td>··········</tr>147 </td>··········</tr>
148 ··········<tr>148 ··········<tr>
149 <td>··············<pre><code·class="language-macaulay2">i8·:·kk·=·ZZ/101;</code></pre>149 <td>··············<pre><code·class="language-macaulay2">i8·:·kk·=·ZZ/101;</code></pre>
150 </td>··········</tr>150 </td>··········</tr>
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162 ··········<tr>162 ··········<tr>
163 <td>··············<pre><code·class="language-macaulay2">i11·:·I·=·minors(3,m);163 <td>··············<pre><code·class="language-macaulay2">i11·:·I·=·minors(3,m);
  
164 o11·:·Ideal·of·S</code></pre>164 o11·:·Ideal·of·S</code></pre>
165 </td>··········</tr>165 </td>··········</tr>
166 ··········<tr>166 ··········<tr>
167 <td>··············<pre><code·class="language-macaulay2">i12·:·time·reesIdeal·(I,·I_0);167 <td>··············<pre><code·class="language-macaulay2">i12·:·time·reesIdeal·(I,·I_0);
168 ·--·used·1.32402s·(cpu);·0.958128s·(thread);·0s·(gc)168 ·--·used·1.68004s·(cpu);·1.20197s·(thread);·0s·(gc)
  
169 o12·:·Ideal·of·S[w·..w·]169 o12·:·Ideal·of·S[w·..w·]
170 ··················0···3</code></pre>170 ··················0···3</code></pre>
171 </td>··········</tr>171 </td>··········</tr>
172 ··········<tr>172 ··········<tr>
173 <td>··············<pre><code·class="language-macaulay2">i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);173 <td>··············<pre><code·class="language-macaulay2">i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);
174 ·--·used·1.14388s·(cpu);·0.99447s·(thread);·0s·(gc)174 ·--·used·1.31339s·(cpu);·1.08531s·(thread);·0s·(gc)
  
175 o13·:·Ideal·of·S[w·..w·]175 o13·:·Ideal·of·S[w·..w·]
176 ··················0···3</code></pre>176 ··················0···3</code></pre>
177 </td>··········</tr>177 </td>··········</tr>
178 ········</table>178 ········</table>
179 ······</div>179 ······</div>
180 ······<div>180 ······<div>
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86 ·············5······486 ·············5······4
87 o5·:·Matrix·S··<--·S87 o5·:·Matrix·S··<--·S
88 i6·:·I·=·minors(n-1,·M);88 i6·:·I·=·minors(n-1,·M);
  
89 o6·:·Ideal·of·S89 o6·:·Ideal·of·S
90 i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·<·1·sec.90 i7·:·time·rI·=·expectedReesIdeal·I;·--·n=·5·case·takes·<·1·sec.
91 ·--·used·1.35211s·(cpu);·0.594392s·(thread);·0s·(gc)91 ·--·used·1.53546s·(cpu);·0.870243s·(thread);·0s·(gc)
  
92 o7·:·Ideal·of·S[w·..w·]92 o7·:·Ideal·of·S[w·..w·]
93 ·················0···493 ·················0···4
94 i8·:·kk·=·ZZ/101;94 i8·:·kk·=·ZZ/101;
95 i9·:·S·=·kk[x,y,z];95 i9·:·S·=·kk[x,y,z];
96 i10·:·m·=·random(S^3,·S^{4:-2});96 i10·:·m·=·random(S^3,·S^{4:-2});
  
97 ··············3······497 ··············3······4
98 o10·:·Matrix·S··<--·S98 o10·:·Matrix·S··<--·S
99 i11·:·I·=·minors(3,m);99 i11·:·I·=·minors(3,m);
  
100 o11·:·Ideal·of·S100 o11·:·Ideal·of·S
101 i12·:·time·reesIdeal·(I,·I_0);101 i12·:·time·reesIdeal·(I,·I_0);
102 ·--·used·1.32402s·(cpu);·0.958128s·(thread);·0s·(gc)102 ·--·used·1.68004s·(cpu);·1.20197s·(thread);·0s·(gc)
  
103 o12·:·Ideal·of·S[w·..w·]103 o12·:·Ideal·of·S[w·..w·]
104 ··················0···3104 ··················0···3
105 i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);105 i13·:·time·reesIdeal·(I,·I_0,·Jacobian·=>false);
106 ·--·used·1.14388s·(cpu);·0.99447s·(thread);·0s·(gc)106 ·--·used·1.31339s·(cpu);·1.08531s·(thread);·0s·(gc)
  
107 o13·:·Ideal·of·S[w·..w·]107 o13·:·Ideal·of·S[w·..w·]
108 ··················0···3108 ··················0···3
109 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*109 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
110 ····*·_\x8s_\x8y_\x8m_\x8m_\x8e_\x8t_\x8r_\x8i_\x8c_\x8A_\x8l_\x8g_\x8e_\x8b_\x8r_\x8a_\x8I_\x8d_\x8e_\x8a_\x8l·--·Ideal·of·the·symmetric·algebra·of·an·ideal·or110 ····*·_\x8s_\x8y_\x8m_\x8m_\x8e_\x8t_\x8r_\x8i_\x8c_\x8A_\x8l_\x8g_\x8e_\x8b_\x8r_\x8a_\x8I_\x8d_\x8e_\x8a_\x8l·--·Ideal·of·the·symmetric·algebra·of·an·ideal·or
111 ······module111 ······module
112 ····*·_\x8j_\x8a_\x8c_\x8o_\x8b_\x8i_\x8a_\x8n_\x8D_\x8u_\x8a_\x8l·--·Computes·the·'jacobian·dual',·part·of·a·method·of·finding112 ····*·_\x8j_\x8a_\x8c_\x8o_\x8b_\x8i_\x8a_\x8n_\x8D_\x8u_\x8a_\x8l·--·Computes·the·'jacobian·dual',·part·of·a·method·of·finding
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113 ·····-·x·x·x·,·x··-·x·x·)113 ·····-·x·x·x·,·x··-·x·x·)
114 ········0·2·4···1····0·4114 ········0·2·4···1····0·4
  
115 o3·:·Ideal·of·S</code></pre>115 o3·:·Ideal·of·S</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i4·:·time·V1·=·reesIdeal·i;118 <td>··············<pre><code·class="language-macaulay2">i4·:·time·V1·=·reesIdeal·i;
119 ·--·used·1.81112s·(cpu);·0.207173s·(thread);·0s·(gc)119 ·--·used·1.44928s·(cpu);·0.222208s·(thread);·0s·(gc)
  
120 o4·:·Ideal·of·S[w·..w·]120 o4·:·Ideal·of·S[w·..w·]
121 ·················0···6</code></pre>121 ·················0···6</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
124 <td>··············<pre><code·class="language-macaulay2">i5·:·time·V2·=·reesIdeal(i,i_0);124 <td>··············<pre><code·class="language-macaulay2">i5·:·time·V2·=·reesIdeal(i,i_0);
125 ·--·used·0.803722s·(cpu);·0.194966s·(thread);·0s·(gc)125 ·--·used·0.701859s·(cpu);·0.200337s·(thread);·0s·(gc)
  
126 o5·:·Ideal·of·S[w·..w·]126 o5·:·Ideal·of·S[w·..w·]
127 ·················0···6</code></pre>127 ·················0···6</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ········</table>129 ········</table>
130 ········<div>130 ········<div>
131 ··········<p>The·following·example·shows·how·we·handle·degrees</p>131 ··········<p>The·following·example·shows·how·we·handle·degrees</p>
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157 ·············2········2157 ·············2········2
158 o8·=·ideal·(a·,·a*b,·b·)158 o8·=·ideal·(a·,·a*b,·b·)
  
159 o8·:·Ideal·of·S</code></pre>159 o8·:·Ideal·of·S</code></pre>
160 </td>··········</tr>160 </td>··········</tr>
161 ··········<tr>161 ··········<tr>
162 <td>··············<pre><code·class="language-macaulay2">i9·:·time·I1·=·reesIdeal·i;162 <td>··············<pre><code·class="language-macaulay2">i9·:·time·I1·=·reesIdeal·i;
163 ·--·used·0.855879s·(cpu);·0.102717s·(thread);·0s·(gc)163 ·--·used·0.785306s·(cpu);·0.116432s·(thread);·0s·(gc)
  
164 o9·:·Ideal·of·S[w·..w·]164 o9·:·Ideal·of·S[w·..w·]
165 ·················0···2</code></pre>165 ·················0···2</code></pre>
166 </td>··········</tr>166 </td>··········</tr>
167 ··········<tr>167 ··········<tr>
168 <td>··············<pre><code·class="language-macaulay2">i10·:·time·I2·=·reesIdeal(i,i_0);168 <td>··············<pre><code·class="language-macaulay2">i10·:·time·I2·=·reesIdeal(i,i_0);
169 ·--·used·0.257167s·(cpu);·0.0465803s·(thread);·0s·(gc)169 ·--·used·0.136396s·(cpu);·0.0182049s·(thread);·0s·(gc)
  
170 o10·:·Ideal·of·S[w·..w·]170 o10·:·Ideal·of·S[w·..w·]
171 ··················0···2</code></pre>171 ··················0···2</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i11·:·transpose·gens·I1174 <td>··············<pre><code·class="language-macaulay2">i11·:·transpose·gens·I1
  
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52 ·····------------------------------------------------------------------------52 ·····------------------------------------------------------------------------
53 ················3····253 ················3····2
54 ·····-·x·x·x·,·x··-·x·x·)54 ·····-·x·x·x·,·x··-·x·x·)
55 ········0·2·4···1····0·455 ········0·2·4···1····0·4
  
56 o3·:·Ideal·of·S56 o3·:·Ideal·of·S
57 i4·:·time·V1·=·reesIdeal·i;57 i4·:·time·V1·=·reesIdeal·i;
58 ·--·used·1.81112s·(cpu);·0.207173s·(thread);·0s·(gc)58 ·--·used·1.44928s·(cpu);·0.222208s·(thread);·0s·(gc)
  
59 o4·:·Ideal·of·S[w·..w·]59 o4·:·Ideal·of·S[w·..w·]
60 ·················0···660 ·················0···6
61 i5·:·time·V2·=·reesIdeal(i,i_0);61 i5·:·time·V2·=·reesIdeal(i,i_0);
62 ·--·used·0.803722s·(cpu);·0.194966s·(thread);·0s·(gc)62 ·--·used·0.701859s·(cpu);·0.200337s·(thread);·0s·(gc)
  
63 o5·:·Ideal·of·S[w·..w·]63 o5·:·Ideal·of·S[w·..w·]
64 ·················0···664 ·················0···6
65 The·following·example·shows·how·we·handle·degrees65 The·following·example·shows·how·we·handle·degrees
66 i6·:·S=kk[a,b,c]66 i6·:·S=kk[a,b,c]
  
67 o6·=·S67 o6·=·S
Offset 82, 20 lines modifiedOffset 82, 20 lines modified
82 i8·:·i=minors(2,m)82 i8·:·i=minors(2,m)
  
83 ·············2········283 ·············2········2
84 o8·=·ideal·(a·,·a*b,·b·)84 o8·=·ideal·(a·,·a*b,·b·)
  
85 o8·:·Ideal·of·S85 o8·:·Ideal·of·S
86 i9·:·time·I1·=·reesIdeal·i;86 i9·:·time·I1·=·reesIdeal·i;
87 ·--·used·0.855879s·(cpu);·0.102717s·(thread);·0s·(gc)87 ·--·used·0.785306s·(cpu);·0.116432s·(thread);·0s·(gc)
  
88 o9·:·Ideal·of·S[w·..w·]88 o9·:·Ideal·of·S[w·..w·]
89 ·················0···289 ·················0···2
90 i10·:·time·I2·=·reesIdeal(i,i_0);90 i10·:·time·I2·=·reesIdeal(i,i_0);
91 ·--·used·0.257167s·(cpu);·0.0465803s·(thread);·0s·(gc)91 ·--·used·0.136396s·(cpu);·0.0182049s·(thread);·0s·(gc)
  
92 o10·:·Ideal·of·S[w·..w·]92 o10·:·Ideal·of·S[w·..w·]
93 ··················0···293 ··················0···2
94 i11·:·transpose·gens·I194 i11·:·transpose·gens·I1
  
95 o11·=·{-1,·-3}·|·aw_1-bw_2····|95 o11·=·{-1,·-3}·|·aw_1-bw_2····|
96 ······{-1,·-3}·|·aw_0-bw_1····|96 ······{-1,·-3}·|·aw_0-bw_1····|
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7 #:len=77 #:len=7
8 S1NFbnRyeQ==8 S1NFbnRyeQ==
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl
11 ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl11 ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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11 b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ11 b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ
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71 ·····x·x·x·,·x··+·x·x··-·x·x··-·x·x·x·,·x··+·x··-·x·x·)71 ·····x·x·x·,·x··+·x·x··-·x·x··-·x·x·x·,·x··+·x··-·x·x·)
72 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·572 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·5
  
73 o7·:·Ideal·of·R73 o7·:·Ideal·of·R
  
74 i8·:·benchmark·"mRegularity·I1"74 i8·:·benchmark·"mRegularity·I1"
  
75 o8·=·.175138187812575 o8·=·.19877517275
  
76 o8·:·RR·(of·precision·53)76 o8·:·RR·(of·precision·53)
  
77 i9·:·R·=·QQ[x_0..x_5]77 i9·:·R·=·QQ[x_0..x_5]
  
78 o9·=·R78 o9·=·R
  
Offset 87, 17 lines modifiedOffset 87, 17 lines modified
  
87 i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,·x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);87 i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,·x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);
  
88 o10·:·Ideal·of·R88 o10·:·Ideal·of·R
  
89 i11·:·benchmark·"·mRegularity·I2"89 i11·:·benchmark·"·mRegularity·I2"
  
90 o11·=·.0477446167391304590 o11·=·.05226691768421053
  
91 o11·:·RR·(of·precision·53)91 o11·:·RR·(of·precision·53)
  
92 i12·:·time·regularity·I2··92 i12·:·time·regularity·I2··
93 ·--·used·0.00364171s·(cpu);·0.00181831s·(thread);·0s·(gc)93 ·--·used·0.00140045s·(cpu);·0.00187346s·(thread);·0s·(gc)
  
94 o12·=·494 o12·=·4
  
95 i13·:·95 i13·:·
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./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html
    
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165 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·5165 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·5
  
166 o7·:·Ideal·of·R</code></pre>166 o7·:·Ideal·of·R</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i8·:·benchmark·&quot;mRegularity·I1&quot;169 <td>··············<pre><code·class="language-macaulay2">i8·:·benchmark·&quot;mRegularity·I1&quot;
  
170 o8·=·.1751381878125170 o8·=·.19877517275
  
171 o8·:·RR·(of·precision·53)</code></pre>171 o8·:·RR·(of·precision·53)</code></pre>
172 </td>··········</tr>172 </td>··········</tr>
173 ········</table>173 ········</table>
174 ········<p>This·is·an·example·where·regularity·is·faster·than·mRegularity.</p>174 ········<p>This·is·an·example·where·regularity·is·faster·than·mRegularity.</p>
175 ········<table·class="examples">175 ········<table·class="examples">
176 ··········<tr>176 ··········<tr>
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187 <td>··············<pre><code·class="language-macaulay2">i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,·x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);187 <td>··············<pre><code·class="language-macaulay2">i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,·x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);
  
188 o10·:·Ideal·of·R</code></pre>188 o10·:·Ideal·of·R</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ··········<tr>190 ··········<tr>
191 <td>··············<pre><code·class="language-macaulay2">i11·:·benchmark·&quot;·mRegularity·I2&quot;191 <td>··············<pre><code·class="language-macaulay2">i11·:·benchmark·&quot;·mRegularity·I2&quot;
  
192 o11·=·.04774461673913045192 o11·=·.05226691768421053
  
193 o11·:·RR·(of·precision·53)</code></pre>193 o11·:·RR·(of·precision·53)</code></pre>
194 </td>··········</tr>194 </td>··········</tr>
195 ··········<tr>195 ··········<tr>
196 <td>··············<pre><code·class="language-macaulay2">i12·:·time·regularity·I2··196 <td>··············<pre><code·class="language-macaulay2">i12·:·time·regularity·I2··
197 ·--·used·0.00364171s·(cpu);·0.00181831s·(thread);·0s·(gc)197 ·--·used·0.00140045s·(cpu);·0.00187346s·(thread);·0s·(gc)
  
198 o12·=·4</code></pre>198 o12·=·4</code></pre>
199 </td>··········</tr>199 </td>··········</tr>
200 ········</table>200 ········</table>
201 ········<p>This·symbol·is·provided·by·the·package·Regularity.</p>201 ········<p>This·symbol·is·provided·by·the·package·Regularity.</p>
202 ······</div>202 ······</div>
203 ······<div>203 ······<div>
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html2text {}
    
Offset 95, 34 lines modifiedOffset 95, 34 lines modified
95 ··············3····2········2············3····3······295 ··············3····2········2············3····3······2
96 ·····x·x·x·,·x··+·x·x··-·x·x··-·x·x·x·,·x··+·x··-·x·x·)96 ·····x·x·x·,·x··+·x·x··-·x·x··-·x·x·x·,·x··+·x··-·x·x·)
97 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·597 ······0·1·3···0····0·1····1·2····0·2·5···0····2····0·5
  
98 o7·:·Ideal·of·R98 o7·:·Ideal·of·R
99 i8·:·benchmark·"mRegularity·I1"99 i8·:·benchmark·"mRegularity·I1"
  
100 o8·=·.1751381878125100 o8·=·.19877517275
  
101 o8·:·RR·(of·precision·53)101 o8·:·RR·(of·precision·53)
102 This·is·an·example·where·regularity·is·faster·than·mRegularity.102 This·is·an·example·where·regularity·is·faster·than·mRegularity.
103 i9·:·R·=·QQ[x_0..x_5]103 i9·:·R·=·QQ[x_0..x_5]
  
104 o9·=·R104 o9·=·R
  
105 o9·:·PolynomialRing105 o9·:·PolynomialRing
106 i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,106 i10·:·I2·=·ideal·(·x_0^2+x_5^2,·x_0^2+x_0*x_3+x_4^2,·x_0^2+x_0*x_5+x_2*x_5,
107 x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);107 x_0^2-x_0*x_3-x_3*x_5,·x_0^2-x_3*x_4,·x_0*x_3);
  
108 o10·:·Ideal·of·R108 o10·:·Ideal·of·R
109 i11·:·benchmark·"·mRegularity·I2"109 i11·:·benchmark·"·mRegularity·I2"
  
110 o11·=·.04774461673913045110 o11·=·.05226691768421053
  
111 o11·:·RR·(of·precision·53)111 o11·:·RR·(of·precision·53)
112 i12·:·time·regularity·I2112 i12·:·time·regularity·I2
113 ·--·used·0.00364171s·(cpu);·0.00181831s·(thread);·0s·(gc)113 ·--·used·0.00140045s·(cpu);·0.00187346s·(thread);·0s·(gc)
  
114 o12·=·4114 o12·=·4
115 This·symbol·is·provided·by·the·package·Regularity.115 This·symbol·is·provided·by·the·package·Regularity.
116 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*116 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
117 ····*·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8i_\x8t_\x8y·--·compute·the·Castelnuovo-Mumford·regularity117 ····*·_\x8r_\x8e_\x8g_\x8u_\x8l_\x8a_\x8r_\x8i_\x8t_\x8y·--·compute·the·Castelnuovo-Mumford·regularity
118 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8mR\x8Re\x8eg\x8gu\x8ul\x8la\x8ar\x8ri\x8it\x8ty\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*118 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·m\x8mR\x8Re\x8eg\x8gu\x8ul\x8la\x8ar\x8ri\x8it\x8ty\x8y:\x8:·*\x8**\x8**\x8**\x8**\x8*
119 ····*·mRegularity(Ideal)119 ····*·mRegularity(Ideal)
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./usr/share/doc/Macaulay2/RelativeCanonicalResolution/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/ResLengthThree/dump/rawdocumentation.dump
    
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs
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./usr/share/doc/Macaulay2/ResidualIntersections/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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7 #:len=157 #:len=15
8 Z2VuZXJpY1Jlc2lkdWFs8 Z2VuZXJpY1Jlc2lkdWFs
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11 bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4NiwgSW5wdXRzID0+11 bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4NiwgSW5wdXRzID0+
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805 B
./usr/share/doc/Macaulay2/ResolutionsOfStanleyReisnerRings/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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8 YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk=8 YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk=
9 #:len=3819 #:len=381
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI4LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp
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./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=357 #:len=35
8 aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik=8 aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik=
9 #:len=2779 #:len=277
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMzMCwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMzMCwgc3ltYm9sIERvY3VtZW50VGFn
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1.64 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_cayley__Trick.out
    
Offset 5, 18 lines modifiedOffset 5, 18 lines modified
5 o2·=·ideal(x·x··-·x·x·)5 o2·=·ideal(x·x··-·x·x·)
6 ············0·1····2·36 ············0·1····2·3
  
7 o2·:·Ideal·of·QQ[x·..x·]7 o2·:·Ideal·of·QQ[x·..x·]
8 ··················0···38 ··················0···3
  
9 i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);9 i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);
10 ·--·used·0.106059s·(cpu);·0.082427s·(thread);·0s·(gc)10 ·--·used·0.110492s·(cpu);·0.0708643s·(thread);·0s·(gc)
  
11 i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)11 i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)
12 ·--·used·0.0399344s·(cpu);·0.0420251s·(thread);·0s·(gc)12 ·--·used·0.064093s·(cpu);·0.0661522s·(thread);·0s·(gc)
  
13 ···········································································13 ···········································································
14 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,14 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
15 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,3·15 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,3·
16 ·····------------------------------------------------------------------------16 ·····------------------------------------------------------------------------
17 ············································································17 ············································································
18 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···18 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
Offset 37, 17 lines modifiedOffset 37, 17 lines modified
37 ··············································2···237 ··············································2···2
38 ·····4x···x···x···x····-·2x···x···x···x····+·x···x···))38 ·····4x···x···x···x····-·2x···x···x···x····+·x···x···))
39 ·······0,0·0,1·1,2·1,3·····0,2·0,3·1,2·1,3····0,2·1,339 ·······0,0·0,1·1,2·1,3·····0,2·0,3·1,2·1,3····0,2·1,3
  
40 o4·:·Sequence40 o4·:·Sequence
  
41 i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);41 i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);
42 ·--·used·0.105846s·(cpu);·0.0798293s·(thread);·0s·(gc)42 ·--·used·0.158759s·(cpu);·0.111621s·(thread);·0s·(gc)
  
43 i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))43 i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))
  
44 i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);44 i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);
45 ·--·used·0.111715s·(cpu);·0.0835642s·(thread);·0s·(gc)45 ·--·used·0.186531s·(cpu);·0.122305s·(thread);·0s·(gc)
  
46 i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))46 i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))
  
47 i9·:·47 i9·:·
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./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Equations.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 o2·=·ideal·(x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x·)9 o2·=·ideal·(x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x·)
10 ·············0····1····2····3···0·1····1·2····2·310 ·············0····1····2····3···0·1····1·2····2·3
  
11 o2·:·Ideal·of·P311 o2·:·Ideal·of·P3
  
12 i3·:·--·Chow·equations·of·C12 i3·:·--·Chow·equations·of·C
13 ·····time·eqsC·=·chowEquations·chowForm·C13 ·····time·eqsC·=·chowEquations·chowForm·C
14 ·--·used·0.07896s·(cpu);·0.0562216s·(thread);·0s·(gc)14 ·--·used·0.12423s·(cpu);·0.0825842s·(thread);·0s·(gc)
  
15 ·············2·2····2·2····2·2····4················2······2·2···2······15 ·············2·2····2·2····2·2····4················2······2·2···2······
16 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+16 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+
17 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·3··17 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·3··
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ······2········3···········2·········2······3···3·········2··········2····2·219 ······2········3···········2·········2······3···3·········2··········2····2·2
20 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x·20 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x·
Offset 72, 15 lines modifiedOffset 72, 15 lines modified
72 o5·=·ideal·(x··-·x·x·,·x··-·x·x·x·,·x·x··-·x·x·)72 o5·=·ideal·(x··-·x·x·,·x··-·x·x·x·,·x·x··-·x·x·)
73 ·············1····0·2···2····0·1·3···1·2····0·373 ·············1····0·2···2····0·1·3···1·2····0·3
  
74 o5·:·Ideal·of·P374 o5·:·Ideal·of·P3
  
75 i6·:·--·Chow·equations·of·D75 i6·:·--·Chow·equations·of·D
76 ·····time·eqsD·=·chowEquations·chowForm·D76 ·····time·eqsD·=·chowEquations·chowForm·D
77 ·--·used·0.0358775s·(cpu);·0.0342206s·(thread);·0s·(gc)77 ·--·used·0.0498103s·(cpu);·0.0494591s·(thread);·0s·(gc)
  
78 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···2·78 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···2·
79 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,79 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,
80 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·3·80 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·3·
81 ·····------------------------------------------------------------------------81 ·····------------------------------------------------------------------------
82 ··········2······3·2···3········3·2···4·········2·········2·2·······3····82 ··········2······3·2···3········3·2···4·········2·········2·2·······3····
83 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-83 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-
Offset 117, 24 lines modifiedOffset 117, 24 lines modified
117 o9·=·ideal(x·x··+·x·x·)117 o9·=·ideal(x·x··+·x·x·)
118 ············0·1····2·3118 ············0·1····2·3
  
119 o9·:·Ideal·of·P3119 o9·:·Ideal·of·P3
  
120 i10·:·--·tangential·Chow·forms·of·Q120 i10·:·--·tangential·Chow·forms·of·Q
121 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))121 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
122 ·--·used·0.110058s·(cpu);·0.0815315s·(thread);·0s·(gc)122 ·--·used·0.111642s·(cpu);·0.0794158s·(thread);·0s·(gc)
  
123 ·····················2······························2123 ·····················2······························2
124 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+124 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+
125 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3··125 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3··
126 ······-----------------------------------------------------------------------126 ······-----------------------------------------------------------------------
127 ······x·····x·····)127 ······x·····x·····)
128 ·······0,2,3·1,2,3128 ·······0,2,3·1,2,3
  
129 o10·:·Sequence129 o10·:·Sequence
  
130 i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))130 i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
131 ·--·used·0.102453s·(cpu);·0.0727086s·(thread);·0s·(gc)131 ·--·used·0.129249s·(cpu);·0.0839007s·(thread);·0s·(gc)
  
132 o11·=·true132 o11·=·true
  
133 i12·:·133 i12·:·
⊟
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./usr/share/doc/Macaulay2/Resultants/example-output/_chow__Form.out
    
Offset 16, 15 lines modifiedOffset 16, 15 lines modified
  
16 ···············ZZ16 ···············ZZ
17 o2·:·Ideal·of·----[x·..x·]17 o2·:·Ideal·of·----[x·..x·]
18 ··············3331··0···518 ··············3331··0···5
  
19 i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an·affine·chart·of·the·Grassmannian)19 i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an·affine·chart·of·the·Grassmannian)
20 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})20 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})
21 ·--·used·4.48232s·(cpu);·4.32064s·(thread);·0s·(gc)21 ·--·used·4.96101s·(cpu);·4.70115s·(thread);·0s·(gc)
  
22 ······4···············2··············2·····2···············2············22 ······4···············2··············2·····2···············2············
23 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+23 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+
24 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,5··24 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,5··
25 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
26 ······2·····2··············2·······································26 ······2·····2··············2·······································
27 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+27 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+
Offset 143, 19 lines modifiedOffset 143, 19 lines modified
143 ··········································································································································································································································································································································································································································································································································································································································································································································································3331··0,1,2···0,1,3···0,2,3···1,2,3···0,1,4···0,2,4···1,2,4···0,3,4···1,3,4···2,3,4···0,1,5···0,2,5···1,2,5···0,3,5···1,3,5···2,3,5···0,4,5···1,4,5···2,4,5···3,4,5143 ··········································································································································································································································································································································································································································································································································································································································································································································································3331··0,1,2···0,1,3···0,2,3···1,2,3···0,1,4···0,2,4···1,2,4···0,3,4···1,3,4···2,3,4···0,1,5···0,2,5···1,2,5···0,3,5···1,3,5···2,3,5···0,4,5···1,4,5···2,4,5···3,4,5
144 o3·:·-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------144 o3·:·-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
145 ·····(x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····)145 ·····(x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····)
146 ·······2,3,5·1,4,5····1,3,5·2,4,5····1,2,5·3,4,5···2,3,4·1,4,5····1,3,4·2,4,5····1,2,4·3,4,5···2,3,5·0,4,5····0,3,5·2,4,5····0,2,5·3,4,5···1,3,5·0,4,5····0,3,5·1,4,5····0,1,5·3,4,5···1,2,5·0,4,5····0,2,5·1,4,5····0,1,5·2,4,5···2,3,4·0,4,5····0,3,4·2,4,5····0,2,4·3,4,5···1,3,4·0,4,5····0,3,4·1,4,5····0,1,4·3,4,5···1,2,4·0,4,5····0,2,4·1,4,5····0,1,4·2,4,5···1,2,3·0,4,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·1,3,5····1,3,4·2,3,5····1,2,3·3,4,5···1,2,5·0,3,5····0,2,5·1,3,5····0,1,5·2,3,5···2,3,4·0,3,5····0,3,4·2,3,5····0,2,3·3,4,5···1,3,4·0,3,5····0,3,4·1,3,5····0,1,3·3,4,5···1,2,4·0,3,5····0,2,4·1,3,5····0,1,4·2,3,5····0,1,2·3,4,5···1,2,3·0,3,5····0,2,3·1,3,5····0,1,3·2,3,5···2,3,4·1,2,5····1,2,4·2,3,5····1,2,3·2,4,5···1,3,4·1,2,5····1,2,4·1,3,5····1,2,3·1,4,5···0,3,4·1,2,5····0,2,4·1,3,5····0,1,4·2,3,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·0,2,5····0,2,4·2,3,5····0,2,3·2,4,5···1,3,4·0,2,5····0,2,4·1,3,5····0,2,3·1,4,5····0,1,2·3,4,5···0,3,4·0,2,5····0,2,4·0,3,5····0,2,3·0,4,5···1,2,4·0,2,5····0,2,4·1,2,5····0,1,2·2,4,5···1,2,3·0,2,5····0,2,3·1,2,5····0,1,2·2,3,5···2,3,4·0,1,5····0,1,4·2,3,5····0,1,3·2,4,5····0,1,2·3,4,5···1,3,4·0,1,5····0,1,4·1,3,5····0,1,3·1,4,5···0,3,4·0,1,5····0,1,4·0,3,5····0,1,3·0,4,5···1,2,4·0,1,5····0,1,4·1,2,5····0,1,2·1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5····0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4146 ·······2,3,5·1,4,5····1,3,5·2,4,5····1,2,5·3,4,5···2,3,4·1,4,5····1,3,4·2,4,5····1,2,4·3,4,5···2,3,5·0,4,5····0,3,5·2,4,5····0,2,5·3,4,5···1,3,5·0,4,5····0,3,5·1,4,5····0,1,5·3,4,5···1,2,5·0,4,5····0,2,5·1,4,5····0,1,5·2,4,5···2,3,4·0,4,5····0,3,4·2,4,5····0,2,4·3,4,5···1,3,4·0,4,5····0,3,4·1,4,5····0,1,4·3,4,5···1,2,4·0,4,5····0,2,4·1,4,5····0,1,4·2,4,5···1,2,3·0,4,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·1,3,5····1,3,4·2,3,5····1,2,3·3,4,5···1,2,5·0,3,5····0,2,5·1,3,5····0,1,5·2,3,5···2,3,4·0,3,5····0,3,4·2,3,5····0,2,3·3,4,5···1,3,4·0,3,5····0,3,4·1,3,5····0,1,3·3,4,5···1,2,4·0,3,5····0,2,4·1,3,5····0,1,4·2,3,5····0,1,2·3,4,5···1,2,3·0,3,5····0,2,3·1,3,5····0,1,3·2,3,5···2,3,4·1,2,5····1,2,4·2,3,5····1,2,3·2,4,5···1,3,4·1,2,5····1,2,4·1,3,5····1,2,3·1,4,5···0,3,4·1,2,5····0,2,4·1,3,5····0,1,4·2,3,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·0,2,5····0,2,4·2,3,5····0,2,3·2,4,5···1,3,4·0,2,5····0,2,4·1,3,5····0,2,3·1,4,5····0,1,2·3,4,5···0,3,4·0,2,5····0,2,4·0,3,5····0,2,3·0,4,5···1,2,4·0,2,5····0,2,4·1,2,5····0,1,2·2,4,5···1,2,3·0,2,5····0,2,3·1,2,5····0,1,2·2,3,5···2,3,4·0,1,5····0,1,4·2,3,5····0,1,3·2,4,5····0,1,2·3,4,5···1,3,4·0,1,5····0,1,4·1,3,5····0,1,3·1,4,5···0,3,4·0,1,5····0,1,4·0,3,5····0,1,3·0,4,5···1,2,4·0,1,5····0,1,4·1,2,5····0,1,2·1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5····0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4
  
147 i4·:·--·equivalently·(but·faster)...147 i4·:·--·equivalently·(but·faster)...
148 ·····time·assert(ChowV·===·chowForm·f)148 ·····time·assert(ChowV·===·chowForm·f)
149 ·--·used·1.151s·(cpu);·1.08375s·(thread);·0s·(gc)149 ·--·used·1.02189s·(cpu);·0.908277s·(thread);·0s·(gc)
  
150 i5·:·--·X-resultant·of·V150 i5·:·--·X-resultant·of·V
151 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;151 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;
152 ·--·used·0.158288s·(cpu);·0.127679s·(thread);·0s·(gc)152 ·--·used·0.215244s·(cpu);·0.15912s·(thread);·0s·(gc)
  
153 i6·:·--·three·generic·ternary·quadrics153 i6·:·--·three·generic·ternary·quadrics
154 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)154 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)
  
155 ·········2···············2························2·····2···············2··155 ·········2···············2························2·····2···············2··
156 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+156 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+
157 ·······0·0····1·0·1····3·1····2·0·2····4·1·2····5·2···0·0····1·0·1····3·1··157 ·······0·0····1·0·1····3·1····2·0·2····4·1·2····5·2···0·0····1·0·1····3·1··
Offset 164, 12 lines modifiedOffset 164, 12 lines modified
164 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}164 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}
165 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2165 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
166 o6·:·List166 o6·:·List
  
167 i7·:·--·resultant·of·the·three·forms167 i7·:·--·resultant·of·the·three·forms
168 ·····time·resF·=·resultant·F;168 ·····time·resF·=·resultant·F;
169 ·--·used·0.170361s·(cpu);·0.138395s·(thread);·0s·(gc)169 ·--·used·0.242248s·(cpu);·0.18567s·(thread);·0s·(gc)
  
170 i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring·Xres))170 i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring·Xres))
  
171 i9·:·171 i9·:·
⊟
1.71 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_discriminant_lp__Ring__Element_rp.out
    
Offset 4, 30 lines modifiedOffset 4, 30 lines modified
  
4 ········2··············24 ········2··············2
5 o2·=·a*x··+·b*x*y·+·c*y5 o2·=·a*x··+·b*x*y·+·c*y
  
6 o2·:·ZZ[a..c][x..y]6 o2·:·ZZ[a..c][x..y]
  
7 i3·:·time·discriminant·F7 i3·:·time·discriminant·F
8 ·--·used·0.0115009s·(cpu);·0.00761948s·(thread);·0s·(gc)8 ·--·used·0.0120058s·(cpu);·0.00883976s·(thread);·0s·(gc)
  
9 ········29 ········2
10 o3·=·-·b··+·4a*c10 o3·=·-·b··+·4a*c
  
11 o3·:·ZZ[a..c]11 o3·:·ZZ[a..c]
  
12 i4·:·ZZ[a,b,c,d][x,y];·F·=·a*x^3+b*x^2*y+c*x*y^2+d*y^312 i4·:·ZZ[a,b,c,d][x,y];·F·=·a*x^3+b*x^2*y+c*x*y^2+d*y^3
  
13 ········3······2·········2······313 ········3······2·········2······3
14 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y14 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y
  
15 o5·:·ZZ[a..d][x..y]15 o5·:·ZZ[a..d][x..y]
  
16 i6·:·time·discriminant·F16 i6·:·time·discriminant·F
17 ·--·used·0.0769243s·(cpu);·0.0520463s·(thread);·0s·(gc)17 ·--·used·0.0596817s·(cpu);·0.0224293s·(thread);·0s·(gc)
  
18 ········2·2·······3·····3···················2·218 ········2·2·······3·····3···················2·2
19 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d19 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d
  
20 o6·:·ZZ[a..d]20 o6·:·ZZ[a..d]
  
21 i7·:·x=symbol·x;·R=ZZ/331[x_0..x_3]21 i7·:·x=symbol·x;·R=ZZ/331[x_0..x_3]
Offset 59, 15 lines modifiedOffset 59, 15 lines modified
59 ················4········3······4·············4········3······459 ················4········3······4·············4········3······4
60 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x60 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x
61 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·361 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·3
  
62 o12·:·R'62 o12·:·R'
  
63 i13·:·time·D=discriminant·pencil63 i13·:·time·D=discriminant·pencil
64 ·--·used·0.284848s·(cpu);·0.288677s·(thread);·0s·(gc)64 ·--·used·0.444882s·(cpu);·0.442355s·(thread);·0s·(gc)
  
65 ···········108······106·2·······102·6······100·8·······98·10·······96·12··65 ···········108······106·2·······102·6······100·8·······98·10·······96·12··
66 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+66 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+
67 ···········0········0···1·······0···1······0···1·······0··1········0··1···67 ···········0········0···1·······0···1······0···1·······0··1········0··1···
68 ······-----------------------------------------------------------------------68 ······-----------------------------------------------------------------------
69 ··········94·14·······92·16······90·18······88·20······86·22·······84·24··69 ··········94·14·······92·16······90·18······88·20······86·22·······84·24··
70 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+70 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+
⊟
1.57 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_dual__Variety.out
    
Offset 9, 25 lines modifiedOffset 9, 25 lines modified
9 ·····x·x·)9 ·····x·x·)
10 ······0·310 ······0·3
  
11 o1·:·Ideal·of·QQ[x·..x·]11 o1·:·Ideal·of·QQ[x·..x·]
12 ··················0···512 ··················0···5
  
13 i2·:·time·V'·=·dualVariety·V13 i2·:·time·V'·=·dualVariety·V
14 ·--·used·0.125978s·(cpu);·0.100075s·(thread);·0s·(gc)14 ·--·used·0.191585s·(cpu);·0.145973s·(thread);·0s·(gc)
  
15 ············2·················2····215 ············2·················2····2
16 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)16 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)
17 ············2·3····1·2·4····0·4····1·5·····0·3·517 ············2·3····1·2·4····0·4····1·5·····0·3·5
  
18 o2·:·Ideal·of·QQ[x·..x·]18 o2·:·Ideal·of·QQ[x·..x·]
19 ··················0···519 ··················0···5
  
20 i3·:·time·V·==·dualVariety·V'20 i3·:·time·V·==·dualVariety·V'
21 ·--·used·0.152069s·(cpu);·0.13056s·(thread);·0s·(gc)21 ·--·used·0.189701s·(cpu);·0.15041s·(thread);·0s·(gc)
  
22 o3·=·true22 o3·=·true
  
23 i4·:·F·=·first·genericPolynomials({3,-1,-1},ZZ/3331)23 i4·:·F·=·first·genericPolynomials({3,-1,-1},ZZ/3331)
  
24 ········3······2··········2······3······2···················2··········2··24 ········3······2··········2······3······2···················2··········2··
25 o4·=·a·x··+·a·x·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x·x··+·a·x·x··+·a·x·x··+25 o4·=·a·x··+·a·x·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x·x··+·a·x·x··+·a·x·x··+
Offset 38, 22 lines modifiedOffset 38, 22 lines modified
38 ······8·1·2····9·238 ······8·1·2····9·2
  
39 ······ZZ39 ······ZZ
40 o4·:·----[a·..a·][x·..x·]40 o4·:·----[a·..a·][x·..x·]
41 ·····3331··0···9···0···241 ·····3331··0···9···0···2
  
42 i5·:·time·discF·=·ideal·discriminant·F;42 i5·:·time·discF·=·ideal·discriminant·F;
43 ·--·used·0.0494826s·(cpu);·0.0474469s·(thread);·0s·(gc)43 ·--·used·0.0697871s·(cpu);·0.0676273s·(thread);·0s·(gc)
  
44 ···············ZZ44 ···············ZZ
45 o5·:·Ideal·of·----[a·..a·]45 o5·:·Ideal·of·----[a·..a·]
46 ··············3331··0···946 ··············3331··0···9
  
47 i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);47 i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
48 ·--·used·0.544585s·(cpu);·0.520612s·(thread);·0s·(gc)48 ·--·used·0.819147s·(cpu);·0.755464s·(thread);·0s·(gc)
  
49 ···············ZZ49 ···············ZZ
50 o6·:·Ideal·of·----[x·..x·]50 o6·:·Ideal·of·----[x·..x·]
51 ··············3331··0···951 ··············3331··0···9
  
52 i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)52 i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)
  
⊟
2.72 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_from__Plucker__To__Stiefel.out
    
Offset 6, 15 lines modifiedOffset 6, 15 lines modified
6 o1·=·ideal·(x··-·x·x·,·x·x··-·x·x·,·x··-·x·x·)6 o1·=·ideal·(x··-·x·x·,·x·x··-·x·x·,·x··-·x·x·)
7 ·············2····1·3···1·2····0·3···1····0·27 ·············2····1·3···1·2····0·3···1····0·2
  
8 o1·:·Ideal·of·QQ[x·..x·]8 o1·:·Ideal·of·QQ[x·..x·]
9 ··················0···39 ··················0···3
  
10 i2·:·time·fromPluckerToStiefel·dualize·chowForm·C10 i2·:·time·fromPluckerToStiefel·dualize·chowForm·C
11 ·--·used·0.0820792s·(cpu);·0.0518427s·(thread);·0s·(gc)11 ·--·used·0.110952s·(cpu);·0.0618939s·(thread);·0s·(gc)
  
12 ········3···3··········2···2··············2·······2··········2···3····12 ········3···3··········2···2··············2·······2··········2···3····
13 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-13 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-
14 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,1··14 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,1··
15 ·····------------------------------------------------------------------------15 ·····------------------------------------------------------------------------
16 ······2·······2···············2···2···································16 ······2·······2···············2···2···································
17 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-17 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 ·····x···x···x···x····-·2x···x···x···x····-·x···x···x···x····+·x···x56 ·····x···x···x···x····-·2x···x···x···x····-·x···x···x···x····+·x···x
57 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,357 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,3
  
58 o2·:·QQ[x···..x···]58 o2·:·QQ[x···..x···]
59 ·········0,0···1,359 ·········0,0···1,3
  
60 i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})60 i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})
61 ·--·used·0.0388993s·(cpu);·0.0380534s·(thread);·0s·(gc)61 ·--·used·0.0378539s·(cpu);·0.0373618s·(thread);·0s·(gc)
  
62 ············3··········2·························2························62 ············3··········2·························2························
63 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-63 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-
64 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,3··64 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,3··
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ···········2······3······266 ···········2······3······2
67 ·····2x···x····+·x····+·x67 ·····2x···x····+·x····+·x
Offset 85, 15 lines modifiedOffset 85, 15 lines modified
  
85 o4·:·QQ[a···..a···]85 o4·:·QQ[a···..a···]
86 ·········0,0···1,186 ·········0,0···1,1
  
87 i5·:·w·=·chowForm·C;87 i5·:·w·=·chowForm·C;
  
88 i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel(w,AffineChartGrass=>s))88 i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel(w,AffineChartGrass=>s))
89 ·--·used·0.0695371s·(cpu);·0.0358889s·(thread);·0s·(gc)89 ·--·used·0.0983379s·(cpu);·0.0445234s·(thread);·0s·(gc)
  
90 ···················3··········2··········3·······················2········90 ···················3··········2··········3·······················2········
91 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+91 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+
92 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3··92 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3··
93 ·····------------------------------------------------------------------------93 ·····------------------------------------------------------------------------
94 ·························2······2············2···3···············2········94 ·························2······2············2···3···············2········
95 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+95 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+
Offset 130, 14 lines modifiedOffset 130, 14 lines modified
130 ···········2······3······2130 ···········2······3······2
131 ·····2x···x····-·x····+·x···)}131 ·····2x···x····-·x····+·x···)}
132 ·······0,0·1,1····1,1····1,0132 ·······0,0·1,1····1,1····1,0
  
133 o6·:·List133 o6·:·List
  
134 i7·:·time·apply(U,u->dim·singularLocus·u)134 i7·:·time·apply(U,u->dim·singularLocus·u)
135 ·--·used·0.0159583s·(cpu);·0.0165241s·(thread);·0s·(gc)135 ·--·used·0.0177967s·(cpu);·0.0188219s·(thread);·0s·(gc)
  
136 o7·=·{2,·2,·2,·2,·2,·2}136 o7·=·{2,·2,·2,·2,·2,·2}
  
137 o7·:·List137 o7·:·List
  
138 i8·:·138 i8·:·
⊟
922 B
./usr/share/doc/Macaulay2/Resultants/example-output/_hurwitz__Form.out
    
Offset 10, 15 lines modifiedOffset 10, 15 lines modified
10 ·····--p··+·p·p··+·-p·p··+·-p·p··+·-p·p··+·7p·)10 ·····--p··+·p·p··+·-p·p··+·-p·p··+·-p·p··+·7p·)
11 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····411 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····4
  
12 o1·:·Ideal·of·QQ[p·..p·]12 o1·:·Ideal·of·QQ[p·..p·]
13 ··················0···413 ··················0···4
  
14 i2·:·time·hurwitzForm·Q14 i2·:·time·hurwitzForm·Q
15 ·--·used·0.0742897s·(cpu);·0.0527309s·(thread);·0s·(gc)15 ·--·used·0.0713701s·(cpu);·0.0467324s·(thread);·0s·(gc)
  
16 ············2····························2··································16 ············2····························2··································
17 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p···17 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p···
18 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,218 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,2
19 ·····------------------------------------------------------------------------19 ·····------------------------------------------------------------------------
20 ···········2··············································2·················20 ···········2··············································2·················
21 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p···21 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p···
⊟
1.06 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_is__Coisotropic.out
    
Offset 22, 15 lines modifiedOffset 22, 15 lines modified
22 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]22 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]
23 ·········0,1···0,2···1,2···0,3···1,3···2,323 ·········0,1···0,2···1,2···0,3···1,3···2,3
24 o1·:·--------------------------------------24 o1·:·--------------------------------------
25 ·········p···p····-·p···p····+·p···p25 ·········p···p····-·p···p····+·p···p
26 ··········1,2·0,3····0,2·1,3····0,1·2,326 ··········1,2·0,3····0,2·1,3····0,1·2,3
  
27 i2·:·time·isCoisotropic·w27 i2·:·time·isCoisotropic·w
28 ·--·used·0.00799718s·(cpu);·0.00601644s·(thread);·0s·(gc)28 ·--·used·0.00835667s·(cpu);·0.00865306s·(thread);·0s·(gc)
  
29 o2·=·true29 o2·=·true
  
30 i3·:·--·random·quadric·in·G(1,3)30 i3·:·--·random·quadric·in·G(1,3)
31 ·····w'·=·random(2,Grass(1,3))31 ·····w'·=·random(2,Grass(1,3))
  
32 ·····7·2··················2·····3·························2·····5··········32 ·····7·2··················2·····3·························2·····5··········
Offset 52, 12 lines modifiedOffset 52, 12 lines modified
52 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]52 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]
53 ·········0,1···0,2···1,2···0,3···1,3···2,353 ·········0,1···0,2···1,2···0,3···1,3···2,3
54 o3·:·--------------------------------------54 o3·:·--------------------------------------
55 ·········p···p····-·p···p····+·p···p55 ·········p···p····-·p···p····+·p···p
56 ··········1,2·0,3····0,2·1,3····0,1·2,356 ··········1,2·0,3····0,2·1,3····0,1·2,3
  
57 i4·:·time·isCoisotropic·w'57 i4·:·time·isCoisotropic·w'
58 ·--·used·0.0034741s·(cpu);·0.00520254s·(thread);·0s·(gc)58 ·--·used·0.00746541s·(cpu);·0.00720071s·(thread);·0s·(gc)
  
59 o4·=·false59 o4·=·false
  
60 i5·:·60 i5·:·
⊟
476 B
./usr/share/doc/Macaulay2/Resultants/example-output/_is__In__Coisotropic.out
    
Offset 31, 12 lines modifiedOffset 31, 12 lines modified
31 ··········4········531 ··········4········5
  
32 ················ZZ32 ················ZZ
33 o3·:·Ideal·of·-----[x·..x·]33 o3·:·Ideal·of·-----[x·..x·]
34 ··············33331··0···534 ··············33331··0···5
  
35 i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))35 i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))
36 ·--·used·0.0120067s·(cpu);·0.0154729s·(thread);·0s·(gc)36 ·--·used·0.026085s·(cpu);·0.0247257s·(thread);·0s·(gc)
  
37 o4·=·true37 o4·=·true
  
38 i5·:·38 i5·:·
⊟
1.64 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_macaulay__Formula.out
    
Offset 13, 15 lines modifiedOffset 13, 15 lines modified
13 ···················2··········2········2······313 ···················2··········2········2······3
14 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}14 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}
15 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·215 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·2
  
16 o1·:·List16 o1·:·List
  
17 i2·:·time·(D,D')·=·macaulayFormula·F17 i2·:·time·(D,D')·=·macaulayFormula·F
18 ·--·used·0.000913945s·(cpu);·0.00275152s·(thread);·0s·(gc)18 ·--·used·0.00400117s·(cpu);·0.0047077s·(thread);·0s·(gc)
  
19 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···0··19 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···0··
20 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···0··20 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···0··
21 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0··21 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0··
22 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···0··22 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···0··
23 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0··23 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0··
24 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0··24 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0··
Offset 78, 15 lines modifiedOffset 78, 15 lines modified
78 ·····6···2·······2·····378 ·····6···2·······2·····3
79 ·····-p·p··+·2p·p··+·6p·}79 ·····-p·p··+·2p·p··+·6p·}
80 ·····7·0·2·····1·2·····280 ·····7·0·2·····1·2·····2
  
81 o3·:·List81 o3·:·List
  
82 i4·:·time·(D,D')·=·macaulayFormula·F82 i4·:·time·(D,D')·=·macaulayFormula·F
83 ·--·used·0.000456377s·(cpu);·0.00212078s·(thread);·0s·(gc)83 ·--·used·0.00210445s·(cpu);·0.0030709s·(thread);·0s·(gc)
  
84 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····0··84 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····0··
85 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0··85 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0··
86 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0··86 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0··
87 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0··87 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0··
88 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0··88 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0··
89 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/489 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/4
⊟
1.94 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_plucker.out
    
Offset 9, 29 lines modifiedOffset 9, 29 lines modified
9 ·····------------------------------------------------------------------------9 ·····------------------------------------------------------------------------
10 ·····664x·)10 ·····664x·)
11 ·········411 ·········4
  
12 o3·:·Ideal·of·P412 o3·:·Ideal·of·P4
  
13 i4·:·time·p·=·plucker·L13 i4·:·time·p·=·plucker·L
14 ·--·used·0.00400072s·(cpu);·0.00347169s·(thread);·0s·(gc)14 ·--·used·0.00438053s·(cpu);·0.004682s·(thread);·0s·(gc)
  
15 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+15 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+
16 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,3··16 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,3··
17 ·····------------------------------------------------------------------------17 ·····------------------------------------------------------------------------
18 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+18 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+
19 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,2··19 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,2··
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ·····11804x···,·x····+·14854x···)21 ·····11804x···,·x····+·14854x···)
22 ···········3,4···0,1·········3,422 ···········3,4···0,1·········3,4
  
23 o4·:·Ideal·of·G'1'423 o4·:·Ideal·of·G'1'4
  
24 i5·:·time·L'·=·plucker·p24 i5·:·time·L'·=·plucker·p
25 ·--·used·0.0637691s·(cpu);·0.0399772s·(thread);·0s·(gc)25 ·--·used·0.0908882s·(cpu);·0.0414548s·(thread);·0s·(gc)
  
26 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-26 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-
27 ·············2········3·········4···1········3·········4···0·········3··27 ·············2········3·········4···1········3·········4···0·········3··
28 ·····------------------------------------------------------------------------28 ·····------------------------------------------------------------------------
29 ·····664x·)29 ·····664x·)
30 ·········430 ·········4
  
Offset 40, 25 lines modifiedOffset 40, 25 lines modified
40 i6·:·assert(L'·==·L)40 i6·:·assert(L'·==·L)
  
41 i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve41 i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve
  
42 o7·:·Ideal·of·G'1'442 o7·:·Ideal·of·G'1'4
  
43 i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y43 i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y
44 ·--·used·0.0258229s·(cpu);·0.0262577s·(thread);·0s·(gc)44 ·--·used·0.0306115s·(cpu);·0.0300526s·(thread);·0s·(gc)
  
45 o8·:·Ideal·of·P445 o8·:·Ideal·of·P4
  
46 i9·:·(codim·W,degree·W)46 i9·:·(codim·W,degree·W)
  
47 o9·=·(2,·5)47 o9·=·(2,·5)
  
48 o9·:·Sequence48 o9·:·Sequence
  
49 i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W49 i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W
50 ·--·used·0.124984s·(cpu);·0.12384s·(thread);·0s·(gc)50 ·--·used·0.170711s·(cpu);·0.170006s·(thread);·0s·(gc)
  
51 o10·:·Ideal·of·G'1'451 o10·:·Ideal·of·G'1'4
  
52 i11·:·assert(Y'·==·Y)52 i11·:·assert(Y'·==·Y)
  
53 i12·:·53 i12·:·
⊟
3.37 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp..._cm__Algorithm_eq_gt..._rp.out
    
Offset 35, 15 lines modifiedOffset 35, 15 lines modified
35 ·····3·····2····9····7·····2····9········3·······1····8····435 ·····3·····2····9····7·····2····9········3·······1····8····4
36 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}36 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}
37 ·····4··········8····8··········7················4····3····537 ·····4··········8····8··········7················4····3····5
  
38 o2·:·List38 o2·:·List
  
39 i3·:·time·resultant(F,Algorithm=>"Poisson2")39 i3·:·time·resultant(F,Algorithm=>"Poisson2")
40 ·--·used·0.202256s·(cpu);·0.151554s·(thread);·0s·(gc)40 ·--·used·0.285233s·(cpu);·0.190647s·(thread);·0s·(gc)
  
41 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···41 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
42 o3·=·-·-----------------------------a··-·-------------------------------a·b·-42 o3·=·-·-----------------------------a··-·-------------------------------a·b·-
43 ···········2222549728809984000000············12700284164628480000000·········43 ···········2222549728809984000000············12700284164628480000000·········
44 ·····------------------------------------------------------------------------44 ·····------------------------------------------------------------------------
45 ·····348237304382147063838108483692249·3·2··45 ·····348237304382147063838108483692249·3·2··
46 ·····---------------------------------a·b··-46 ·····---------------------------------a·b··-
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 ·····1146977327343523453866040839029···4···194441910898734675845094443·556 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
57 ·····-------------------------------a*b··-·---------------------------b57 ·····-------------------------------a*b··-·---------------------------b
58 ··········1119954511872000000000················89596360949760000058 ··········1119954511872000000000················895963609497600000
  
59 o3·:·QQ[a..b]59 o3·:·QQ[a..b]
  
60 i4·:·time·resultant(F,Algorithm=>"Macaulay2")60 i4·:·time·resultant(F,Algorithm=>"Macaulay2")
61 ·--·used·0.0985538s·(cpu);·0.0741696s·(thread);·0s·(gc)61 ·--·used·0.101548s·(cpu);·0.0716955s·(thread);·0s·(gc)
  
62 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···62 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
63 o4·=·-·-----------------------------a··-·-------------------------------a·b·-63 o4·=·-·-----------------------------a··-·-------------------------------a·b·-
64 ···········2222549728809984000000············12700284164628480000000·········64 ···········2222549728809984000000············12700284164628480000000·········
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ·····348237304382147063838108483692249·3·2··66 ·····348237304382147063838108483692249·3·2··
67 ·····---------------------------------a·b··-67 ·····---------------------------------a·b··-
Offset 77, 15 lines modifiedOffset 77, 15 lines modified
77 ·····1146977327343523453866040839029···4···194441910898734675845094443·577 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
78 ·····-------------------------------a*b··-·---------------------------b78 ·····-------------------------------a*b··-·---------------------------b
79 ··········1119954511872000000000················89596360949760000079 ··········1119954511872000000000················895963609497600000
  
80 o4·:·QQ[a..b]80 o4·:·QQ[a..b]
  
81 i5·:·time·resultant(F,Algorithm=>"Poisson")81 i5·:·time·resultant(F,Algorithm=>"Poisson")
82 ·--·used·0.224459s·(cpu);·0.224151s·(thread);·0s·(gc)82 ·--·used·0.248571s·(cpu);·0.249901s·(thread);·0s·(gc)
  
83 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···83 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
84 o5·=·-·-----------------------------a··-·-------------------------------a·b·-84 o5·=·-·-----------------------------a··-·-------------------------------a·b·-
85 ···········2222549728809984000000············12700284164628480000000·········85 ···········2222549728809984000000············12700284164628480000000·········
86 ·····------------------------------------------------------------------------86 ·····------------------------------------------------------------------------
87 ·····348237304382147063838108483692249·3·2··87 ·····348237304382147063838108483692249·3·2··
88 ·····---------------------------------a·b··-88 ·····---------------------------------a·b··-
Offset 98, 15 lines modifiedOffset 98, 15 lines modified
98 ·····1146977327343523453866040839029···4···194441910898734675845094443·598 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
99 ·····-------------------------------a*b··-·---------------------------b99 ·····-------------------------------a*b··-·---------------------------b
100 ··········1119954511872000000000················895963609497600000100 ··········1119954511872000000000················895963609497600000
  
101 o5·:·QQ[a..b]101 o5·:·QQ[a..b]
  
102 i6·:·time·resultant(F,Algorithm=>"Macaulay")102 i6·:·time·resultant(F,Algorithm=>"Macaulay")
103 ·--·used·0.456844s·(cpu);·0.437105s·(thread);·0s·(gc)103 ·--·used·0.56392s·(cpu);·0.535167s·(thread);·0s·(gc)
  
104 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···104 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
105 o6·=·-·-----------------------------a··-·-------------------------------a·b·-105 o6·=·-·-----------------------------a··-·-------------------------------a·b·-
106 ···········2222549728809984000000············12700284164628480000000·········106 ···········2222549728809984000000············12700284164628480000000·········
107 ·····------------------------------------------------------------------------107 ·····------------------------------------------------------------------------
108 ·····348237304382147063838108483692249·3·2··108 ·····348237304382147063838108483692249·3·2··
109 ·····---------------------------------a·b··-109 ·····---------------------------------a·b··-
⊟
1.95 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_resultant_lp__Matrix_rp.out
    
Offset 10, 15 lines modifiedOffset 10, 15 lines modified
  
10 ·······2···············2····························3······2·········410 ·······2···············2····························3······2·········4
11 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}11 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}
  
12 o2·:·List12 o2·:·List
  
13 i3·:·time·resultant·F13 i3·:·time·resultant·F
14 ·--·used·0.0167989s·(cpu);·0.0196189s·(thread);·0s·(gc)14 ·--·used·0.0205937s·(cpu);·0.023252s·(thread);·0s·(gc)
  
15 ··········12·········11·2·········10·3·········9·4··········8·5··········7·615 ··········12·········11·2·········10·3·········9·4··········8·5··········7·6
16 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u·16 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u·
17 ·····------------------------------------------------------------------------17 ·····------------------------------------------------------------------------
18 ··············6·7··········5·8·······11··········10·2·········9·3··18 ··············6·7··········5·8·······11··········10·2·········9·3··
19 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+19 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
Offset 64, 15 lines modifiedOffset 64, 15 lines modified
64 ··········364 ··········3
65 ·····+·c·x·}65 ·····+·c·x·}
66 ········9·266 ········9·2
  
67 o4·:·List67 o4·:·List
  
68 i5·:·time·resultant·F68 i5·:·time·resultant·F
69 ·--·used·1.70599s·(cpu);·1.45931s·(thread);·0s·(gc)69 ·--·used·1.93709s·(cpu);·1.61928s·(thread);·0s·(gc)
  
70 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···2··70 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···2··
71 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-71 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-
72 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·0··72 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·0··
73 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
74 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·2··74 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·2··
75 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-75 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-
Offset 1690, 12 lines modifiedOffset 1690, 12 lines modified
1690 ··························2·····2···············2························21690 ··························2·····2···············2························2
1691 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}1691 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}
1692 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·21692 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
1693 o6·:·List1693 o6·:·List
  
1694 i7·:·time·#·terms·resultant·F1694 i7·:·time·#·terms·resultant·F
1695 ·--·used·0.37424s·(cpu);·0.321845s·(thread);·0s·(gc)1695 ·--·used·0.407066s·(cpu);·0.342191s·(thread);·0s·(gc)
  
1696 o7·=·218941696 o7·=·21894
  
1697 i8·:·1697 i8·:·
⊟
4.0 KB
./usr/share/doc/Macaulay2/Resultants/example-output/_tangential__Chow__Form.out
    
Offset 8, 15 lines modifiedOffset 8, 15 lines modified
8 ···············1·2····0·3·····1·3····0·4·····3····2·48 ···············1·2····0·3·····1·3····0·4·····3····2·4
  
9 o2·:·Ideal·of·QQ[p·..p·]9 o2·:·Ideal·of·QQ[p·..p·]
10 ··················0···410 ··················0···4
  
11 i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)11 i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)
12 ·····time·tangentialChowForm(S,0)12 ·····time·tangentialChowForm(S,0)
13 ·--·used·0.0666742s·(cpu);·0.0409446s·(thread);·0s·(gc)13 ·--·used·0.0876343s·(cpu);·0.046571s·(thread);·0s·(gc)
  
14 ······2·······················································2········14 ······2·······················································2········
15 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+15 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+
16 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,4··16 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,4··
17 ·····------------------------------------------------------------------------17 ·····------------------------------------------------------------------------
18 ······218 ······2
19 ·····p···p····-·2p···p···p····-·p···p···p19 ·····p···p····-·2p···p···p····-·p···p···p
Offset 26, 15 lines modifiedOffset 26, 15 lines modified
26 ··························································0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,426 ··························································0,1···0,2···1,2···0,3···1,3···2,3···0,4···1,4···2,4···3,4
27 o3·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------27 o3·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------
28 ·····(p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···)28 ·····(p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···)
29 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,4····0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,1·2,329 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,4····0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,1·2,3
  
30 i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)30 i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)
31 ·····time·tangentialChowForm(S,1)31 ·····time·tangentialChowForm(S,1)
32 ·--·used·0.0413244s·(cpu);·0.0436383s·(thread);·0s·(gc)32 ·--·used·0.0584356s·(cpu);·0.0577753s·(thread);·0s·(gc)
  
33 ······2·····2········2·····2···············3········2·····2······33 ······2·····2········2·····2···············3········2·····2······
34 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-34 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-
35 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,4··35 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,4··
36 ·····------------------------------------------------------------------------36 ·····------------------------------------------------------------------------
37 ·············3·········3···············3············37 ·············3·········3···············3············
38 ·····4p·····p······-·4p·····p······-·2p·····p······+38 ·····4p·····p······-·4p·····p······-·2p·····p······+
Offset 68, 32 lines modifiedOffset 68, 32 lines modified
68 ··············································································0,1,2···0,1,3···0,2,3···1,2,3···0,1,4···0,2,4···1,2,4···0,3,4···1,3,4···2,3,468 ··············································································0,1,2···0,1,3···0,2,3···1,2,3···0,1,4···0,2,4···1,2,4···0,3,4···1,3,4···2,3,4
69 o4·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------69 o4·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
70 ·····(p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)70 ·····(p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)
71 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,471 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4
  
72 i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent·hyperplanes·to·S)72 i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent·hyperplanes·to·S)
73 ·····time·tangentialChowForm(S,2)73 ·····time·tangentialChowForm(S,2)
74 ·--·used·0.0248222s·(cpu);·0.0259189s·(thread);·0s·(gc)74 ·--·used·0.0361974s·(cpu);·0.0361604s·(thread);·0s·(gc)
  
75 ··············2·············································275 ··············2·············································2
76 o5·=·p·······p········-·p·······p·······p········+·p·······p76 o5·=·p·······p········-·p·······p·······p········+·p·······p
77 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,477 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,4
  
78 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]78 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]
79 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,479 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,4
  
80 i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing80 i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing
81 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)81 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)
82 ·--·used·0.075193s·(cpu);·0.0492194s·(thread);·0s·(gc)82 ·--·used·0.0955263s·(cpu);·0.0648529s·(thread);·0s·(gc)
  
83 ············2···············283 ············2···············2
84 o6·=·ideal(p·p··-·p·p·p··+·p·p·)84 o6·=·ideal(p·p··-·p·p·p··+·p·p·)
85 ············1·2····0·1·3····0·485 ············1·2····0·1·3····0·4
  
86 o6·:·Ideal·of·QQ[p·..p·]86 o6·:·Ideal·of·QQ[p·..p·]
87 ··················0···487 ··················0···4
  
88 i7·:·--·we·then·can·recover·S88 i7·:·--·we·then·can·recover·S
89 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)89 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)
90 ·--·used·0.112612s·(cpu);·0.086262s·(thread);·0s·(gc)90 ·--·used·0.136925s·(cpu);·0.0963347s·(thread);·0s·(gc)
  
91 i8·:·91 i8·:·
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87 ············0·1····2·387 ············0·1····2·3
  
88 o2·:·Ideal·of·QQ[x·..x·]88 o2·:·Ideal·of·QQ[x·..x·]
89 ··················0···3</code></pre>89 ··················0···3</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);92 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);
93 ·--·used·0.106059s·(cpu);·0.082427s·(thread);·0s·(gc)</code></pre>93 ·--·used·0.110492s·(cpu);·0.0708643s·(thread);·0s·(gc)</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ········</table>95 ········</table>
96 ········<p>In·the·next·example,·we·calculate·the·defining·ideal·of·$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$·and·that·of·its·dual·variety.</p>96 ········<p>In·the·next·example,·we·calculate·the·defining·ideal·of·$\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$·and·that·of·its·dual·variety.</p>
97 ········<table·class="examples">97 ········<table·class="examples">
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)99 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)
100 ·--·used·0.0399344s·(cpu);·0.0420251s·(thread);·0s·(gc)100 ·--·used·0.064093s·(cpu);·0.0661522s·(thread);·0s·(gc)
  
101 ···········································································101 ···········································································
102 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,102 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
103 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,3·103 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,3·
104 ·····------------------------------------------------------------------------104 ·····------------------------------------------------------------------------
105 ············································································105 ············································································
106 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···106 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x···
Offset 127, 22 lines modifiedOffset 127, 22 lines modified
127 o4·:·Sequence</code></pre>127 o4·:·Sequence</code></pre>
128 </td>··········</tr>128 </td>··········</tr>
129 ········</table>129 ········</table>
130 ········<p>If·the·option·<span·class="tt">Duality</span>·is·set·to·<span·class="tt">true</span>,·then·the·method·applies·the·so-called·&quot;dual·Cayley·trick&quot;.</p>130 ········<p>If·the·option·<span·class="tt">Duality</span>·is·set·to·<span·class="tt">true</span>,·then·the·method·applies·the·so-called·&quot;dual·Cayley·trick&quot;.</p>
131 ········<table·class="examples">131 ········<table·class="examples">
132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);133 <td>··············<pre><code·class="language-macaulay2">i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);
134 ·--·used·0.105846s·(cpu);·0.0798293s·(thread);·0s·(gc)</code></pre>134 ·--·used·0.158759s·(cpu);·0.111621s·(thread);·0s·(gc)</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))</code></pre>137 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);140 <td>··············<pre><code·class="language-macaulay2">i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);
141 ·--·used·0.111715s·(cpu);·0.0835642s·(thread);·0s·(gc)</code></pre>141 ·--·used·0.186531s·(cpu);·0.122305s·(thread);·0s·(gc)</code></pre>
142 </td>··········</tr>142 </td>··········</tr>
143 ··········<tr>143 ··········<tr>
144 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))</code></pre>144 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ········</table>146 ········</table>
147 ······</div>147 ······</div>
148 ······<div>148 ······<div>
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39 o2·=·ideal(x·x··-·x·x·)39 o2·=·ideal(x·x··-·x·x·)
40 ············0·1····2·340 ············0·1····2·3
  
41 o2·:·Ideal·of·QQ[x·..x·]41 o2·:·Ideal·of·QQ[x·..x·]
42 ··················0···342 ··················0···3
43 i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);43 i3·:·time·(P1xP1xP2,P1xP1xP2')·=·cayleyTrick(P1xP1,2);
44 ·--·used·0.106059s·(cpu);·0.082427s·(thread);·0s·(gc)44 ·--·used·0.110492s·(cpu);·0.0708643s·(thread);·0s·(gc)
45 In·the·next·example,·we·calculate·the·defining·ideal·of·$\mathbb45 In·the·next·example,·we·calculate·the·defining·ideal·of·$\mathbb
46 {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$·and·that·of·its46 {P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$·and·that·of·its
47 dual·variety.47 dual·variety.
48 i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)48 i4·:·time·(P1xP1xP1,P1xP1xP1')·=·cayleyTrick(P1xP1,1)
49 ·--·used·0.0399344s·(cpu);·0.0420251s·(thread);·0s·(gc)49 ·--·used·0.064093s·(cpu);·0.0661522s·(thread);·0s·(gc)
  
  
50 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,50 o4·=·(ideal·(x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,
51 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,351 ··············0,3·1,2····0,2·1,3···1,0·1,1····1,2·1,3···0,3·1,1····0,1·1,3
52 ·····------------------------------------------------------------------------52 ·····------------------------------------------------------------------------
  
53 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x53 ·····x···x····-·x···x···,·x···x····-·x···x···,·x···x····-·x···x···,·x···x
Offset 74, 18 lines modifiedOffset 74, 18 lines modified
74 ·····4x···x···x···x····-·2x···x···x···x····+·x···x···))74 ·····4x···x···x···x····-·2x···x···x···x····+·x···x···))
75 ·······0,0·0,1·1,2·1,3·····0,2·0,3·1,2·1,3····0,2·1,375 ·······0,0·0,1·1,2·1,3·····0,2·0,3·1,2·1,3····0,2·1,3
  
76 o4·:·Sequence76 o4·:·Sequence
77 If·the·option·Duality·is·set·to·true,·then·the·method·applies·the·so-called77 If·the·option·Duality·is·set·to·true,·then·the·method·applies·the·so-called
78 "dual·Cayley·trick".78 "dual·Cayley·trick".
79 i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);79 i5·:·time·cayleyTrick(P1xP1,1,Duality=>true);
80 ·--·used·0.105846s·(cpu);·0.0798293s·(thread);·0s·(gc)80 ·--·used·0.158759s·(cpu);·0.111621s·(thread);·0s·(gc)
81 i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))81 i6·:·assert(oo·==·(P1xP1xP1,P1xP1xP1'))
82 i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);82 i7·:·time·cayleyTrick(P1xP1,2,Duality=>true);
83 ·--·used·0.111715s·(cpu);·0.0835642s·(thread);·0s·(gc)83 ·--·used·0.186531s·(cpu);·0.122305s·(thread);·0s·(gc)
84 i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))84 i8·:·assert(oo·==·(P1xP1xP2,P1xP1xP2'))
85 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*85 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
86 ····*·_\x8d_\x8u_\x8a_\x8l_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·projective·dual·variety86 ····*·_\x8d_\x8u_\x8a_\x8l_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y·--·projective·dual·variety
87 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ca\x8ay\x8yl\x8le\x8ey\x8yT\x8Tr\x8ri\x8ic\x8ck\x8k:\x8:·*\x8**\x8**\x8**\x8**\x8*87 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ca\x8ay\x8yl\x8le\x8ey\x8yT\x8Tr\x8ri\x8ic\x8ck\x8k:\x8:·*\x8**\x8**\x8**\x8**\x8*
88 ····*·cayleyTrick(Ideal,ZZ)88 ····*·cayleyTrick(Ideal,ZZ)
89 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*89 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
90 The·object·_\x8c_\x8a_\x8y_\x8l_\x8e_\x8y_\x8T_\x8r_\x8i_\x8c_\x8k·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.90 The·object·_\x8c_\x8a_\x8y_\x8l_\x8e_\x8y_\x8T_\x8r_\x8i_\x8c_\x8k·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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87 ·············0····1····2····3···0·1····1·2····2·387 ·············0····1····2····3···0·1····1·2····2·3
  
88 o2·:·Ideal·of·P3</code></pre>88 o2·:·Ideal·of·P3</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i3·:·--·Chow·equations·of·C91 <td>··············<pre><code·class="language-macaulay2">i3·:·--·Chow·equations·of·C
92 ·····time·eqsC·=·chowEquations·chowForm·C92 ·····time·eqsC·=·chowEquations·chowForm·C
93 ·--·used·0.07896s·(cpu);·0.0562216s·(thread);·0s·(gc)93 ·--·used·0.12423s·(cpu);·0.0825842s·(thread);·0s·(gc)
  
94 ·············2·2····2·2····2·2····4················2······2·2···2······94 ·············2·2····2·2····2·2····4················2······2·2···2······
95 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+95 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+
96 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·3··96 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·3··
97 ·····------------------------------------------------------------------------97 ·····------------------------------------------------------------------------
98 ······2········3···········2·········2······3···3·········2··········2····2·298 ······2········3···········2·········2······3···3·········2··········2····2·2
99 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x·99 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x·
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153 ·············1····0·2···2····0·1·3···1·2····0·3153 ·············1····0·2···2····0·1·3···1·2····0·3
  
154 o5·:·Ideal·of·P3</code></pre>154 o5·:·Ideal·of·P3</code></pre>
155 </td>··········</tr>155 </td>··········</tr>
156 ··········<tr>156 ··········<tr>
157 <td>··············<pre><code·class="language-macaulay2">i6·:·--·Chow·equations·of·D157 <td>··············<pre><code·class="language-macaulay2">i6·:·--·Chow·equations·of·D
158 ·····time·eqsD·=·chowEquations·chowForm·D158 ·····time·eqsD·=·chowEquations·chowForm·D
159 ·--·used·0.0358775s·(cpu);·0.0342206s·(thread);·0s·(gc)159 ·--·used·0.0498103s·(cpu);·0.0494591s·(thread);·0s·(gc)
  
160 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···2·160 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···2·
161 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,161 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,
162 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·3·162 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·3·
163 ·····------------------------------------------------------------------------163 ·····------------------------------------------------------------------------
164 ··········2······3·2···3········3·2···4·········2·········2·2·······3····164 ··········2······3·2···3········3·2···4·········2·········2·2·······3····
165 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-165 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-
Offset 205, 28 lines modifiedOffset 205, 28 lines modified
205 ············0·1····2·3205 ············0·1····2·3
  
206 o9·:·Ideal·of·P3</code></pre>206 o9·:·Ideal·of·P3</code></pre>
207 </td>··········</tr>207 </td>··········</tr>
208 ··········<tr>208 ··········<tr>
209 <td>··············<pre><code·class="language-macaulay2">i10·:·--·tangential·Chow·forms·of·Q209 <td>··············<pre><code·class="language-macaulay2">i10·:·--·tangential·Chow·forms·of·Q
210 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))210 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
211 ·--·used·0.110058s·(cpu);·0.0815315s·(thread);·0s·(gc)211 ·--·used·0.111642s·(cpu);·0.0794158s·(thread);·0s·(gc)
  
212 ·····················2······························2212 ·····················2······························2
213 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+213 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+
214 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3··214 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3··
215 ······-----------------------------------------------------------------------215 ······-----------------------------------------------------------------------
216 ······x·····x·····)216 ······x·····x·····)
217 ·······0,2,3·1,2,3217 ·······0,2,3·1,2,3
  
218 o10·:·Sequence</code></pre>218 o10·:·Sequence</code></pre>
219 </td>··········</tr>219 </td>··········</tr>
220 ··········<tr>220 ··········<tr>
221 <td>··············<pre><code·class="language-macaulay2">i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))221 <td>··············<pre><code·class="language-macaulay2">i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations(W2,2))
222 ·--·used·0.102453s·(cpu);·0.0727086s·(thread);·0s·(gc)222 ·--·used·0.129249s·(cpu);·0.0839007s·(thread);·0s·(gc)
  
223 o11·=·true</code></pre>223 o11·=·true</code></pre>
224 </td>··········</tr>224 </td>··········</tr>
225 ········</table>225 ········</table>
226 ········<p>Note·that·<span·class="tt">chowEquations(W,0)</span>·is·not·the·same·as·<span·class="tt">chowEquations·W</span>.</p>226 ········<p>Note·that·<span·class="tt">chowEquations(W,0)</span>·is·not·the·same·as·<span·class="tt">chowEquations·W</span>.</p>
227 ······</div>227 ······</div>
228 ······<div·class="waystouse">228 ······<div·class="waystouse">
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Offset 29, 15 lines modifiedOffset 29, 15 lines modified
29 ·············2····2····2····229 ·············2····2····2····2
30 o2·=·ideal·(x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x·)30 o2·=·ideal·(x··+·x··+·x··+·x·,·x·x··+·x·x··+·x·x·)
31 ·············0····1····2····3···0·1····1·2····2·331 ·············0····1····2····3···0·1····1·2····2·3
  
32 o2·:·Ideal·of·P332 o2·:·Ideal·of·P3
33 i3·:·--·Chow·equations·of·C33 i3·:·--·Chow·equations·of·C
34 ·····time·eqsC·=·chowEquations·chowForm·C34 ·····time·eqsC·=·chowEquations·chowForm·C
35 ·--·used·0.07896s·(cpu);·0.0562216s·(thread);·0s·(gc)35 ·--·used·0.12423s·(cpu);·0.0825842s·(thread);·0s·(gc)
  
36 ·············2·2····2·2····2·2····4················2······2·2···236 ·············2·2····2·2····2·2····4················2······2·2···2
37 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+37 o3·=·ideal·(x·x··+·x·x··+·x·x··+·x·,·x·x·x·x··+·x·x·x··+·x·x·,·x·x·x··+
38 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·338 ·············0·3····1·3····2·3····3···0·1·2·3····1·2·3····2·3···0·2·3
39 ·····------------------------------------------------------------------------39 ·····------------------------------------------------------------------------
40 ······2········3···········2·········2······3···3·········2··········2····2·240 ······2········3···········2·········2······3···3·········2··········2····2·2
41 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x41 ·····x·x·x··+·x·x··-·2x·x·x··-·2x·x·x··-·x·x·,·x·x··+·2x·x·x··-·x·x·x··+·x·x
Offset 89, 15 lines modifiedOffset 89, 15 lines modified
89 ·············2··········3··············2····289 ·············2··········3··············2····2
90 o5·=·ideal·(x··-·x·x·,·x··-·x·x·x·,·x·x··-·x·x·)90 o5·=·ideal·(x··-·x·x·,·x··-·x·x·x·,·x·x··-·x·x·)
91 ·············1····0·2···2····0·1·3···1·2····0·391 ·············1····0·2···2····0·1·3···1·2····0·3
  
92 o5·:·Ideal·of·P392 o5·:·Ideal·of·P3
93 i6·:·--·Chow·equations·of·D93 i6·:·--·Chow·equations·of·D
94 ·····time·eqsD·=·chowEquations·chowForm·D94 ·····time·eqsD·=·chowEquations·chowForm·D
95 ·--·used·0.0358775s·(cpu);·0.0342206s·(thread);·0s·(gc)95 ·--·used·0.0498103s·(cpu);·0.0494591s·(thread);·0s·(gc)
  
96 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···296 ·············4······3·2·····3········2·2·····3······2···2···2·2······2···2
97 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,97 o6·=·ideal·(x·x··-·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,·x·x·x··-·x·x·x·,
98 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·398 ·············2·3····1·3···1·2·3····0·1·3···0·2·3····0·1·3···1·2·3····0·1·3
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ··········2······3·2···3········3·2···4·········2·········2·2·······3100 ··········2······3·2···3········3·2···4·········2·········2·2·······3
101 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-101 ·····x·x·x·x··-·x·x·,·x·x·x··-·x·x·,·x·x··-·4x·x·x·x··+·3x·x·x·,·x·x·x··-
Offset 136, 27 lines modifiedOffset 136, 27 lines modified
136 o9·=·ideal(x·x··+·x·x·)136 o9·=·ideal(x·x··+·x·x·)
137 ············0·1····2·3137 ············0·1····2·3
  
138 o9·:·Ideal·of·P3138 o9·:·Ideal·of·P3
139 i10·:·--·tangential·Chow·forms·of·Q139 i10·:·--·tangential·Chow·forms·of·Q
140 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm140 ······time·(W0,W1,W2)·=·(tangentialChowForm(Q,0),tangentialChowForm
141 (Q,1),tangentialChowForm(Q,2))141 (Q,1),tangentialChowForm(Q,2))
142 ·--·used·0.110058s·(cpu);·0.0815315s·(thread);·0s·(gc)142 ·--·used·0.111642s·(cpu);·0.0794158s·(thread);·0s·(gc)
  
143 ·····················2······························2143 ·····················2······························2
144 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+144 o10·=·(x·x··+·x·x·,·x····-·4x···x····+·2x···x····+·x···,·x·····x······+
145 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3145 ········0·1····2·3···0,1·····0,2·1,3·····0,1·2,3····2,3···0,1,2·0,1,3
146 ······-----------------------------------------------------------------------146 ······-----------------------------------------------------------------------
147 ······x·····x·····)147 ······x·····x·····)
148 ·······0,2,3·1,2,3148 ·······0,2,3·1,2,3
  
149 o10·:·Sequence149 o10·:·Sequence
150 i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations150 i11·:·time·(Q,Q,Q)·==·(chowEquations(W0,0),chowEquations(W1,1),chowEquations
151 (W2,2))151 (W2,2))
152 ·--·used·0.102453s·(cpu);·0.0727086s·(thread);·0s·(gc)152 ·--·used·0.129249s·(cpu);·0.0839007s·(thread);·0s·(gc)
  
153 o11·=·true153 o11·=·true
154 Note·that·chowEquations(W,0)·is·not·the·same·as·chowEquations·W.154 Note·that·chowEquations(W,0)·is·not·the·same·as·chowEquations·W.
155 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ch\x8ho\x8ow\x8wE\x8Eq\x8qu\x8ua\x8at\x8ti\x8io\x8on\x8ns\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*155 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ch\x8ho\x8ow\x8wE\x8Eq\x8qu\x8ua\x8at\x8ti\x8io\x8on\x8ns\x8s:\x8:·*\x8**\x8**\x8**\x8**\x8*
156 ····*·chowEquations(RingElement)156 ····*·chowEquations(RingElement)
157 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*157 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
158 The·object·_\x8c_\x8h_\x8o_\x8w_\x8E_\x8q_\x8u_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.158 The·object·_\x8c_\x8h_\x8o_\x8w_\x8E_\x8q_\x8u_\x8a_\x8t_\x8i_\x8o_\x8n_\x8s·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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Offset 99, 15 lines modifiedOffset 99, 15 lines modified
99 ···············ZZ99 ···············ZZ
100 o2·:·Ideal·of·----[x·..x·]100 o2·:·Ideal·of·----[x·..x·]
101 ··············3331··0···5</code></pre>101 ··············3331··0···5</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an·affine·chart·of·the·Grassmannian)104 <td>··············<pre><code·class="language-macaulay2">i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an·affine·chart·of·the·Grassmannian)
105 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})105 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})
106 ·--·used·4.48232s·(cpu);·4.32064s·(thread);·0s·(gc)106 ·--·used·4.96101s·(cpu);·4.70115s·(thread);·0s·(gc)
  
107 ······4···············2··············2·····2···············2············107 ······4···············2··············2·····2···············2············
108 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+108 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+
109 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,5··109 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,5··
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
111 ······2·····2··············2·······································111 ······2·····2··············2·······································
112 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+112 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+
Offset 227, 20 lines modifiedOffset 227, 20 lines modified
227 o3·:·-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------227 o3·:·-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
228 ·····(x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····)228 ·····(x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x······-·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····,·x·····x······-·x·····x······+·x·····x·····)
229 ·······2,3,5·1,4,5····1,3,5·2,4,5····1,2,5·3,4,5···2,3,4·1,4,5····1,3,4·2,4,5····1,2,4·3,4,5···2,3,5·0,4,5····0,3,5·2,4,5····0,2,5·3,4,5···1,3,5·0,4,5····0,3,5·1,4,5····0,1,5·3,4,5···1,2,5·0,4,5····0,2,5·1,4,5····0,1,5·2,4,5···2,3,4·0,4,5····0,3,4·2,4,5····0,2,4·3,4,5···1,3,4·0,4,5····0,3,4·1,4,5····0,1,4·3,4,5···1,2,4·0,4,5····0,2,4·1,4,5····0,1,4·2,4,5···1,2,3·0,4,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·1,3,5····1,3,4·2,3,5····1,2,3·3,4,5···1,2,5·0,3,5····0,2,5·1,3,5····0,1,5·2,3,5···2,3,4·0,3,5····0,3,4·2,3,5····0,2,3·3,4,5···1,3,4·0,3,5····0,3,4·1,3,5····0,1,3·3,4,5···1,2,4·0,3,5····0,2,4·1,3,5····0,1,4·2,3,5····0,1,2·3,4,5···1,2,3·0,3,5····0,2,3·1,3,5····0,1,3·2,3,5···2,3,4·1,2,5····1,2,4·2,3,5····1,2,3·2,4,5···1,3,4·1,2,5····1,2,4·1,3,5····1,2,3·1,4,5···0,3,4·1,2,5····0,2,4·1,3,5····0,1,4·2,3,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·0,2,5····0,2,4·2,3,5····0,2,3·2,4,5···1,3,4·0,2,5····0,2,4·1,3,5····0,2,3·1,4,5····0,1,2·3,4,5···0,3,4·0,2,5····0,2,4·0,3,5····0,2,3·0,4,5···1,2,4·0,2,5····0,2,4·1,2,5····0,1,2·2,4,5···1,2,3·0,2,5····0,2,3·1,2,5····0,1,2·2,3,5···2,3,4·0,1,5····0,1,4·2,3,5····0,1,3·2,4,5····0,1,2·3,4,5···1,3,4·0,1,5····0,1,4·1,3,5····0,1,3·1,4,5···0,3,4·0,1,5····0,1,4·0,3,5····0,1,3·0,4,5···1,2,4·0,1,5····0,1,4·1,2,5····0,1,2·1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5····0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4</code></pre>229 ·······2,3,5·1,4,5····1,3,5·2,4,5····1,2,5·3,4,5···2,3,4·1,4,5····1,3,4·2,4,5····1,2,4·3,4,5···2,3,5·0,4,5····0,3,5·2,4,5····0,2,5·3,4,5···1,3,5·0,4,5····0,3,5·1,4,5····0,1,5·3,4,5···1,2,5·0,4,5····0,2,5·1,4,5····0,1,5·2,4,5···2,3,4·0,4,5····0,3,4·2,4,5····0,2,4·3,4,5···1,3,4·0,4,5····0,3,4·1,4,5····0,1,4·3,4,5···1,2,4·0,4,5····0,2,4·1,4,5····0,1,4·2,4,5···1,2,3·0,4,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·1,3,5····1,3,4·2,3,5····1,2,3·3,4,5···1,2,5·0,3,5····0,2,5·1,3,5····0,1,5·2,3,5···2,3,4·0,3,5····0,3,4·2,3,5····0,2,3·3,4,5···1,3,4·0,3,5····0,3,4·1,3,5····0,1,3·3,4,5···1,2,4·0,3,5····0,2,4·1,3,5····0,1,4·2,3,5····0,1,2·3,4,5···1,2,3·0,3,5····0,2,3·1,3,5····0,1,3·2,3,5···2,3,4·1,2,5····1,2,4·2,3,5····1,2,3·2,4,5···1,3,4·1,2,5····1,2,4·1,3,5····1,2,3·1,4,5···0,3,4·1,2,5····0,2,4·1,3,5····0,1,4·2,3,5····0,2,3·1,4,5····0,1,3·2,4,5····0,1,2·3,4,5···2,3,4·0,2,5····0,2,4·2,3,5····0,2,3·2,4,5···1,3,4·0,2,5····0,2,4·1,3,5····0,2,3·1,4,5····0,1,2·3,4,5···0,3,4·0,2,5····0,2,4·0,3,5····0,2,3·0,4,5···1,2,4·0,2,5····0,2,4·1,2,5····0,1,2·2,4,5···1,2,3·0,2,5····0,2,3·1,2,5····0,1,2·2,3,5···2,3,4·0,1,5····0,1,4·2,3,5····0,1,3·2,4,5····0,1,2·3,4,5···1,3,4·0,1,5····0,1,4·1,3,5····0,1,3·1,4,5···0,3,4·0,1,5····0,1,4·0,3,5····0,1,3·0,4,5···1,2,4·0,1,5····0,1,4·1,2,5····0,1,2·1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5····0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4</code></pre>
230 </td>··········</tr>230 </td>··········</tr>
231 ··········<tr>231 ··········<tr>
232 <td>··············<pre><code·class="language-macaulay2">i4·:·--·equivalently·(but·faster)...232 <td>··············<pre><code·class="language-macaulay2">i4·:·--·equivalently·(but·faster)...
233 ·····time·assert(ChowV·===·chowForm·f)233 ·····time·assert(ChowV·===·chowForm·f)
234 ·--·used·1.151s·(cpu);·1.08375s·(thread);·0s·(gc)</code></pre>234 ·--·used·1.02189s·(cpu);·0.908277s·(thread);·0s·(gc)</code></pre>
235 </td>··········</tr>235 </td>··········</tr>
236 ··········<tr>236 ··········<tr>
237 <td>··············<pre><code·class="language-macaulay2">i5·:·--·X-resultant·of·V237 <td>··············<pre><code·class="language-macaulay2">i5·:·--·X-resultant·of·V
238 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;238 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;
239 ·--·used·0.158288s·(cpu);·0.127679s·(thread);·0s·(gc)</code></pre>239 ·--·used·0.215244s·(cpu);·0.15912s·(thread);·0s·(gc)</code></pre>
240 </td>··········</tr>240 </td>··········</tr>
241 ··········<tr>241 ··········<tr>
242 <td>··············<pre><code·class="language-macaulay2">i6·:·--·three·generic·ternary·quadrics242 <td>··············<pre><code·class="language-macaulay2">i6·:·--·three·generic·ternary·quadrics
243 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)243 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)
  
244 ·········2···············2························2·····2···············2··244 ·········2···············2························2·····2···············2··
245 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+245 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+
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251 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2251 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
252 o6·:·List</code></pre>252 o6·:·List</code></pre>
253 </td>··········</tr>253 </td>··········</tr>
254 ··········<tr>254 ··········<tr>
255 <td>··············<pre><code·class="language-macaulay2">i7·:·--·resultant·of·the·three·forms255 <td>··············<pre><code·class="language-macaulay2">i7·:·--·resultant·of·the·three·forms
256 ·····time·resF·=·resultant·F;256 ·····time·resF·=·resultant·F;
257 ·--·used·0.170361s·(cpu);·0.138395s·(thread);·0s·(gc)</code></pre>257 ·--·used·0.242248s·(cpu);·0.18567s·(thread);·0s·(gc)</code></pre>
258 </td>··········</tr>258 </td>··········</tr>
259 ··········<tr>259 ··········<tr>
260 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring·Xres))</code></pre>260 <td>··············<pre><code·class="language-macaulay2">i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring·Xres))</code></pre>
261 </td>··········</tr>261 </td>··········</tr>
262 ········</table>262 ········</table>
263 ······</div>263 ······</div>
264 ······<div>264 ······<div>
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42 ···············ZZ42 ···············ZZ
43 o2·:·Ideal·of·----[x·..x·]43 o2·:·Ideal·of·----[x·..x·]
44 ··············3331··0···544 ··············3331··0···5
45 i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an45 i3·:·--·Chow·form·of·V·in·Grass(2,5)·(performing·internal·computations·on·an
46 affine·chart·of·the·Grassmannian)46 affine·chart·of·the·Grassmannian)
47 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})47 ·····time·ChowV·=·chowForm(V,AffineChartGrass=>{1,2,3})
48 ·--·used·4.48232s·(cpu);·4.32064s·(thread);·0s·(gc)48 ·--·used·4.96101s·(cpu);·4.70115s·(thread);·0s·(gc)
  
49 ······4···············2··············2·····2···············249 ······4···············2··············2·····2···············2
50 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+50 o3·=·x······+·2x·····x·····x······+·x·····x······-·2x·····x·····x······+
51 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,551 ······1,2,4·····0,2,4·1,2,4·2,3,4····0,2,4·2,3,4·····1,2,3·1,2,4·1,2,5
52 ·····------------------------------------------------------------------------52 ·····------------------------------------------------------------------------
53 ······2·····2··············253 ······2·····2··············2
54 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+54 ·····x·····x······-·x·····x·····x······+·x·····x·····x·····x······+
Offset 235, 33 lines modifiedOffset 235, 33 lines modified
235 1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5235 1,4,5···0,2,4·0,1,5····0,1,4·0,2,5····0,1,2·0,4,5···1,2,3·0,1,5····0,1,3·1,2,5
236 0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4236 0,1,2·1,3,5···0,2,3·0,1,5····0,1,3·0,2,5····0,1,2·0,3,5···1,2,4·0,3,4····0,2,4
237 1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4237 1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4
238 0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3238 0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3
239 0,1,4····0,1,3·0,2,4····0,1,2·0,3,4239 0,1,4····0,1,3·0,2,4····0,1,2·0,3,4
240 i4·:·--·equivalently·(but·faster)...240 i4·:·--·equivalently·(but·faster)...
241 ·····time·assert(ChowV·===·chowForm·f)241 ·····time·assert(ChowV·===·chowForm·f)
242 ·--·used·1.151s·(cpu);·1.08375s·(thread);·0s·(gc)242 ·--·used·1.02189s·(cpu);·0.908277s·(thread);·0s·(gc)
243 i5·:·--·X-resultant·of·V243 i5·:·--·X-resultant·of·V
244 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;244 ·····time·Xres·=·fromPluckerToStiefel·dualize·ChowV;
245 ·--·used·0.158288s·(cpu);·0.127679s·(thread);·0s·(gc)245 ·--·used·0.215244s·(cpu);·0.15912s·(thread);·0s·(gc)
246 i6·:·--·three·generic·ternary·quadrics246 i6·:·--·three·generic·ternary·quadrics
247 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)247 ·····F·=·genericPolynomials({2,2,2},ZZ/3331)
  
248 ·········2···············2························2·····2···············2248 ·········2···············2························2·····2···············2
249 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+249 o6·=·{a·x··+·a·x·x··+·a·x··+·a·x·x··+·a·x·x··+·a·x·,·b·x··+·b·x·x··+·b·x··+
250 ·······0·0····1·0·1····3·1····2·0·2····4·1·2····5·2···0·0····1·0·1····3·1250 ·······0·0····1·0·1····3·1····2·0·2····4·1·2····5·2···0·0····1·0·1····3·1
251 ·····------------------------------------------------------------------------251 ·····------------------------------------------------------------------------
252 ··························2·····2···············2························2252 ··························2·····2···············2························2
253 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}253 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}
254 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2254 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
255 o6·:·List255 o6·:·List
256 i7·:·--·resultant·of·the·three·forms256 i7·:·--·resultant·of·the·three·forms
257 ·····time·resF·=·resultant·F;257 ·····time·resF·=·resultant·F;
258 ·--·used·0.170361s·(cpu);·0.138395s·(thread);·0s·(gc)258 ·--·used·0.242248s·(cpu);·0.18567s·(thread);·0s·(gc)
259 i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring259 i8·:·assert(resF·===·sub(Xres,vars·ring·resF)·and·Xres·===·sub(resF,vars·ring
260 Xres))260 Xres))
261 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*261 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
262 ····*·_\x8t_\x8a_\x8n_\x8g_\x8e_\x8n_\x8t_\x8i_\x8a_\x8l_\x8C_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·higher·Chow·forms·of·a·projective·variety262 ····*·_\x8t_\x8a_\x8n_\x8g_\x8e_\x8n_\x8t_\x8i_\x8a_\x8l_\x8C_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·higher·Chow·forms·of·a·projective·variety
263 ····*·_\x8h_\x8u_\x8r_\x8w_\x8i_\x8t_\x8z_\x8F_\x8o_\x8r_\x8m·--·Hurwitz·form·of·a·projective·variety263 ····*·_\x8h_\x8u_\x8r_\x8w_\x8i_\x8t_\x8z_\x8F_\x8o_\x8r_\x8m·--·Hurwitz·form·of·a·projective·variety
264 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ch\x8ho\x8ow\x8wF\x8Fo\x8or\x8rm\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*264 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·c\x8ch\x8ho\x8ow\x8wF\x8Fo\x8or\x8rm\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*
265 ····*·chowForm(Ideal)265 ····*·chowForm(Ideal)
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83 ········2··············283 ········2··············2
84 o2·=·a*x··+·b*x*y·+·c*y84 o2·=·a*x··+·b*x*y·+·c*y
  
85 o2·:·ZZ[a..c][x..y]</code></pre>85 o2·:·ZZ[a..c][x..y]</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i3·:·time·discriminant·F88 <td>··············<pre><code·class="language-macaulay2">i3·:·time·discriminant·F
89 ·--·used·0.0115009s·(cpu);·0.00761948s·(thread);·0s·(gc)89 ·--·used·0.0120058s·(cpu);·0.00883976s·(thread);·0s·(gc)
  
90 ········290 ········2
91 o3·=·-·b··+·4a*c91 o3·=·-·b··+·4a*c
  
92 o3·:·ZZ[a..c]</code></pre>92 o3·:·ZZ[a..c]</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
Offset 100, 15 lines modifiedOffset 100, 15 lines modified
100 ········3······2·········2······3100 ········3······2·········2······3
101 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y101 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y
  
102 o5·:·ZZ[a..d][x..y]</code></pre>102 o5·:·ZZ[a..d][x..y]</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i6·:·time·discriminant·F105 <td>··············<pre><code·class="language-macaulay2">i6·:·time·discriminant·F
106 ·--·used·0.0769243s·(cpu);·0.0520463s·(thread);·0s·(gc)106 ·--·used·0.0596817s·(cpu);·0.0224293s·(thread);·0s·(gc)
  
107 ········2·2·······3·····3···················2·2107 ········2·2·······3·····3···················2·2
108 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d108 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d
  
109 o6·:·ZZ[a..d]</code></pre>109 o6·:·ZZ[a..d]</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ········</table>111 ········</table>
Offset 149, 15 lines modifiedOffset 149, 15 lines modified
149 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x149 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x
150 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·3150 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·3
  
151 o12·:·R'</code></pre>151 o12·:·R'</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i13·:·time·D=discriminant·pencil154 <td>··············<pre><code·class="language-macaulay2">i13·:·time·D=discriminant·pencil
155 ·--·used·0.284848s·(cpu);·0.288677s·(thread);·0s·(gc)155 ·--·used·0.444882s·(cpu);·0.442355s·(thread);·0s·(gc)
  
156 ···········108······106·2·······102·6······100·8·······98·10·······96·12··156 ···········108······106·2·······102·6······100·8·······98·10·······96·12··
157 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+157 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+
158 ···········0········0···1·······0···1······0···1·······0··1········0··1···158 ···········0········0···1·······0···1······0···1·······0··1········0··1···
159 ······-----------------------------------------------------------------------159 ······-----------------------------------------------------------------------
160 ··········94·14·······92·16······90·18······88·20······86·22·······84·24··160 ··········94·14·······92·16······90·18······88·20······86·22·······84·24··
161 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+161 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+
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24 i1·:·ZZ[a,b,c][x,y];·F·=·a*x^2+b*x*y+c*y^224 i1·:·ZZ[a,b,c][x,y];·F·=·a*x^2+b*x*y+c*y^2
  
25 ········2··············225 ········2··············2
26 o2·=·a*x··+·b*x*y·+·c*y26 o2·=·a*x··+·b*x*y·+·c*y
  
27 o2·:·ZZ[a..c][x..y]27 o2·:·ZZ[a..c][x..y]
28 i3·:·time·discriminant·F28 i3·:·time·discriminant·F
29 ·--·used·0.0115009s·(cpu);·0.00761948s·(thread);·0s·(gc)29 ·--·used·0.0120058s·(cpu);·0.00883976s·(thread);·0s·(gc)
  
30 ········230 ········2
31 o3·=·-·b··+·4a*c31 o3·=·-·b··+·4a*c
  
32 o3·:·ZZ[a..c]32 o3·:·ZZ[a..c]
33 i4·:·ZZ[a,b,c,d][x,y];·F·=·a*x^3+b*x^2*y+c*x*y^2+d*y^333 i4·:·ZZ[a,b,c,d][x,y];·F·=·a*x^3+b*x^2*y+c*x*y^2+d*y^3
  
34 ········3······2·········2······334 ········3······2·········2······3
35 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y35 o5·=·a*x··+·b*x·y·+·c*x*y··+·d*y
  
36 o5·:·ZZ[a..d][x..y]36 o5·:·ZZ[a..d][x..y]
37 i6·:·time·discriminant·F37 i6·:·time·discriminant·F
38 ·--·used·0.0769243s·(cpu);·0.0520463s·(thread);·0s·(gc)38 ·--·used·0.0596817s·(cpu);·0.0224293s·(thread);·0s·(gc)
  
39 ········2·2·······3·····3···················2·239 ········2·2·······3·····3···················2·2
40 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d40 o6·=·-·b·c··+·4a*c··+·4b·d·-·18a*b*c*d·+·27a·d
  
41 o6·:·ZZ[a..d]41 o6·:·ZZ[a..d]
42 The·next·example·illustrates·how·computing·the·intersection·of·a·pencil42 The·next·example·illustrates·how·computing·the·intersection·of·a·pencil
43 generated·by·two·degree·$d$·forms·$F(x_0,\ldots,x_n),·G(x_0,\ldots,x_n)$·with43 generated·by·two·degree·$d$·forms·$F(x_0,\ldots,x_n),·G(x_0,\ldots,x_n)$·with
Offset 75, 15 lines modifiedOffset 75, 15 lines modified
  
75 ················4········3······4·············4········3······475 ················4········3······4·············4········3······4
76 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x76 o12·=·(t··+·t·)x··-·t·x·x··+·t·x··+·(t··-·t·)x··+·t·x·x··+·t·x
77 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·377 ········0····1··0····1·0·1····0·1·····0····1··2····1·2·3····0·3
  
78 o12·:·R'78 o12·:·R'
79 i13·:·time·D=discriminant·pencil79 i13·:·time·D=discriminant·pencil
80 ·--·used·0.284848s·(cpu);·0.288677s·(thread);·0s·(gc)80 ·--·used·0.444882s·(cpu);·0.442355s·(thread);·0s·(gc)
  
81 ···········108······106·2·······102·6······100·8·······98·10·······96·1281 ···········108······106·2·······102·6······100·8·······98·10·······96·12
82 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+82 o13·=·-·62t····+·19t···t··+·160t···t··+·91t···t··+·129t··t···+·117t··t···+
83 ···········0········0···1·······0···1······0···1·······0··1········0··183 ···········0········0···1·······0···1······0···1·······0··1········0··1
84 ······-----------------------------------------------------------------------84 ······-----------------------------------------------------------------------
85 ··········94·14·······92·16······90·18······88·20······86·22·······84·2485 ··········94·14·······92·16······90·18······88·20······86·22·······84·24
86 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+86 ······161t··t···+·124t··t···-·82t··t···-·21t··t···-·49t··t···-·123t··t···+
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Offset 91, 26 lines modifiedOffset 91, 26 lines modified
91 ······0·391 ······0·3
  
92 o1·:·Ideal·of·QQ[x·..x·]92 o1·:·Ideal·of·QQ[x·..x·]
93 ··················0···5</code></pre>93 ··················0···5</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i2·:·time·V'·=·dualVariety·V96 <td>··············<pre><code·class="language-macaulay2">i2·:·time·V'·=·dualVariety·V
97 ·--·used·0.125978s·(cpu);·0.100075s·(thread);·0s·(gc)97 ·--·used·0.191585s·(cpu);·0.145973s·(thread);·0s·(gc)
  
98 ············2·················2····298 ············2·················2····2
99 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)99 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)
100 ············2·3····1·2·4····0·4····1·5·····0·3·5100 ············2·3····1·2·4····0·4····1·5·····0·3·5
  
101 o2·:·Ideal·of·QQ[x·..x·]101 o2·:·Ideal·of·QQ[x·..x·]
102 ··················0···5</code></pre>102 ··················0···5</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i3·:·time·V·==·dualVariety·V'105 <td>··············<pre><code·class="language-macaulay2">i3·:·time·V·==·dualVariety·V'
106 ·--·used·0.152069s·(cpu);·0.13056s·(thread);·0s·(gc)106 ·--·used·0.189701s·(cpu);·0.15041s·(thread);·0s·(gc)
  
107 o3·=·true</code></pre>107 o3·=·true</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ········</table>109 ········</table>
110 ········<p>In·the·next·example,·we·verify·that·the·discriminant·of·a·generic·ternary·cubic·form·coincides·with·the·dual·variety·of·the·3-th·Veronese·embedding·of·the·plane,·which·is·a·hypersurface·of·degree·12·in·$\mathbb{P}^9$</p>110 ········<p>In·the·next·example,·we·verify·that·the·discriminant·of·a·generic·ternary·cubic·form·coincides·with·the·dual·variety·of·the·3-th·Veronese·embedding·of·the·plane,·which·is·a·hypersurface·of·degree·12·in·$\mathbb{P}^9$</p>
111 ········<table·class="examples">111 ········<table·class="examples">
112 ··········<tr>112 ··········<tr>
Offset 126, 23 lines modifiedOffset 126, 23 lines modified
  
126 ······ZZ126 ······ZZ
127 o4·:·----[a·..a·][x·..x·]127 o4·:·----[a·..a·][x·..x·]
128 ·····3331··0···9···0···2</code></pre>128 ·····3331··0···9···0···2</code></pre>
129 </td>··········</tr>129 </td>··········</tr>
130 ··········<tr>130 ··········<tr>
131 <td>··············<pre><code·class="language-macaulay2">i5·:·time·discF·=·ideal·discriminant·F;131 <td>··············<pre><code·class="language-macaulay2">i5·:·time·discF·=·ideal·discriminant·F;
132 ·--·used·0.0494826s·(cpu);·0.0474469s·(thread);·0s·(gc)132 ·--·used·0.0697871s·(cpu);·0.0676273s·(thread);·0s·(gc)
  
133 ···············ZZ133 ···············ZZ
134 o5·:·Ideal·of·----[a·..a·]134 o5·:·Ideal·of·----[a·..a·]
135 ··············3331··0···9</code></pre>135 ··············3331··0···9</code></pre>
136 </td>··········</tr>136 </td>··········</tr>
137 ··········<tr>137 ··········<tr>
138 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);138 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
139 ·--·used·0.544585s·(cpu);·0.520612s·(thread);·0s·(gc)139 ·--·used·0.819147s·(cpu);·0.755464s·(thread);·0s·(gc)
  
140 ···············ZZ140 ···············ZZ
141 o6·:·Ideal·of·----[x·..x·]141 o6·:·Ideal·of·----[x·..x·]
142 ··············3331··0···9</code></pre>142 ··············3331··0···9</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)145 <td>··············<pre><code·class="language-macaulay2">i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)
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32 ·····------------------------------------------------------------------------32 ·····------------------------------------------------------------------------
33 ·····x·x·)33 ·····x·x·)
34 ······0·334 ······0·3
  
35 o1·:·Ideal·of·QQ[x·..x·]35 o1·:·Ideal·of·QQ[x·..x·]
36 ··················0···536 ··················0···5
37 i2·:·time·V'·=·dualVariety·V37 i2·:·time·V'·=·dualVariety·V
38 ·--·used·0.125978s·(cpu);·0.100075s·(thread);·0s·(gc)38 ·--·used·0.191585s·(cpu);·0.145973s·(thread);·0s·(gc)
  
39 ············2·················2····239 ············2·················2····2
40 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)40 o2·=·ideal(x·x··-·x·x·x··+·x·x··+·x·x··-·4x·x·x·)
41 ············2·3····1·2·4····0·4····1·5·····0·3·541 ············2·3····1·2·4····0·4····1·5·····0·3·5
  
42 o2·:·Ideal·of·QQ[x·..x·]42 o2·:·Ideal·of·QQ[x·..x·]
43 ··················0···543 ··················0···5
44 i3·:·time·V·==·dualVariety·V'44 i3·:·time·V·==·dualVariety·V'
45 ·--·used·0.152069s·(cpu);·0.13056s·(thread);·0s·(gc)45 ·--·used·0.189701s·(cpu);·0.15041s·(thread);·0s·(gc)
  
46 o3·=·true46 o3·=·true
47 In·the·next·example,·we·verify·that·the·discriminant·of·a·generic·ternary·cubic47 In·the·next·example,·we·verify·that·the·discriminant·of·a·generic·ternary·cubic
48 form·coincides·with·the·dual·variety·of·the·3-th·Veronese·embedding·of·the48 form·coincides·with·the·dual·variety·of·the·3-th·Veronese·embedding·of·the
49 plane,·which·is·a·hypersurface·of·degree·12·in·$\mathbb{P}^9$49 plane,·which·is·a·hypersurface·of·degree·12·in·$\mathbb{P}^9$
50 i4·:·F·=·first·genericPolynomials({3,-1,-1},ZZ/3331)50 i4·:·F·=·first·genericPolynomials({3,-1,-1},ZZ/3331)
  
Offset 61, 21 lines modifiedOffset 61, 21 lines modified
61 ·····a·x·x··+·a·x61 ·····a·x·x··+·a·x
62 ······8·1·2····9·262 ······8·1·2····9·2
  
63 ······ZZ63 ······ZZ
64 o4·:·----[a·..a·][x·..x·]64 o4·:·----[a·..a·][x·..x·]
65 ·····3331··0···9···0···265 ·····3331··0···9···0···2
66 i5·:·time·discF·=·ideal·discriminant·F;66 i5·:·time·discF·=·ideal·discriminant·F;
67 ·--·used·0.0494826s·(cpu);·0.0474469s·(thread);·0s·(gc)67 ·--·used·0.0697871s·(cpu);·0.0676273s·(thread);·0s·(gc)
  
68 ···············ZZ68 ···············ZZ
69 o5·:·Ideal·of·----[a·..a·]69 o5·:·Ideal·of·----[a·..a·]
70 ··············3331··0···970 ··············3331··0···9
71 i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);71 i6·:·time·Z·=·dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
72 ·--·used·0.544585s·(cpu);·0.520612s·(thread);·0s·(gc)72 ·--·used·0.819147s·(cpu);·0.755464s·(thread);·0s·(gc)
  
73 ···············ZZ73 ···············ZZ
74 o6·:·Ideal·of·----[x·..x·]74 o6·:·Ideal·of·----[x·..x·]
75 ··············3331··0···975 ··············3331··0···9
76 i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)76 i7·:·discF·==·sub(Z,vars·ring·discF)·and·Z·==·sub(discF,vars·ring·Z)
  
77 o7·=·true77 o7·=·true
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Offset 85, 15 lines modifiedOffset 85, 15 lines modified
85 ·············2····1·3···1·2····0·3···1····0·285 ·············2····1·3···1·2····0·3···1····0·2
  
86 o1·:·Ideal·of·QQ[x·..x·]86 o1·:·Ideal·of·QQ[x·..x·]
87 ··················0···3</code></pre>87 ··················0···3</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i2·:·time·fromPluckerToStiefel·dualize·chowForm·C90 <td>··············<pre><code·class="language-macaulay2">i2·:·time·fromPluckerToStiefel·dualize·chowForm·C
91 ·--·used·0.0820792s·(cpu);·0.0518427s·(thread);·0s·(gc)91 ·--·used·0.110952s·(cpu);·0.0618939s·(thread);·0s·(gc)
  
92 ········3···3··········2···2··············2·······2··········2···3····92 ········3···3··········2···2··············2·······2··········2···3····
93 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-93 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-
94 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,1··94 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,1··
95 ·····------------------------------------------------------------------------95 ·····------------------------------------------------------------------------
96 ······2·······2···············2···2···································96 ······2·······2···············2···2···································
97 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-97 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-
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136 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,3136 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,3
  
137 o2·:·QQ[x···..x···]137 o2·:·QQ[x···..x···]
138 ·········0,0···1,3</code></pre>138 ·········0,0···1,3</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})141 <td>··············<pre><code·class="language-macaulay2">i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})
142 ·--·used·0.0388993s·(cpu);·0.0380534s·(thread);·0s·(gc)142 ·--·used·0.0378539s·(cpu);·0.0373618s·(thread);·0s·(gc)
  
143 ············3··········2·························2························143 ············3··········2·························2························
144 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-144 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-
145 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,3··145 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,3··
146 ·····------------------------------------------------------------------------146 ·····------------------------------------------------------------------------
147 ···········2······3······2147 ···········2······3······2
148 ·····2x···x····+·x····+·x148 ·····2x···x····+·x····+·x
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171 ········<p>As·another·application,·we·check·that·the·singular·locus·of·the·Chow·form·of·the·twisted·cubic·has·dimension·2·(on·each·standard·chart).</p>171 ········<p>As·another·application,·we·check·that·the·singular·locus·of·the·Chow·form·of·the·twisted·cubic·has·dimension·2·(on·each·standard·chart).</p>
172 ········<table·class="examples">172 ········<table·class="examples">
173 ··········<tr>173 ··········<tr>
174 <td>··············<pre><code·class="language-macaulay2">i5·:·w·=·chowForm·C;</code></pre>174 <td>··············<pre><code·class="language-macaulay2">i5·:·w·=·chowForm·C;</code></pre>
175 </td>··········</tr>175 </td>··········</tr>
176 ··········<tr>176 ··········<tr>
177 <td>··············<pre><code·class="language-macaulay2">i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel(w,AffineChartGrass=>s))177 <td>··············<pre><code·class="language-macaulay2">i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel(w,AffineChartGrass=>s))
178 ·--·used·0.0695371s·(cpu);·0.0358889s·(thread);·0s·(gc)178 ·--·used·0.0983379s·(cpu);·0.0445234s·(thread);·0s·(gc)
  
179 ···················3··········2··········3·······················2········179 ···················3··········2··········3·······················2········
180 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+180 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+
181 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3··181 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3··
182 ·····------------------------------------------------------------------------182 ·····------------------------------------------------------------------------
183 ·························2······2············2···3···············2········183 ·························2······2············2···3···············2········
184 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+184 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+
Offset 217, 15 lines modifiedOffset 217, 15 lines modified
217 ·····2x···x····-·x····+·x···)}217 ·····2x···x····-·x····+·x···)}
218 ·······0,0·1,1····1,1····1,0218 ·······0,0·1,1····1,1····1,0
  
219 o6·:·List</code></pre>219 o6·:·List</code></pre>
220 </td>··········</tr>220 </td>··········</tr>
221 ··········<tr>221 ··········<tr>
222 <td>··············<pre><code·class="language-macaulay2">i7·:·time·apply(U,u->dim·singularLocus·u)222 <td>··············<pre><code·class="language-macaulay2">i7·:·time·apply(U,u->dim·singularLocus·u)
223 ·--·used·0.0159583s·(cpu);·0.0165241s·(thread);·0s·(gc)223 ·--·used·0.0177967s·(cpu);·0.0188219s·(thread);·0s·(gc)
  
224 o7·=·{2,·2,·2,·2,·2,·2}224 o7·=·{2,·2,·2,·2,·2,·2}
  
225 o7·:·List</code></pre>225 o7·:·List</code></pre>
226 </td>··········</tr>226 </td>··········</tr>
227 ········</table>227 ········</table>
228 ······</div>228 ······</div>
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27 ·············2·······················227 ·············2·······················2
28 o1·=·ideal·(x··-·x·x·,·x·x··-·x·x·,·x··-·x·x·)28 o1·=·ideal·(x··-·x·x·,·x·x··-·x·x·,·x··-·x·x·)
29 ·············2····1·3···1·2····0·3···1····0·229 ·············2····1·3···1·2····0·3···1····0·2
  
30 o1·:·Ideal·of·QQ[x·..x·]30 o1·:·Ideal·of·QQ[x·..x·]
31 ··················0···331 ··················0···3
32 i2·:·time·fromPluckerToStiefel·dualize·chowForm·C32 i2·:·time·fromPluckerToStiefel·dualize·chowForm·C
33 ·--·used·0.0820792s·(cpu);·0.0518427s·(thread);·0s·(gc)33 ·--·used·0.110952s·(cpu);·0.0618939s·(thread);·0s·(gc)
  
34 ········3···3··········2···2··············2·······2··········2···334 ········3···3··········2···2··············2·······2··········2···3
35 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-35 o2·=·-·x···x····+·x···x···x···x····-·x···x···x···x····+·x···x···x····-
36 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,136 ········0,3·1,0····0,2·0,3·1,0·1,1····0,1·0,3·1,0·1,1····0,0·0,3·1,1
37 ·····------------------------------------------------------------------------37 ·····------------------------------------------------------------------------
38 ······2·······2···············2···238 ······2·······2···············2···2
39 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-39 ·····x···x···x···x····+·2x···x···x···x····+·x···x···x···x···x···x····-
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76 ··········2·······2·······2···········2······2···········2······3···376 ··········2·······2·······2···········2······2···········2······3···3
77 ·····x···x···x···x····-·2x···x···x···x····-·x···x···x···x····+·x···x77 ·····x···x···x···x····-·2x···x···x···x····-·x···x···x···x····+·x···x
78 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,378 ······0,0·0,1·1,1·1,3·····0,0·0,2·1,1·1,3····0,0·0,1·1,2·1,3····0,0·1,3
  
79 o2·:·QQ[x···..x···]79 o2·:·QQ[x···..x···]
80 ·········0,0···1,380 ·········0,0···1,3
81 i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})81 i3·:·time·fromPluckerToStiefel(dualize·chowForm·C,AffineChartGrass=>{0,1})
82 ·--·used·0.0388993s·(cpu);·0.0380534s·(thread);·0s·(gc)82 ·--·used·0.0378539s·(cpu);·0.0373618s·(thread);·0s·(gc)
  
83 ············3··········2·························283 ············3··········2·························2
84 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-84 o3·=·-·x···x····+·x···x···x····-·x···x···x····+·x···x····+·3x···x···x····-
85 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,385 ········0,3·1,2····0,2·1,2·1,3····0,2·0,3·1,2····0,2·1,3·····0,3·1,2·1,3
86 ·····------------------------------------------------------------------------86 ·····------------------------------------------------------------------------
87 ···········2······3······287 ···········2······3······2
88 ·····2x···x····+·x····+·x88 ·····2x···x····+·x····+·x
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106 o4·:·QQ[a···..a···]106 o4·:·QQ[a···..a···]
107 ·········0,0···1,1107 ·········0,0···1,1
108 As·another·application,·we·check·that·the·singular·locus·of·the·Chow·form·of108 As·another·application,·we·check·that·the·singular·locus·of·the·Chow·form·of
109 the·twisted·cubic·has·dimension·2·(on·each·standard·chart).109 the·twisted·cubic·has·dimension·2·(on·each·standard·chart).
110 i5·:·w·=·chowForm·C;110 i5·:·w·=·chowForm·C;
111 i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel111 i6·:·time·U·=·apply(subsets(4,2),s->ideal·fromPluckerToStiefel
112 (w,AffineChartGrass=>s))112 (w,AffineChartGrass=>s))
113 ·--·used·0.0695371s·(cpu);·0.0358889s·(thread);·0s·(gc)113 ·--·used·0.0983379s·(cpu);·0.0445234s·(thread);·0s·(gc)
  
114 ···················3··········2··········3·······················2114 ···················3··········2··········3·······················2
115 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+115 o6·=·{ideal(-·x···x····+·x···x···x····-·x····-·3x···x···x····+·2x···x····+
116 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3116 ···············0,3·1,2····0,2·1,2·1,3····0,2·····0,2·0,3·1,2·····0,2·1,3
117 ·····------------------------------------------------------------------------117 ·····------------------------------------------------------------------------
118 ·························2······2············2···3···············2118 ·························2······2············2···3···············2
119 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+119 ·····x···x···x····-·x···x····+·x···),·ideal(x···x····-·2x···x···x···x····+
Offset 150, 15 lines modifiedOffset 150, 15 lines modified
150 ·····------------------------------------------------------------------------150 ·····------------------------------------------------------------------------
151 ···········2······3······2151 ···········2······3······2
152 ·····2x···x····-·x····+·x···)}152 ·····2x···x····-·x····+·x···)}
153 ·······0,0·1,1····1,1····1,0153 ·······0,0·1,1····1,1····1,0
  
154 o6·:·List154 o6·:·List
155 i7·:·time·apply(U,u->dim·singularLocus·u)155 i7·:·time·apply(U,u->dim·singularLocus·u)
156 ·--·used·0.0159583s·(cpu);·0.0165241s·(thread);·0s·(gc)156 ·--·used·0.0177967s·(cpu);·0.0188219s·(thread);·0s·(gc)
  
157 o7·=·{2,·2,·2,·2,·2,·2}157 o7·=·{2,·2,·2,·2,·2,·2}
  
158 o7·:·List158 o7·:·List
159 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·f\x8fr\x8ro\x8om\x8mP\x8Pl\x8lu\x8uc\x8ck\x8ke\x8er\x8rT\x8To\x8oS\x8St\x8ti\x8ie\x8ef\x8fe\x8el\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*159 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·f\x8fr\x8ro\x8om\x8mP\x8Pl\x8lu\x8uc\x8ck\x8ke\x8er\x8rT\x8To\x8oS\x8St\x8ti\x8ie\x8ef\x8fe\x8el\x8l:\x8:·*\x8**\x8**\x8**\x8**\x8*
160 ····*·fromPluckerToStiefel(Ideal)160 ····*·fromPluckerToStiefel(Ideal)
161 ····*·fromPluckerToStiefel(Matrix)161 ····*·fromPluckerToStiefel(Matrix)
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Offset 95, 15 lines modifiedOffset 95, 15 lines modified
95 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····495 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····4
  
96 o1·:·Ideal·of·QQ[p·..p·]96 o1·:·Ideal·of·QQ[p·..p·]
97 ··················0···4</code></pre>97 ··················0···4</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i2·:·time·hurwitzForm·Q100 <td>··············<pre><code·class="language-macaulay2">i2·:·time·hurwitzForm·Q
101 ·--·used·0.0742897s·(cpu);·0.0527309s·(thread);·0s·(gc)101 ·--·used·0.0713701s·(cpu);·0.0467324s·(thread);·0s·(gc)
  
102 ············2····························2··································102 ············2····························2··································
103 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p···103 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p···
104 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,2104 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,2
105 ·····------------------------------------------------------------------------105 ·····------------------------------------------------------------------------
106 ···········2··············································2·················106 ···········2··············································2·················
107 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p···107 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p···
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35 ······7·2··········7·······1·······7·········235 ······7·2··········7·······1·······7·········2
36 ·····--p··+·p·p··+·-p·p··+·-p·p··+·-p·p··+·7p·)36 ·····--p··+·p·p··+·-p·p··+·-p·p··+·-p·p··+·7p·)
37 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····437 ·····10·3····0·4···4·1·4···2·2·4···3·3·4·····4
  
38 o1·:·Ideal·of·QQ[p·..p·]38 o1·:·Ideal·of·QQ[p·..p·]
39 ··················0···439 ··················0···4
40 i2·:·time·hurwitzForm·Q40 i2·:·time·hurwitzForm·Q
41 ·--·used·0.0742897s·(cpu);·0.0527309s·(thread);·0s·(gc)41 ·--·used·0.0713701s·(cpu);·0.0467324s·(thread);·0s·(gc)
  
42 ············2····························242 ············2····························2
43 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p43 o2·=·143100p····+·267300p···p····+·96525p····-·56700p···p····-·56100p···p
44 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,244 ············0,1··········0,1·0,2·········0,2·········0,1·1,2·········0,2·1,2
45 ·····------------------------------------------------------------------------45 ·····------------------------------------------------------------------------
46 ···········2··············································246 ···········2··············································2
47 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p47 ·····+·900p····+·140400p···p····+·111780p···p····+·133380p····-·8100p···p
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99 ·········0,1···0,2···1,2···0,3···1,3···2,399 ·········0,1···0,2···1,2···0,3···1,3···2,3
100 o1·:·--------------------------------------100 o1·:·--------------------------------------
101 ·········p···p····-·p···p····+·p···p101 ·········p···p····-·p···p····+·p···p
102 ··········1,2·0,3····0,2·1,3····0,1·2,3</code></pre>102 ··········1,2·0,3····0,2·1,3····0,1·2,3</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i2·:·time·isCoisotropic·w105 <td>··············<pre><code·class="language-macaulay2">i2·:·time·isCoisotropic·w
106 ·--·used·0.00799718s·(cpu);·0.00601644s·(thread);·0s·(gc)106 ·--·used·0.00835667s·(cpu);·0.00865306s·(thread);·0s·(gc)
  
107 o2·=·true</code></pre>107 o2·=·true</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i3·:·--·random·quadric·in·G(1,3)110 <td>··············<pre><code·class="language-macaulay2">i3·:·--·random·quadric·in·G(1,3)
111 ·····w'·=·random(2,Grass(1,3))111 ·····w'·=·random(2,Grass(1,3))
  
Offset 131, 15 lines modifiedOffset 131, 15 lines modified
131 ·········0,1···0,2···1,2···0,3···1,3···2,3131 ·········0,1···0,2···1,2···0,3···1,3···2,3
132 o3·:·--------------------------------------132 o3·:·--------------------------------------
133 ·········p···p····-·p···p····+·p···p133 ·········p···p····-·p···p····+·p···p
134 ··········1,2·0,3····0,2·1,3····0,1·2,3</code></pre>134 ··········1,2·0,3····0,2·1,3····0,1·2,3</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isCoisotropic·w'137 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isCoisotropic·w'
138 ·--·used·0.0034741s·(cpu);·0.00520254s·(thread);·0s·(gc)138 ·--·used·0.00746541s·(cpu);·0.00720071s·(thread);·0s·(gc)
  
139 o4·=·false</code></pre>139 o4·=·false</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ········</table>141 ········</table>
142 ······</div>142 ······</div>
143 ······<div·class="waystouse">143 ······<div·class="waystouse">
144 ········<h2>Ways·to·use·<span·class="tt">isCoisotropic</span>:</h2>144 ········<h2>Ways·to·use·<span·class="tt">isCoisotropic</span>:</h2>
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41 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]41 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]
42 ·········0,1···0,2···1,2···0,3···1,3···2,342 ·········0,1···0,2···1,2···0,3···1,3···2,3
43 o1·:·--------------------------------------43 o1·:·--------------------------------------
44 ·········p···p····-·p···p····+·p···p44 ·········p···p····-·p···p····+·p···p
45 ··········1,2·0,3····0,2·1,3····0,1·2,345 ··········1,2·0,3····0,2·1,3····0,1·2,3
46 i2·:·time·isCoisotropic·w46 i2·:·time·isCoisotropic·w
47 ·--·used·0.00799718s·(cpu);·0.00601644s·(thread);·0s·(gc)47 ·--·used·0.00835667s·(cpu);·0.00865306s·(thread);·0s·(gc)
  
48 o2·=·true48 o2·=·true
49 i3·:·--·random·quadric·in·G(1,3)49 i3·:·--·random·quadric·in·G(1,3)
50 ·····w'·=·random(2,Grass(1,3))50 ·····w'·=·random(2,Grass(1,3))
  
51 ·····7·2··················2·····3·························2·····551 ·····7·2··················2·····3·························2·····5
52 o3·=·-p····+·7p···p····+·p····+·-p···p····+·2p···p····+·5p····+·-p···p····+52 o3·=·-p····+·7p···p····+·p····+·-p···p····+·2p···p····+·5p····+·-p···p····+
Offset 69, 14 lines modifiedOffset 69, 14 lines modified
  
69 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]69 ·····QQ[p···..p···,·p···,·p···,·p···,·p···]
70 ·········0,1···0,2···1,2···0,3···1,3···2,370 ·········0,1···0,2···1,2···0,3···1,3···2,3
71 o3·:·--------------------------------------71 o3·:·--------------------------------------
72 ·········p···p····-·p···p····+·p···p72 ·········p···p····-·p···p····+·p···p
73 ··········1,2·0,3····0,2·1,3····0,1·2,373 ··········1,2·0,3····0,2·1,3····0,1·2,3
74 i4·:·time·isCoisotropic·w'74 i4·:·time·isCoisotropic·w'
75 ·--·used·0.0034741s·(cpu);·0.00520254s·(thread);·0s·(gc)75 ·--·used·0.00746541s·(cpu);·0.00720071s·(thread);·0s·(gc)
  
76 o4·=·false76 o4·=·false
77 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sC\x8Co\x8oi\x8is\x8so\x8ot\x8tr\x8ro\x8op\x8pi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*77 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sC\x8Co\x8oi\x8is\x8so\x8ot\x8tr\x8ro\x8op\x8pi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*
78 ····*·isCoisotropic(RingElement)78 ····*·isCoisotropic(RingElement)
79 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*79 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
80 The·object·_\x8i_\x8s_\x8C_\x8o_\x8i_\x8s_\x8o_\x8t_\x8r_\x8o_\x8p_\x8i_\x8c·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.80 The·object·_\x8i_\x8s_\x8C_\x8o_\x8i_\x8s_\x8o_\x8t_\x8r_\x8o_\x8p_\x8i_\x8c·is·a·_\x8m_\x8e_\x8t_\x8h_\x8o_\x8d_\x8·_\x8f_\x8u_\x8n_\x8c_\x8t_\x8i_\x8o_\x8n_\x8·_\x8w_\x8i_\x8t_\x8h_\x8·_\x8o_\x8p_\x8t_\x8i_\x8o_\x8n_\x8s.
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114 ················ZZ114 ················ZZ
115 o3·:·Ideal·of·-----[x·..x·]115 o3·:·Ideal·of·-----[x·..x·]
116 ··············33331··0···5</code></pre>116 ··············33331··0···5</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))119 <td>··············<pre><code·class="language-macaulay2">i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))
120 ·--·used·0.0120067s·(cpu);·0.0154729s·(thread);·0s·(gc)120 ·--·used·0.026085s·(cpu);·0.0247257s·(thread);·0s·(gc)
  
121 o4·=·true</code></pre>121 o4·=·true</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ········</table>123 ········</table>
124 ······</div>124 ······</div>
125 ······<div>125 ······<div>
126 ········<h2>See·also</h2>126 ········<h2>See·also</h2>
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55 ·····2380x··+·9482x·)55 ·····2380x··+·9482x·)
56 ··········4········556 ··········4········5
  
57 ················ZZ57 ················ZZ
58 o3·:·Ideal·of·-----[x·..x·]58 o3·:·Ideal·of·-----[x·..x·]
59 ··············33331··0···559 ··············33331··0···5
60 i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))60 i4·:·time·isInCoisotropic(L,I)·--·whether·L·belongs·to·Z_1(V(I))
61 ·--·used·0.0120067s·(cpu);·0.0154729s·(thread);·0s·(gc)61 ·--·used·0.026085s·(cpu);·0.0247257s·(thread);·0s·(gc)
  
62 o4·=·true62 o4·=·true
63 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*63 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
64 ····*·_\x8t_\x8a_\x8n_\x8g_\x8e_\x8n_\x8t_\x8i_\x8a_\x8l_\x8C_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·higher·Chow·forms·of·a·projective·variety64 ····*·_\x8t_\x8a_\x8n_\x8g_\x8e_\x8n_\x8t_\x8i_\x8a_\x8l_\x8C_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·higher·Chow·forms·of·a·projective·variety
65 ····*·_\x8p_\x8l_\x8u_\x8c_\x8k_\x8e_\x8r·--·get·the·Plücker·coordinates·of·a·linear·subspace65 ····*·_\x8p_\x8l_\x8u_\x8c_\x8k_\x8e_\x8r·--·get·the·Plücker·coordinates·of·a·linear·subspace
66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sI\x8In\x8nC\x8Co\x8oi\x8is\x8so\x8ot\x8tr\x8ro\x8op\x8pi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*66 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sI\x8In\x8nC\x8Co\x8oi\x8is\x8so\x8ot\x8tr\x8ro\x8op\x8pi\x8ic\x8c:\x8:·*\x8**\x8**\x8**\x8**\x8*
67 ····*·isInCoisotropic(Ideal,Ideal)67 ····*·isInCoisotropic(Ideal,Ideal)
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84 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}84 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}
85 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·285 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·2
  
86 o1·:·List</code></pre>86 o1·:·List</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i2·:·time·(D,D')·=·macaulayFormula·F89 <td>··············<pre><code·class="language-macaulay2">i2·:·time·(D,D')·=·macaulayFormula·F
90 ·--·used·0.000913945s·(cpu);·0.00275152s·(thread);·0s·(gc)90 ·--·used·0.00400117s·(cpu);·0.0047077s·(thread);·0s·(gc)
  
91 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···0··91 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···0··
92 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···0··92 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···0··
93 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0··93 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0··
94 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···0··94 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···0··
95 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0··95 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0··
96 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0··96 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0··
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151 ·····-p·p··+·2p·p··+·6p·}151 ·····-p·p··+·2p·p··+·6p·}
152 ·····7·0·2·····1·2·····2152 ·····7·0·2·····1·2·····2
  
153 o3·:·List</code></pre>153 o3·:·List</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(D,D')·=·macaulayFormula·F156 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(D,D')·=·macaulayFormula·F
157 ·--·used·0.000456377s·(cpu);·0.00212078s·(thread);·0s·(gc)157 ·--·used·0.00210445s·(cpu);·0.0030709s·(thread);·0s·(gc)
  
158 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····0··158 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····0··
159 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0··159 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0··
160 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0··160 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0··
161 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0··161 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0··
162 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0··162 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0··
163 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/4163 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/4
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29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
30 ···················2··········2········2······330 ···················2··········2········2······3
31 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}31 ·····c·x·x·x··+·c·x·x··+·c·x·x··+·c·x·x··+·c·x·}
32 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·232 ······4·0·1·2····7·1·2····5·0·2····8·1·2····9·2
  
33 o1·:·List33 o1·:·List
34 i2·:·time·(D,D')·=·macaulayFormula·F34 i2·:·time·(D,D')·=·macaulayFormula·F
35 ·--·used·0.000913945s·(cpu);·0.00275152s·(thread);·0s·(gc)35 ·--·used·0.00400117s·(cpu);·0.0047077s·(thread);·0s·(gc)
  
36 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···036 o2·=·(|·a_0·a_1·a_2·a_3·a_4·a_5·0···0···0···0···0···0···0···0···0···0···0
37 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···037 ······|·0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0···0
38 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···038 ······|·0···0···a_0·0···a_1·a_2·0···a_3·a_4·a_5·0···0···0···0···0···0···0
39 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···039 ······|·0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0···0
40 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···040 ······|·0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0···0
41 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···041 ······|·0···0···0···0···0···a_0·0···0···a_1·a_2·0···0···a_3·a_4·a_5·0···0
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92 ·····------------------------------------------------------------------------92 ·····------------------------------------------------------------------------
93 ·····6···2·······2·····393 ·····6···2·······2·····3
94 ·····-p·p··+·2p·p··+·6p·}94 ·····-p·p··+·2p·p··+·6p·}
95 ·····7·0·2·····1·2·····295 ·····7·0·2·····1·2·····2
  
96 o3·:·List96 o3·:·List
97 i4·:·time·(D,D')·=·macaulayFormula·F97 i4·:·time·(D,D')·=·macaulayFormula·F
98 ·--·used·0.000456377s·(cpu);·0.00212078s·(thread);·0s·(gc)98 ·--·used·0.00210445s·(cpu);·0.0030709s·(thread);·0s·(gc)
  
99 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····099 o4·=·(|·9/2·1/2··9/4··1/2·1····3/4··0···0···0···0···0···0····0····0····0
100 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0100 ······|·0···9/2··0····1/2·9/4··0····1/2·1···3/4·0···0···0····0····0····0
101 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0101 ······|·0···0····9/2··0···1/2··9/4··0···1/2·1···3/4·0···0····0····0····0
102 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0102 ······|·0···0····0····9/2·0····0····1/2·9/4·0···0···1/2·1····3/4··0····0
103 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0103 ······|·0···0····0····0···9/2··0····0···1/2·9/4·0···0···1/2··1····3/4··0
104 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/4104 ······|·0···0····0····0···0····9/2··0···0···1/2·9/4·0···0····1/2··1····3/4
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91 ·····664x·)91 ·····664x·)
92 ·········492 ·········4
  
93 o3·:·Ideal·of·P4</code></pre>93 o3·:·Ideal·of·P4</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i4·:·time·p·=·plucker·L96 <td>··············<pre><code·class="language-macaulay2">i4·:·time·p·=·plucker·L
97 ·--·used·0.00400072s·(cpu);·0.00347169s·(thread);·0s·(gc)97 ·--·used·0.00438053s·(cpu);·0.004682s·(thread);·0s·(gc)
  
98 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+98 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+
99 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,3··99 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,3··
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
101 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+101 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+
102 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,2··102 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,2··
103 ·····------------------------------------------------------------------------103 ·····------------------------------------------------------------------------
104 ·····11804x···,·x····+·14854x···)104 ·····11804x···,·x····+·14854x···)
105 ···········3,4···0,1·········3,4105 ···········3,4···0,1·········3,4
  
106 o4·:·Ideal·of·G'1'4</code></pre>106 o4·:·Ideal·of·G'1'4</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i5·:·time·L'·=·plucker·p109 <td>··············<pre><code·class="language-macaulay2">i5·:·time·L'·=·plucker·p
110 ·--·used·0.0637691s·(cpu);·0.0399772s·(thread);·0s·(gc)110 ·--·used·0.0908882s·(cpu);·0.0414548s·(thread);·0s·(gc)
  
111 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-111 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-
112 ·············2········3·········4···1········3·········4···0·········3··112 ·············2········3·········4···1········3·········4···0·········3··
113 ·····------------------------------------------------------------------------113 ·····------------------------------------------------------------------------
114 ·····664x·)114 ·····664x·)
115 ·········4115 ·········4
  
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129 ··········<tr>129 ··········<tr>
130 <td>··············<pre><code·class="language-macaulay2">i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve130 <td>··············<pre><code·class="language-macaulay2">i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve
  
131 o7·:·Ideal·of·G'1'4</code></pre>131 o7·:·Ideal·of·G'1'4</code></pre>
132 </td>··········</tr>132 </td>··········</tr>
133 ··········<tr>133 ··········<tr>
134 <td>··············<pre><code·class="language-macaulay2">i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y134 <td>··············<pre><code·class="language-macaulay2">i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y
135 ·--·used·0.0258229s·(cpu);·0.0262577s·(thread);·0s·(gc)135 ·--·used·0.0306115s·(cpu);·0.0300526s·(thread);·0s·(gc)
  
136 o8·:·Ideal·of·P4</code></pre>136 o8·:·Ideal·of·P4</code></pre>
137 </td>··········</tr>137 </td>··········</tr>
138 ··········<tr>138 ··········<tr>
139 <td>··············<pre><code·class="language-macaulay2">i9·:·(codim·W,degree·W)139 <td>··············<pre><code·class="language-macaulay2">i9·:·(codim·W,degree·W)
  
140 o9·=·(2,·5)140 o9·=·(2,·5)
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145 o9·:·Sequence</code></pre>145 o9·:·Sequence</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ········</table>147 ········</table>
148 ········<p>In·this·example,·we·can·recover·the·subvariety·$Y\subset\mathbb{G}(k,\mathbb{P}^n)$·by·computing·the·Fano·variety·of·$k$-planes·contained·in·$W$.</p>148 ········<p>In·this·example,·we·can·recover·the·subvariety·$Y\subset\mathbb{G}(k,\mathbb{P}^n)$·by·computing·the·Fano·variety·of·$k$-planes·contained·in·$W$.</p>
149 ········<table·class="examples">149 ········<table·class="examples">
150 ··········<tr>150 ··········<tr>
151 <td>··············<pre><code·class="language-macaulay2">i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W151 <td>··············<pre><code·class="language-macaulay2">i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W
152 ·--·used·0.124984s·(cpu);·0.12384s·(thread);·0s·(gc)152 ·--·used·0.170711s·(cpu);·0.170006s·(thread);·0s·(gc)
  
153 o10·:·Ideal·of·G'1'4</code></pre>153 o10·:·Ideal·of·G'1'4</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i11·:·assert(Y'·==·Y)</code></pre>156 <td>··············<pre><code·class="language-macaulay2">i11·:·assert(Y'·==·Y)</code></pre>
157 </td>··········</tr>157 </td>··········</tr>
158 ········</table>158 ········</table>
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29 ·············2········3·········4···1········3·········4···0·········329 ·············2········3·········4···1········3·········4···0·········3
30 ·····------------------------------------------------------------------------30 ·····------------------------------------------------------------------------
31 ·····664x·)31 ·····664x·)
32 ·········432 ·········4
  
33 o3·:·Ideal·of·P433 o3·:·Ideal·of·P4
34 i4·:·time·p·=·plucker·L34 i4·:·time·p·=·plucker·L
35 ·--·used·0.00400072s·(cpu);·0.00347169s·(thread);·0s·(gc)35 ·--·used·0.00438053s·(cpu);·0.004682s·(thread);·0s·(gc)
  
36 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+36 o4·=·ideal·(x····+·8480x···,·x····-·6727x···,·x····+·15777x···,·x····+
37 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,337 ·············2,4········3,4···1,4········3,4···0,4·········3,4···2,3
38 ·····------------------------------------------------------------------------38 ·····------------------------------------------------------------------------
39 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+39 ·····11656x···,·x····-·14853x···,·x····+·664x···,·x····+·13522x···,·x····+
40 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,240 ···········3,4···1,3·········3,4···0,3·······3,4···1,2·········3,4···0,2
41 ·····------------------------------------------------------------------------41 ·····------------------------------------------------------------------------
42 ·····11804x···,·x····+·14854x···)42 ·····11804x···,·x····+·14854x···)
43 ···········3,4···0,1·········3,443 ···········3,4···0,1·········3,4
  
44 o4·:·Ideal·of·G'1'444 o4·:·Ideal·of·G'1'4
45 i5·:·time·L'·=·plucker·p45 i5·:·time·L'·=·plucker·p
46 ·--·used·0.0637691s·(cpu);·0.0399772s·(thread);·0s·(gc)46 ·--·used·0.0908882s·(cpu);·0.0414548s·(thread);·0s·(gc)
  
47 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-47 o5·=·ideal·(x··+·8480x··-·11656x·,·x··-·6727x··+·14853x·,·x··+·15777x··-
48 ·············2········3·········4···1········3·········4···0·········348 ·············2········3·········4···1········3·········4···0·········3
49 ·····------------------------------------------------------------------------49 ·····------------------------------------------------------------------------
50 ·····664x·)50 ·····664x·)
51 ·········451 ·········4
  
Offset 61, 26 lines modifiedOffset 61, 26 lines modified
61 $W\subset\mathbb{P}^n$·swept·out·by·the·linear·spaces·corresponding·to·points61 $W\subset\mathbb{P}^n$·swept·out·by·the·linear·spaces·corresponding·to·points
62 of·$Y$.·As·an·example,·we·now·compute·a·surface·scroll·$W\subset\mathbb{P}^4$62 of·$Y$.·As·an·example,·we·now·compute·a·surface·scroll·$W\subset\mathbb{P}^4$
63 over·an·elliptic·curve·$Y\subset\mathbb{G}(1,\mathbb{P}^4)$.63 over·an·elliptic·curve·$Y\subset\mathbb{G}(1,\mathbb{P}^4)$.
64 i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve64 i7·:·Y·=·ideal·apply(5,i->random(1,G'1'4));·--·an·elliptic·curve
  
65 o7·:·Ideal·of·G'1'465 o7·:·Ideal·of·G'1'4
66 i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y66 i8·:·time·W·=·plucker·Y;·--·surface·swept·out·by·the·lines·of·Y
67 ·--·used·0.0258229s·(cpu);·0.0262577s·(thread);·0s·(gc)67 ·--·used·0.0306115s·(cpu);·0.0300526s·(thread);·0s·(gc)
  
68 o8·:·Ideal·of·P468 o8·:·Ideal·of·P4
69 i9·:·(codim·W,degree·W)69 i9·:·(codim·W,degree·W)
  
70 o9·=·(2,·5)70 o9·=·(2,·5)
  
71 o9·:·Sequence71 o9·:·Sequence
72 In·this·example,·we·can·recover·the·subvariety·$Y\subset\mathbb{G}(k,\mathbb72 In·this·example,·we·can·recover·the·subvariety·$Y\subset\mathbb{G}(k,\mathbb
73 {P}^n)$·by·computing·the·Fano·variety·of·$k$-planes·contained·in·$W$.73 {P}^n)$·by·computing·the·Fano·variety·of·$k$-planes·contained·in·$W$.
74 i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W74 i10·:·time·Y'·=·plucker(W,1);·--·variety·of·lines·contained·in·W
75 ·--·used·0.124984s·(cpu);·0.12384s·(thread);·0s·(gc)75 ·--·used·0.170711s·(cpu);·0.170006s·(thread);·0s·(gc)
  
76 o10·:·Ideal·of·G'1'476 o10·:·Ideal·of·G'1'4
77 i11·:·assert(Y'·==·Y)77 i11·:·assert(Y'·==·Y)
78 W\x8Wa\x8ar\x8rn\x8ni\x8in\x8ng\x8g:·Notice·that,·by·default,·the·computation·is·done·on·a·randomly·chosen78 W\x8Wa\x8ar\x8rn\x8ni\x8in\x8ng\x8g:·Notice·that,·by·default,·the·computation·is·done·on·a·randomly·chosen
79 affine·chart·on·the·Grassmannian.·To·change·this·behavior,·you·can·use·the79 affine·chart·on·the·Grassmannian.·To·change·this·behavior,·you·can·use·the
80 _\x8A_\x8f_\x8f_\x8i_\x8n_\x8e_\x8C_\x8h_\x8a_\x8r_\x8t_\x8G_\x8r_\x8a_\x8s_\x8s·option.80 _\x8A_\x8f_\x8f_\x8i_\x8n_\x8e_\x8C_\x8h_\x8a_\x8r_\x8t_\x8G_\x8r_\x8a_\x8s_\x8s·option.
81 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8pl\x8lu\x8uc\x8ck\x8ke\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*81 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·p\x8pl\x8lu\x8uc\x8ck\x8ke\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
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99 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}99 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}
100 ·····4··········8····8··········7················4····3····5100 ·····4··········8····8··········7················4····3····5
  
101 o2·:·List</code></pre>101 o2·:·List</code></pre>
102 </td>··········</tr>102 </td>··········</tr>
103 ··········<tr>103 ··········<tr>
104 <td>··············<pre><code·class="language-macaulay2">i3·:·time·resultant(F,Algorithm=>&quot;Poisson2&quot;)104 <td>··············<pre><code·class="language-macaulay2">i3·:·time·resultant(F,Algorithm=>&quot;Poisson2&quot;)
105 ·--·used·0.202256s·(cpu);·0.151554s·(thread);·0s·(gc)105 ·--·used·0.285233s·(cpu);·0.190647s·(thread);·0s·(gc)
  
106 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···106 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
107 o3·=·-·-----------------------------a··-·-------------------------------a·b·-107 o3·=·-·-----------------------------a··-·-------------------------------a·b·-
108 ···········2222549728809984000000············12700284164628480000000·········108 ···········2222549728809984000000············12700284164628480000000·········
109 ·····------------------------------------------------------------------------109 ·····------------------------------------------------------------------------
110 ·····348237304382147063838108483692249·3·2··110 ·····348237304382147063838108483692249·3·2··
111 ·····---------------------------------a·b··-111 ·····---------------------------------a·b··-
Offset 121, 15 lines modifiedOffset 121, 15 lines modified
121 ·····-------------------------------a*b··-·---------------------------b121 ·····-------------------------------a*b··-·---------------------------b
122 ··········1119954511872000000000················895963609497600000122 ··········1119954511872000000000················895963609497600000
  
123 o3·:·QQ[a..b]</code></pre>123 o3·:·QQ[a..b]</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i4·:·time·resultant(F,Algorithm=>&quot;Macaulay2&quot;)126 <td>··············<pre><code·class="language-macaulay2">i4·:·time·resultant(F,Algorithm=>&quot;Macaulay2&quot;)
127 ·--·used·0.0985538s·(cpu);·0.0741696s·(thread);·0s·(gc)127 ·--·used·0.101548s·(cpu);·0.0716955s·(thread);·0s·(gc)
  
128 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···128 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
129 o4·=·-·-----------------------------a··-·-------------------------------a·b·-129 o4·=·-·-----------------------------a··-·-------------------------------a·b·-
130 ···········2222549728809984000000············12700284164628480000000·········130 ···········2222549728809984000000············12700284164628480000000·········
131 ·····------------------------------------------------------------------------131 ·····------------------------------------------------------------------------
132 ·····348237304382147063838108483692249·3·2··132 ·····348237304382147063838108483692249·3·2··
133 ·····---------------------------------a·b··-133 ·····---------------------------------a·b··-
Offset 143, 15 lines modifiedOffset 143, 15 lines modified
143 ·····-------------------------------a*b··-·---------------------------b143 ·····-------------------------------a*b··-·---------------------------b
144 ··········1119954511872000000000················895963609497600000144 ··········1119954511872000000000················895963609497600000
  
145 o4·:·QQ[a..b]</code></pre>145 o4·:·QQ[a..b]</code></pre>
146 </td>··········</tr>146 </td>··········</tr>
147 ··········<tr>147 ··········<tr>
148 <td>··············<pre><code·class="language-macaulay2">i5·:·time·resultant(F,Algorithm=>&quot;Poisson&quot;)148 <td>··············<pre><code·class="language-macaulay2">i5·:·time·resultant(F,Algorithm=>&quot;Poisson&quot;)
149 ·--·used·0.224459s·(cpu);·0.224151s·(thread);·0s·(gc)149 ·--·used·0.248571s·(cpu);·0.249901s·(thread);·0s·(gc)
  
150 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···150 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
151 o5·=·-·-----------------------------a··-·-------------------------------a·b·-151 o5·=·-·-----------------------------a··-·-------------------------------a·b·-
152 ···········2222549728809984000000············12700284164628480000000·········152 ···········2222549728809984000000············12700284164628480000000·········
153 ·····------------------------------------------------------------------------153 ·····------------------------------------------------------------------------
154 ·····348237304382147063838108483692249·3·2··154 ·····348237304382147063838108483692249·3·2··
155 ·····---------------------------------a·b··-155 ·····---------------------------------a·b··-
Offset 165, 15 lines modifiedOffset 165, 15 lines modified
165 ·····-------------------------------a*b··-·---------------------------b165 ·····-------------------------------a*b··-·---------------------------b
166 ··········1119954511872000000000················895963609497600000166 ··········1119954511872000000000················895963609497600000
  
167 o5·:·QQ[a..b]</code></pre>167 o5·:·QQ[a..b]</code></pre>
168 </td>··········</tr>168 </td>··········</tr>
169 ··········<tr>169 ··········<tr>
170 <td>··············<pre><code·class="language-macaulay2">i6·:·time·resultant(F,Algorithm=>&quot;Macaulay&quot;)170 <td>··············<pre><code·class="language-macaulay2">i6·:·time·resultant(F,Algorithm=>&quot;Macaulay&quot;)
171 ·--·used·0.456844s·(cpu);·0.437105s·(thread);·0s·(gc)171 ·--·used·0.56392s·(cpu);·0.535167s·(thread);·0s·(gc)
  
172 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···172 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4···
173 o6·=·-·-----------------------------a··-·-------------------------------a·b·-173 o6·=·-·-----------------------------a··-·-------------------------------a·b·-
174 ···········2222549728809984000000············12700284164628480000000·········174 ···········2222549728809984000000············12700284164628480000000·········
175 ·····------------------------------------------------------------------------175 ·····------------------------------------------------------------------------
176 ·····348237304382147063838108483692249·3·2··176 ·····348237304382147063838108483692249·3·2··
177 ·····---------------------------------a·b··-177 ·····---------------------------------a·b··-
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Offset 59, 15 lines modifiedOffset 59, 15 lines modified
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····3·····2····9····7·····2····9········3·······1····8····460 ·····3·····2····9····7·····2····9········3·······1····8····4
61 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}61 ·····-b)y*w··+·(-a·+·-b)z*w··+·(-a·+·2b)w·,·2x·+·-y·+·-z·+·-w}
62 ·····4··········8····8··········7················4····3····562 ·····4··········8····8··········7················4····3····5
  
63 o2·:·List63 o2·:·List
64 i3·:·time·resultant(F,Algorithm=>"Poisson2")64 i3·:·time·resultant(F,Algorithm=>"Poisson2")
65 ·--·used·0.202256s·(cpu);·0.151554s·(thread);·0s·(gc)65 ·--·used·0.285233s·(cpu);·0.190647s·(thread);·0s·(gc)
  
66 ·······21002161660529014459938925799·5···2085933800619238998825958079203·466 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4
67 o3·=·-·-----------------------------a··-·-------------------------------a·b·-67 o3·=·-·-----------------------------a··-·-------------------------------a·b·-
68 ···········2222549728809984000000············1270028416462848000000068 ···········2222549728809984000000············12700284164628480000000
69 ·····------------------------------------------------------------------------69 ·····------------------------------------------------------------------------
70 ·····348237304382147063838108483692249·3·270 ·····348237304382147063838108483692249·3·2
71 ·····---------------------------------a·b··-71 ·····---------------------------------a·b··-
Offset 79, 15 lines modifiedOffset 79, 15 lines modified
79 ·····------------------------------------------------------------------------79 ·····------------------------------------------------------------------------
80 ·····1146977327343523453866040839029···4···194441910898734675845094443·580 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
81 ·····-------------------------------a*b··-·---------------------------b81 ·····-------------------------------a*b··-·---------------------------b
82 ··········1119954511872000000000················89596360949760000082 ··········1119954511872000000000················895963609497600000
  
83 o3·:·QQ[a..b]83 o3·:·QQ[a..b]
84 i4·:·time·resultant(F,Algorithm=>"Macaulay2")84 i4·:·time·resultant(F,Algorithm=>"Macaulay2")
85 ·--·used·0.0985538s·(cpu);·0.0741696s·(thread);·0s·(gc)85 ·--·used·0.101548s·(cpu);·0.0716955s·(thread);·0s·(gc)
  
86 ·······21002161660529014459938925799·5···2085933800619238998825958079203·486 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4
87 o4·=·-·-----------------------------a··-·-------------------------------a·b·-87 o4·=·-·-----------------------------a··-·-------------------------------a·b·-
88 ···········2222549728809984000000············1270028416462848000000088 ···········2222549728809984000000············12700284164628480000000
89 ·····------------------------------------------------------------------------89 ·····------------------------------------------------------------------------
90 ·····348237304382147063838108483692249·3·290 ·····348237304382147063838108483692249·3·2
91 ·····---------------------------------a·b··-91 ·····---------------------------------a·b··-
Offset 99, 15 lines modifiedOffset 99, 15 lines modified
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ·····1146977327343523453866040839029···4···194441910898734675845094443·5100 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
101 ·····-------------------------------a*b··-·---------------------------b101 ·····-------------------------------a*b··-·---------------------------b
102 ··········1119954511872000000000················895963609497600000102 ··········1119954511872000000000················895963609497600000
  
103 o4·:·QQ[a..b]103 o4·:·QQ[a..b]
104 i5·:·time·resultant(F,Algorithm=>"Poisson")104 i5·:·time·resultant(F,Algorithm=>"Poisson")
105 ·--·used·0.224459s·(cpu);·0.224151s·(thread);·0s·(gc)105 ·--·used·0.248571s·(cpu);·0.249901s·(thread);·0s·(gc)
  
106 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4106 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4
107 o5·=·-·-----------------------------a··-·-------------------------------a·b·-107 o5·=·-·-----------------------------a··-·-------------------------------a·b·-
108 ···········2222549728809984000000············12700284164628480000000108 ···········2222549728809984000000············12700284164628480000000
109 ·····------------------------------------------------------------------------109 ·····------------------------------------------------------------------------
110 ·····348237304382147063838108483692249·3·2110 ·····348237304382147063838108483692249·3·2
111 ·····---------------------------------a·b··-111 ·····---------------------------------a·b··-
Offset 119, 15 lines modifiedOffset 119, 15 lines modified
119 ·····------------------------------------------------------------------------119 ·····------------------------------------------------------------------------
120 ·····1146977327343523453866040839029···4···194441910898734675845094443·5120 ·····1146977327343523453866040839029···4···194441910898734675845094443·5
121 ·····-------------------------------a*b··-·---------------------------b121 ·····-------------------------------a*b··-·---------------------------b
122 ··········1119954511872000000000················895963609497600000122 ··········1119954511872000000000················895963609497600000
  
123 o5·:·QQ[a..b]123 o5·:·QQ[a..b]
124 i6·:·time·resultant(F,Algorithm=>"Macaulay")124 i6·:·time·resultant(F,Algorithm=>"Macaulay")
125 ·--·used·0.456844s·(cpu);·0.437105s·(thread);·0s·(gc)125 ·--·used·0.56392s·(cpu);·0.535167s·(thread);·0s·(gc)
  
126 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4126 ·······21002161660529014459938925799·5···2085933800619238998825958079203·4
127 o6·=·-·-----------------------------a··-·-------------------------------a·b·-127 o6·=·-·-----------------------------a··-·-------------------------------a·b·-
128 ···········2222549728809984000000············12700284164628480000000128 ···········2222549728809984000000············12700284164628480000000
129 ·····------------------------------------------------------------------------129 ·····------------------------------------------------------------------------
130 ·····348237304382147063838108483692249·3·2130 ·····348237304382147063838108483692249·3·2
131 ·····---------------------------------a·b··-131 ·····---------------------------------a·b··-
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./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
    
Offset 90, 15 lines modifiedOffset 90, 15 lines modified
90 ·······2···············2····························3······2·········490 ·······2···············2····························3······2·········4
91 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}91 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}
  
92 o2·:·List</code></pre>92 o2·:·List</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·resultant·F95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·resultant·F
96 ·--·used·0.0167989s·(cpu);·0.0196189s·(thread);·0s·(gc)96 ·--·used·0.0205937s·(cpu);·0.023252s·(thread);·0s·(gc)
  
97 ··········12·········11·2·········10·3·········9·4··········8·5··········7·697 ··········12·········11·2·········10·3·········9·4··········8·5··········7·6
98 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u·98 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u·
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ··············6·7··········5·8·······11··········10·2·········9·3··100 ··············6·7··········5·8·······11··········10·2·········9·3··
101 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+101 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+
102 ·····------------------------------------------------------------------------102 ·····------------------------------------------------------------------------
Offset 149, 15 lines modifiedOffset 149, 15 lines modified
149 ·····+·c·x·}149 ·····+·c·x·}
150 ········9·2150 ········9·2
  
151 o4·:·List</code></pre>151 o4·:·List</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i5·:·time·resultant·F154 <td>··············<pre><code·class="language-macaulay2">i5·:·time·resultant·F
155 ·--·used·1.70599s·(cpu);·1.45931s·(thread);·0s·(gc)155 ·--·used·1.93709s·(cpu);·1.61928s·(thread);·0s·(gc)
  
156 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···2··156 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···2··
157 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-157 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-
158 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·0··158 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·0··
159 ·····------------------------------------------------------------------------159 ·····------------------------------------------------------------------------
160 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·2··160 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·2··
161 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-161 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-
Offset 1780, 15 lines modifiedOffset 1780, 15 lines modified
1780 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}1780 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}
1781 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·21781 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
1782 o6·:·List</code></pre>1782 o6·:·List</code></pre>
1783 </td>··········</tr>1783 </td>··········</tr>
1784 ··········<tr>1784 ··········<tr>
1785 <td>··············<pre><code·class="language-macaulay2">i7·:·time·#·terms·resultant·F1785 <td>··············<pre><code·class="language-macaulay2">i7·:·time·#·terms·resultant·F
1786 ·--·used·0.37424s·(cpu);·0.321845s·(thread);·0s·(gc)1786 ·--·used·0.407066s·(cpu);·0.342191s·(thread);·0s·(gc)
  
1787 o7·=·21894</code></pre>1787 o7·=·21894</code></pre>
1788 </td>··········</tr>1788 </td>··········</tr>
1789 ········</table>1789 ········</table>
1790 ······</div>1790 ······</div>
1791 ······<div>1791 ······<div>
1792 ········<h2>See·also</h2>1792 ········<h2>See·also</h2>
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Offset 33, 15 lines modifiedOffset 33, 15 lines modified
33 i2·:·F·=·{x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4}33 i2·:·F·=·{x^2+3*t*y*z-u*z^2,(t+3*u-1)*x-y,-t*x*y^3+t*x^2*y*z+u*z^4}
  
34 ·······2···············2····························3······2·········434 ·······2···············2····························3······2·········4
35 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}35 o2·=·{x··+·3t*y*z·-·u*z·,·(t·+·3u·-·1)x·-·y,·-·t*x*y··+·t*x·y*z·+·u*z·}
  
36 o2·:·List36 o2·:·List
37 i3·:·time·resultant·F37 i3·:·time·resultant·F
38 ·--·used·0.0167989s·(cpu);·0.0196189s·(thread);·0s·(gc)38 ·--·used·0.0205937s·(cpu);·0.023252s·(thread);·0s·(gc)
  
39 ··········12·········11·2·········10·3·········9·4··········8·5··········7·639 ··········12·········11·2·········10·3·········9·4··········8·5··········7·6
40 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u40 o3·=·-·81t··u·-·1701t··u··-·15309t··u··-·76545t·u··-·229635t·u··-·413343t·u
41 ·····------------------------------------------------------------------------41 ·····------------------------------------------------------------------------
42 ··············6·7··········5·8·······11··········10·2·········9·342 ··············6·7··········5·8·······11··········10·2·········9·3
43 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+43 ·····-·413343t·u··-·177147t·u··+·567t··u·+·10206t··u··+·76545t·u··+
44 ·····------------------------------------------------------------------------44 ·····------------------------------------------------------------------------
Offset 87, 15 lines modifiedOffset 87, 15 lines modified
87 ·····------------------------------------------------------------------------87 ·····------------------------------------------------------------------------
88 ··········388 ··········3
89 ·····+·c·x·}89 ·····+·c·x·}
90 ········9·290 ········9·2
  
91 o4·:·List91 o4·:·List
92 i5·:·time·resultant·F92 i5·:·time·resultant·F
93 ·--·used·1.70599s·(cpu);·1.45931s·(thread);·0s·(gc)93 ·--·used·1.93709s·(cpu);·1.61928s·(thread);·0s·(gc)
  
94 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···294 ······6·3·2·······5·2···2·····2·4···2·2····3·3·3·2·····2·4·2···2
95 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-95 o5·=·a·b·c··-·3a·a·b·b·c··+·3a·a·b·b·c··-·a·a·b·c··+·3a·a·b·b·c··-
96 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·096 ······2·3·0·····1·2·3·4·0·····1·2·3·4·0····1·2·4·0·····1·2·3·5·0
97 ·····------------------------------------------------------------------------97 ·····------------------------------------------------------------------------
98 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·298 ·······3·3·······2·····4·2·2···2·····4·2···2·2·····5·····2·2····6·3·2
99 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-99 ·····6a·a·b·b·b·c··+·3a·a·b·b·c··+·3a·a·b·b·c··-·3a·a·b·b·c··+·a·b·c··-
Offset 1713, 15 lines modifiedOffset 1713, 15 lines modified
1713 ·····------------------------------------------------------------------------1713 ·····------------------------------------------------------------------------
1714 ··························2·····2···············2························21714 ··························2·····2···············2························2
1715 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}1715 ·····b·x·x··+·b·x·x··+·b·x·,·c·x··+·c·x·x··+·c·x··+·c·x·x··+·c·x·x··+·c·x·}
1716 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·21716 ······2·0·2····4·1·2····5·2···0·0····1·0·1····3·1····2·0·2····4·1·2····5·2
  
1717 o6·:·List1717 o6·:·List
1718 i7·:·time·#·terms·resultant·F1718 i7·:·time·#·terms·resultant·F
1719 ·--·used·0.37424s·(cpu);·0.321845s·(thread);·0s·(gc)1719 ·--·used·0.407066s·(cpu);·0.342191s·(thread);·0s·(gc)
  
1720 o7·=·218941720 o7·=·21894
1721 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*1721 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
1722 ····*·_\x8c_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·Chow·form·of·a·projective·variety1722 ····*·_\x8c_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·Chow·form·of·a·projective·variety
1723 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8R_\x8i_\x8n_\x8g_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8)1723 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8R_\x8i_\x8n_\x8g_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8)
1724 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*1724 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
1725 ····*·resultant(List)1725 ····*·resultant(List)
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Offset 102, 15 lines modifiedOffset 102, 15 lines modified
  
102 o2·:·Ideal·of·QQ[p·..p·]102 o2·:·Ideal·of·QQ[p·..p·]
103 ··················0···4</code></pre>103 ··················0···4</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)106 <td>··············<pre><code·class="language-macaulay2">i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)
107 ·····time·tangentialChowForm(S,0)107 ·····time·tangentialChowForm(S,0)
108 ·--·used·0.0666742s·(cpu);·0.0409446s·(thread);·0s·(gc)108 ·--·used·0.0876343s·(cpu);·0.046571s·(thread);·0s·(gc)
  
109 ······2·······················································2········109 ······2·······················································2········
110 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+110 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+
111 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,4··111 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,4··
112 ·····------------------------------------------------------------------------112 ·····------------------------------------------------------------------------
113 ······2113 ······2
114 ·····p···p····-·2p···p···p····-·p···p···p114 ·····p···p····-·2p···p···p····-·p···p···p
Offset 121, 15 lines modifiedOffset 121, 15 lines modified
121 o3·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------121 o3·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------
122 ·····(p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···)122 ·····(p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p···p···)
123 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,4····0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,1·2,3</code></pre>123 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,4····0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,1·2,3</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
126 <td>··············<pre><code·class="language-macaulay2">i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)126 <td>··············<pre><code·class="language-macaulay2">i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)
127 ·····time·tangentialChowForm(S,1)127 ·····time·tangentialChowForm(S,1)
128 ·--·used·0.0413244s·(cpu);·0.0436383s·(thread);·0s·(gc)128 ·--·used·0.0584356s·(cpu);·0.0577753s·(thread);·0s·(gc)
  
129 ······2·····2········2·····2···············3········2·····2······129 ······2·····2········2·····2···············3········2·····2······
130 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-130 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-
131 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,4··131 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,4··
132 ·····------------------------------------------------------------------------132 ·····------------------------------------------------------------------------
133 ·············3·········3···············3············133 ·············3·········3···············3············
134 ·····4p·····p······-·4p·····p······-·2p·····p······+134 ·····4p·····p······-·4p·····p······-·2p·····p······+
Offset 164, 39 lines modifiedOffset 164, 39 lines modified
164 o4·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------164 o4·:·----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
165 ·····(p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)165 ·····(p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)
166 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4</code></pre>166 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4····0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3·1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent·hyperplanes·to·S)169 <td>··············<pre><code·class="language-macaulay2">i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent·hyperplanes·to·S)
170 ·····time·tangentialChowForm(S,2)170 ·····time·tangentialChowForm(S,2)
171 ·--·used·0.0248222s·(cpu);·0.0259189s·(thread);·0s·(gc)171 ·--·used·0.0361974s·(cpu);·0.0361604s·(thread);·0s·(gc)
  
172 ··············2·············································2172 ··············2·············································2
173 o5·=·p·······p········-·p·······p·······p········+·p·······p173 o5·=·p·······p········-·p·······p·······p········+·p·······p
174 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,4174 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,4
  
175 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]175 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]
176 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,4</code></pre>176 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,4</code></pre>
177 </td>··········</tr>177 </td>··········</tr>
178 ··········<tr>178 ··········<tr>
179 <td>··············<pre><code·class="language-macaulay2">i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing179 <td>··············<pre><code·class="language-macaulay2">i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing
180 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)180 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)
181 ·--·used·0.075193s·(cpu);·0.0492194s·(thread);·0s·(gc)181 ·--·used·0.0955263s·(cpu);·0.0648529s·(thread);·0s·(gc)
  
182 ············2···············2182 ············2···············2
183 o6·=·ideal(p·p··-·p·p·p··+·p·p·)183 o6·=·ideal(p·p··-·p·p·p··+·p·p·)
184 ············1·2····0·1·3····0·4184 ············1·2····0·1·3····0·4
  
185 o6·:·Ideal·of·QQ[p·..p·]185 o6·:·Ideal·of·QQ[p·..p·]
186 ··················0···4</code></pre>186 ··················0···4</code></pre>
187 </td>··········</tr>187 </td>··········</tr>
188 ··········<tr>188 ··········<tr>
189 <td>··············<pre><code·class="language-macaulay2">i7·:·--·we·then·can·recover·S189 <td>··············<pre><code·class="language-macaulay2">i7·:·--·we·then·can·recover·S
190 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)190 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)
191 ·--·used·0.112612s·(cpu);·0.086262s·(thread);·0s·(gc)</code></pre>191 ·--·used·0.136925s·(cpu);·0.0963347s·(thread);·0s·(gc)</code></pre>
192 </td>··········</tr>192 </td>··········</tr>
193 ········</table>193 ········</table>
194 ······</div>194 ······</div>
195 ······<div>195 ······<div>
196 ········<h2>See·also</h2>196 ········<h2>See·also</h2>
197 ········<ul>197 ········<ul>
198 ··········<li>198 ··········<li>
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Offset 64, 15 lines modifiedOffset 64, 15 lines modified
64 o2·=·ideal·(-·p·p··+·p·p·,·-·p·p··+·p·p·,·-·p··+·p·p·)64 o2·=·ideal·(-·p·p··+·p·p·,·-·p·p··+·p·p·,·-·p··+·p·p·)
65 ···············1·2····0·3·····1·3····0·4·····3····2·465 ···············1·2····0·3·····1·3····0·4·····3····2·4
  
66 o2·:·Ideal·of·QQ[p·..p·]66 o2·:·Ideal·of·QQ[p·..p·]
67 ··················0···467 ··················0···4
68 i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)68 i3·:·--·0-th·associated·hypersurface·of·S·in·G(1,4)·(Chow·form)
69 ·····time·tangentialChowForm(S,0)69 ·····time·tangentialChowForm(S,0)
70 ·--·used·0.0666742s·(cpu);·0.0409446s·(thread);·0s·(gc)70 ·--·used·0.0876343s·(cpu);·0.046571s·(thread);·0s·(gc)
  
71 ······2·······················································271 ······2·······················································2
72 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+72 o3·=·p···p····-·p···p···p····-·p···p···p····+·p···p···p····+·p···p····+
73 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,473 ······1,3·2,3····1,2·1,3·2,4····0,3·1,3·2,4····0,2·1,4·2,4····1,2·3,4
74 ·····------------------------------------------------------------------------74 ·····------------------------------------------------------------------------
75 ······275 ······2
76 ·····p···p····-·2p···p···p····-·p···p···p76 ·····p···p····-·2p···p···p····-·p···p···p
Offset 89, 15 lines modifiedOffset 89, 15 lines modified
89 -·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p89 -·p···p····+·p···p···,·p···p····-·p···p····+·p···p···,·p···p····-·p···p····+·p
90 p···)90 p···)
91 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,491 ·······2,3·1,4····1,3·2,4····1,2·3,4···2,3·0,4····0,3·2,4····0,2·3,4···1,3·0,4
92 0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,192 0,3·1,4····0,1·3,4···1,2·0,4····0,2·1,4····0,1·2,4···1,2·0,3····0,2·1,3····0,1
93 2,393 2,3
94 i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)94 i4·:·--·1-th·associated·hypersurface·of·S·in·G(2,4)
95 ·····time·tangentialChowForm(S,1)95 ·····time·tangentialChowForm(S,1)
96 ·--·used·0.0413244s·(cpu);·0.0436383s·(thread);·0s·(gc)96 ·--·used·0.0584356s·(cpu);·0.0577753s·(thread);·0s·(gc)
  
97 ······2·····2········2·····2···············3········2·····297 ······2·····2········2·····2···············3········2·····2
98 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-98 o4·=·p·····p······+·p·····p······-·2p·····p······+·p·····p······-
99 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,499 ······1,2,3·1,2,4····0,2,4·1,2,4·····0,2,3·1,2,4····0,2,4·0,3,4
100 ·····------------------------------------------------------------------------100 ·····------------------------------------------------------------------------
101 ·············3·········3···············3101 ·············3·········3···············3
102 ·····4p·····p······-·4p·····p······-·2p·····p······+102 ·····4p·····p······-·4p·····p······-·2p·····p······+
Offset 139, 35 lines modifiedOffset 139, 35 lines modified
139 p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)139 p······+·p·····p·····,·p·····p······-·p·····p······+·p·····p·····)
140 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4140 ·······1,2,4·0,3,4····0,2,4·1,3,4····0,1,4·2,3,4···1,2,3·0,3,4····0,2,3·1,3,4
141 0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3141 0,1,3·2,3,4···1,2,3·0,2,4····0,2,3·1,2,4····0,1,2·2,3,4···1,2,3·0,1,4····0,1,3
142 1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4142 1,2,4····0,1,2·1,3,4···0,2,3·0,1,4····0,1,3·0,2,4····0,1,2·0,3,4
143 i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent143 i5·:·--·2-th·associated·hypersurface·of·S·in·G(3,4)·(parameterizing·tangent
144 hyperplanes·to·S)144 hyperplanes·to·S)
145 ·····time·tangentialChowForm(S,2)145 ·····time·tangentialChowForm(S,2)
146 ·--·used·0.0248222s·(cpu);·0.0259189s·(thread);·0s·(gc)146 ·--·used·0.0361974s·(cpu);·0.0361604s·(thread);·0s·(gc)
  
147 ··············2·············································2147 ··············2·············································2
148 o5·=·p·······p········-·p·······p·······p········+·p·······p148 o5·=·p·······p········-·p·······p·······p········+·p·······p
149 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,4149 ······0,1,3,4·0,2,3,4····0,1,2,4·0,2,3,4·1,2,3,4····0,1,2,3·1,2,3,4
  
150 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]150 o5·:·QQ[p·······..p·······,·p·······,·p·······,·p·······]
151 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,4151 ·········0,1,2,3···0,1,2,4···0,1,3,4···0,2,3,4···1,2,3,4
152 i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing152 i6·:·--·we·get·the·dual·hypersurface·of·S·in·G(0,4)·by·dualizing
153 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)153 ·····time·S'·=·ideal·dualize·tangentialChowForm(S,2)
154 ·--·used·0.075193s·(cpu);·0.0492194s·(thread);·0s·(gc)154 ·--·used·0.0955263s·(cpu);·0.0648529s·(thread);·0s·(gc)
  
155 ············2···············2155 ············2···············2
156 o6·=·ideal(p·p··-·p·p·p··+·p·p·)156 o6·=·ideal(p·p··-·p·p·p··+·p·p·)
157 ············1·2····0·1·3····0·4157 ············1·2····0·1·3····0·4
  
158 o6·:·Ideal·of·QQ[p·..p·]158 o6·:·Ideal·of·QQ[p·..p·]
159 ··················0···4159 ··················0···4
160 i7·:·--·we·then·can·recover·S160 i7·:·--·we·then·can·recover·S
161 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)161 ·····time·assert(dualize·tangentialChowForm(S',3)·==·S)
162 ·--·used·0.112612s·(cpu);·0.086262s·(thread);·0s·(gc)162 ·--·used·0.136925s·(cpu);·0.0963347s·(thread);·0s·(gc)
163 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*163 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
164 ····*·_\x8i_\x8s_\x8C_\x8o_\x8i_\x8s_\x8o_\x8t_\x8r_\x8o_\x8p_\x8i_\x8c·--·whether·a·hypersurface·of·a·Grassmannian·is·a·tangential164 ····*·_\x8i_\x8s_\x8C_\x8o_\x8i_\x8s_\x8o_\x8t_\x8r_\x8o_\x8p_\x8i_\x8c·--·whether·a·hypersurface·of·a·Grassmannian·is·a·tangential
165 ······Chow·form165 ······Chow·form
166 ····*·_\x8c_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·Chow·form·of·a·projective·variety166 ····*·_\x8c_\x8h_\x8o_\x8w_\x8F_\x8o_\x8r_\x8m·--·Chow·form·of·a·projective·variety
167 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8ta\x8an\x8ng\x8ge\x8en\x8nt\x8ti\x8ia\x8al\x8lC\x8Ch\x8ho\x8ow\x8wF\x8Fo\x8or\x8rm\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*167 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8ta\x8an\x8ng\x8ge\x8en\x8nt\x8ti\x8ia\x8al\x8lC\x8Ch\x8ho\x8ow\x8wF\x8Fo\x8or\x8rm\x8m:\x8:·*\x8**\x8**\x8**\x8**\x8*
168 ····*·tangentialChowForm(Ideal,ZZ)168 ····*·tangentialChowForm(Ideal,ZZ)
169 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*169 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
⊟
736 B
./usr/share/doc/Macaulay2/RunExternalM2/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=187 #:len=18
8 aXNFeHRlcm5hbE0yUGFyZW508 aXNFeHRlcm5hbE0yUGFyZW50
9 #:len=10439 #:len=1043
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz
11 IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l11 IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l
⊟
1.39 KB
./usr/share/doc/Macaulay2/RunExternalM2/example-output/_resource_splimits.out
    
Offset 1, 19 lines modifiedOffset 1, 24 lines modified
1 --·-*-·M2-comint·-*-·hash:·13323964676561 --·-*-·M2-comint·-*-·hash:·1332396467656
  
2 i1·:·run("ulimit·-a")2 i1·:·run("ulimit·-a")
3 time(seconds)········700 
4 file(blocks)·········unlimited 
5 data(kbytes)·········unlimited 
6 stack(kbytes)········8192 
7 coredump(blocks)·····0 
8 memory(kbytes)·······850000 
9 locked·memory(kbytes)·3077552 
10 process··············96079 
11 nofiles··············512 
12 vmemory(kbytes)······unlimited3 real-time·non-blocking·time··(microseconds,·-R)·unlimited
 4 core·file·size··············(blocks,·-c)·0
 5 data·seg·size···············(kbytes,·-d)·unlimited
 6 scheduling·priority·················(-e)·0
 7 file·size···················(blocks,·-f)·unlimited
 8 pending·signals·····················(-i)·95981
 9 max·locked·memory···········(kbytes,·-l)·3076784
 10 max·memory·size·············(kbytes,·-m)·850000
 11 open·files··························(-n)·512
 12 pipe·size················(512·bytes,·-p)·8
 13 POSIX·message·queues·········(bytes,·-q)·819200
 14 real-time·priority··················(-r)·0
 15 stack·size··················(kbytes,·-s)·8192
 16 cpu·time···················(seconds,·-t)·700
 17 max·user·processes··················(-u)·95981
 18 virtual·memory··············(kbytes,·-v)·unlimited
13 locks················unlimited19 file·locks··························(-x)·unlimited
14 rtprio···············0 
  
15 o1·=·020 o1·=·0
  
16 i2·:·21 i2·:·
⊟
9.04 KB
./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out
    
Offset 1, 23 lines modifiedOffset 1, 23 lines modified
1 --·-*-·M2-comint·-*-·hash:·29279780664557873951 --·-*-·M2-comint·-*-·hash:·2927978066455787395
  
2 i1·:·fn=temporaryFileName()|".m2"2 i1·:·fn=temporaryFileName()|".m2"
  
3 o1·=·/tmp/M2-3830577-0/0.m23 o1·=·/tmp/M2-2801784-0/0.m2
  
4 i2·:·fn<<///·square·=·(x)·->·(stderr<<"Running"<<endl;·sleep(1);·x^2);·///<<endl;4 i2·:·fn<<///·square·=·(x)·->·(stderr<<"Running"<<endl;·sleep(1);·x^2);·///<<endl;
  
5 i3·:·fn<<///·justexit·=·()·->·(·exit(27);·);·///<<endl;5 i3·:·fn<<///·justexit·=·()·->·(·exit(27);·);·///<<endl;
  
6 i4·:·fn<<///·spin·=·(x)·->·(stderr<<"Spinning!!"<<endl;·startTime:=cpuTime();·while(cpuTime()-startTime<x)·do·for·i·to·10000000·do·i;·return(x););·///<<endl;6 i4·:·fn<<///·spin·=·(x)·->·(stderr<<"Spinning!!"<<endl;·startTime:=cpuTime();·while(cpuTime()-startTime<x)·do·for·i·to·10000000·do·i;·return(x););·///<<endl;
  
7 i5·:·fn<<flush;7 i5·:·fn<<flush;
  
8 i6·:·h=runExternalM2(fn,"square",(4));8 i6·:·h=runExternalM2(fn,"square",(4));
9 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/1.m2"·>"/tmp/M2-3830577-0/1.out"·2>&1·))9 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/1.m2"·>"/tmp/M2-2801784-0/1.out"·2>&1·))
10 Finished·running.10 Finished·running.
  
11 i7·:·h11 i7·:·h
  
12 o7·=·HashTable{"answer·file"·=>·null}12 o7·=·HashTable{"answer·file"·=>·null}
13 ···············"exit·code"·=>·013 ···············"exit·code"·=>·0
14 ···············"output·file"·=>·null14 ···············"output·file"·=>·null
Offset 33, 105 lines modifiedOffset 33, 104 lines modified
33 o8·=·true33 o8·=·true
  
34 i9·:·h#"exit·code"===034 i9·:·h#"exit·code"===0
  
35 o9·=·true35 o9·=·true
  
36 i10·:·h=runExternalM2(fn,"justexit",());36 i10·:·h=runExternalM2(fn,"justexit",());
37 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/2.m2"·>"/tmp/M2-3830577-0/2.out"·2>&1·))37 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/2.m2"·>"/tmp/M2-2801784-0/2.out"·2>&1·))
38 Finished·running.38 Finished·running.
39 RunExternalM2:·expected·answer·file·does·not·exist39 RunExternalM2:·expected·answer·file·does·not·exist
  
40 i11·:·h40 i11·:·h
  
41 o11·=·HashTable{"answer·file"·=>·/tmp/M2-3830577-0/2.ans}41 o11·=·HashTable{"answer·file"·=>·/tmp/M2-2801784-0/2.ans}
42 ················"exit·code"·=>·2742 ················"exit·code"·=>·27
43 ················"output·file"·=>·/tmp/M2-3830577-0/2.out43 ················"output·file"·=>·/tmp/M2-2801784-0/2.out
44 ················"return·code"·=>·691244 ················"return·code"·=>·6912
45 ················"statistics"·=>·null45 ················"statistics"·=>·null
46 ················"time·used"·=>·246 ················"time·used"·=>·1
47 ················value·=>·null47 ················value·=>·null
  
48 o11·:·HashTable48 o11·:·HashTable
  
49 i12·:·fileExists(h#"output·file")49 i12·:·fileExists(h#"output·file")
  
50 o12·=·true50 o12·=·true
  
51 i13·:·fileExists(h#"answer·file")51 i13·:·fileExists(h#"answer·file")
  
52 o13·=·false52 o13·=·false
  
53 i14·:·h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit·-t·2");53 i14·:·h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit·-t·2");
54 Running·(ulimit·-t·2·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/3.m2"·>"/tmp/M2-3830577-0/3.out"·2>&1·))54 Running·(ulimit·-t·2·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/3.m2"·>"/tmp/M2-2801784-0/3.out"·2>&1·))
55 Killed 
56 Finished·running.55 Finished·running.
57 RunExternalM2:·expected·answer·file·does·not·exist56 RunExternalM2:·expected·answer·file·does·not·exist
  
58 i15·:·h57 i15·:·h
  
59 o15·=·HashTable{"answer·file"·=>·/tmp/M2-3830577-0/3.ans}58 o15·=·HashTable{"answer·file"·=>·/tmp/M2-2801784-0/3.ans}
60 ················"exit·code"·=>·059 ················"exit·code"·=>·0
61 ················"output·file"·=>·/tmp/M2-3830577-0/3.out60 ················"output·file"·=>·/tmp/M2-2801784-0/3.out
62 ················"return·code"·=>·961 ················"return·code"·=>·9
63 ················"statistics"·=>·null62 ················"statistics"·=>·null
64 ················"time·used"·=>·463 ················"time·used"·=>·3
65 ················value·=>·null64 ················value·=>·null
  
66 o15·:·HashTable65 o15·:·HashTable
  
67 i16·:·if·h#"output·file"·=!=·null·and·fileExists(h#"output·file")·then·get(h#"output·file")66 i16·:·if·h#"output·file"·=!=·null·and·fileExists(h#"output·file")·then·get(h#"output·file")
  
68 o16·=·67 o16·=·
69 ······i1·:·--·Script·/tmp/M2-3830577-0/3.m2·automatically·generated·by·RunExternalM268 ······i1·:·--·Script·/tmp/M2-2801784-0/3.m2·automatically·generated·by·RunExternalM2
70 ···········needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});69 ···········needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
  
71 ······i2·:·load·"/tmp/M2-3830577-0/0.m2";70 ······i2·:·load·"/tmp/M2-2801784-0/0.m2";
  
72 ······i3·:·runExternalM2ReturnAnswer("/tmp/M2-3830577-0/3.ans",spin·(10));71 ······i3·:·runExternalM2ReturnAnswer("/tmp/M2-2801784-0/3.ans",spin·(10));
73 ······Spinning!!72 ······Spinning!!
  
  
74 i17·:·if·h#"answer·file"·=!=·null·and·fileExists(h#"answer·file")·then·get(h#"answer·file")73 i17·:·if·h#"answer·file"·=!=·null·and·fileExists(h#"answer·file")·then·get(h#"answer·file")
  
75 i18·:·h=runExternalM2(fn,"spin",3,KeepStatistics=>true);74 i18·:·h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
76 Running·(true·&&·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/4.m2"·>"/tmp/M2-3830577-0/4.out"·2>&1')·>"/tmp/M2-3830577-0/4.stat"·2>&1·))75 Running·(true·&&·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/4.m2"·>"/tmp/M2-2801784-0/4.out"·2>&1')·>"/tmp/M2-2801784-0/4.stat"·2>&1·))
77 Finished·running.76 Finished·running.
  
78 i19·:·h#"statistics"77 i19·:·h#"statistics"
  
79 o19·=·········Command·being·timed:·"sh·-c·/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/4.m2"·>"/tmp/M2-3830577-0/4.out"·2>&1"78 o19·=·········Command·being·timed:·"sh·-c·/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/4.m2"·>"/tmp/M2-2801784-0/4.out"·2>&1"
80 ··············User·time·(seconds):·4.2579 ··············User·time·(seconds):·4.84
81 ··············System·time·(seconds):·0.0480 ··············System·time·(seconds):·0.33
82 ··············Percent·of·CPU·this·job·got:·53%81 ··············Percent·of·CPU·this·job·got:·88%
83 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:08.0182 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:05.86
84 ··············Average·shared·text·size·(kbytes):·083 ··············Average·shared·text·size·(kbytes):·0
85 ··············Average·unshared·data·size·(kbytes):·084 ··············Average·unshared·data·size·(kbytes):·0
86 ··············Average·stack·size·(kbytes):·085 ··············Average·stack·size·(kbytes):·0
87 ··············Average·total·size·(kbytes):·086 ··············Average·total·size·(kbytes):·0
88 ··············Maximum·resident·set·size·(kbytes):·30051287 ··············Maximum·resident·set·size·(kbytes):·296888
89 ··············Average·resident·set·size·(kbytes):·088 ··············Average·resident·set·size·(kbytes):·0
90 ··············Major·(requiring·I/O)·page·faults:·089 ··············Major·(requiring·I/O)·page·faults:·0
91 ··············Minor·(reclaiming·a·frame)·page·faults:·1017990 ··············Minor·(reclaiming·a·frame)·page·faults:·71400
92 ··············Voluntary·context·switches:·259391 ··············Voluntary·context·switches:·6148
93 ··············Involuntary·context·switches:·72092 ··············Involuntary·context·switches:·2185
94 ··············Swaps:·093 ··············Swaps:·0
95 ··············File·system·inputs:·094 ··············File·system·inputs:·0
96 ··············File·system·outputs:·1695 ··············File·system·outputs:·24
97 ··············Socket·messages·sent:·096 ··············Socket·messages·sent:·0
98 ··············Socket·messages·received:·097 ··············Socket·messages·received:·0
99 ··············Signals·delivered:·098 ··············Signals·delivered:·0
100 ··············Page·size·(bytes):·409699 ··············Page·size·(bytes):·4096
101 ··············Exit·status:·0100 ··············Exit·status:·0
  
  
102 i20·:·v=///·A·complicated·string^%&C@#CERQVASDFQ#BQBSDH"'·ewrjwklsf///;101 i20·:·v=///·A·complicated·string^%&C@#CERQVASDFQ#BQBSDH"'·ewrjwklsf///;
  
103 i21·:·(runExternalM2(fn,identity,v))#value===v102 i21·:·(runExternalM2(fn,identity,v))#value===v
104 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-3830577-0/6.m2"·>"/tmp/M2-3830577-0/6.out"·2>&1·))103 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/M2-2801784-0/6.m2"·>"/tmp/M2-2801784-0/6.out"·2>&1·))
105 Finished·running.104 Finished·running.
  
106 o21·=·true105 o21·=·true
  
107 i22·:·R=QQ[x,y];106 i22·:·R=QQ[x,y];
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60 ········<div>60 ········<div>
61 ··········<p>which·will·run·M2·with·these·limits·while·leaving·the·shell's·ulimits·unchanged.</p>61 ··········<p>which·will·run·M2·with·these·limits·while·leaving·the·shell's·ulimits·unchanged.</p>
62 ··········<p><b>From·within·Macaulay2</b>,·you·can·<b>view</b>·the·ulimits·currently·available·to·the·Macaulay2·process·by·using·the·<a·title="run·an·external·command"·href="../../Macaulay2Doc/html/_run.html">run</a>·command:</p>62 ··········<p><b>From·within·Macaulay2</b>,·you·can·<b>view</b>·the·ulimits·currently·available·to·the·Macaulay2·process·by·using·the·<a·title="run·an·external·command"·href="../../Macaulay2Doc/html/_run.html">run</a>·command:</p>
63 ········</div>63 ········</div>
64 ········<table·class="examples">64 ········<table·class="examples">
65 ··········<tr>65 ··········<tr>
66 <td>··············<pre><code·class="language-macaulay2">i1·:·run(&quot;ulimit·-a&quot;)66 <td>··············<pre><code·class="language-macaulay2">i1·:·run(&quot;ulimit·-a&quot;)
67 time(seconds)········700 
68 file(blocks)·········unlimited 
69 data(kbytes)·········unlimited 
70 stack(kbytes)········8192 
71 coredump(blocks)·····0 
72 memory(kbytes)·······850000 
73 locked·memory(kbytes)·3077552 
74 process··············96079 
75 nofiles··············512 
76 vmemory(kbytes)······unlimited67 real-time·non-blocking·time··(microseconds,·-R)·unlimited
 68 core·file·size··············(blocks,·-c)·0
 69 data·seg·size···············(kbytes,·-d)·unlimited
 70 scheduling·priority·················(-e)·0
 71 file·size···················(blocks,·-f)·unlimited
 72 pending·signals·····················(-i)·95981
 73 max·locked·memory···········(kbytes,·-l)·3076784
 74 max·memory·size·············(kbytes,·-m)·850000
 75 open·files··························(-n)·512
 76 pipe·size················(512·bytes,·-p)·8
 77 POSIX·message·queues·········(bytes,·-q)·819200
 78 real-time·priority··················(-r)·0
 79 stack·size··················(kbytes,·-s)·8192
 80 cpu·time···················(seconds,·-t)·700
 81 max·user·processes··················(-u)·95981
 82 virtual·memory··············(kbytes,·-v)·unlimited
77 locks················unlimited83 file·locks··························(-x)·unlimited
78 rtprio···············0 
  
79 o1·=·0</code></pre>84 o1·=·0</code></pre>
80 </td>··········</tr>85 </td>··········</tr>
81 ········</table>86 ········</table>
82 ········<div>87 ········<div>
83 ··········<p>This·starts·a·new·shell·and·executes·the·command·given,·which·in·this·case·provides·the·list·of·ulimits·of·the·shell.·Since·ulimits·are·inherited,·this·should·be·the·same·as·the·ulimits·of·Macaulay2·itself.</p>88 ··········<p>This·starts·a·new·shell·and·executes·the·command·given,·which·in·this·case·provides·the·list·of·ulimits·of·the·shell.·Since·ulimits·are·inherited,·this·should·be·the·same·as·the·ulimits·of·Macaulay2·itself.</p>
84 ··········<p><b>From·within·Macaulay2</b>,·it·is·not·possible·to·set·ulimits·on·the·current·Macaulay2·process·because·Macaulay2·does·not·provide·access·to·the·<span·class="tt">setrlimit</span>·system·call·(other·than·the·<a·href="../../Macaulay2Doc/html/_limit__Files.html">limitFiles</a>·and·<a·href="../../Macaulay2Doc/html/_limit__Processes.html">limitProcesses</a>·commands).·However,·it·is·possible·to·<b>set</b>·ulimits·on·child·Macaulay2·processes·that·are·started·by·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">runExternalM2</a>,·by·using·the·<a·title="a·way·to·impose·resource·limits"·href="___Pre__Run__Script.html">PreRunScript</a>·option·of·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">runExternalM2</a>.</p>89 ··········<p><b>From·within·Macaulay2</b>,·it·is·not·possible·to·set·ulimits·on·the·current·Macaulay2·process·because·Macaulay2·does·not·provide·access·to·the·<span·class="tt">setrlimit</span>·system·call·(other·than·the·<a·href="../../Macaulay2Doc/html/_limit__Files.html">limitFiles</a>·and·<a·href="../../Macaulay2Doc/html/_limit__Processes.html">limitProcesses</a>·commands).·However,·it·is·possible·to·<b>set</b>·ulimits·on·child·Macaulay2·processes·that·are·started·by·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">runExternalM2</a>,·by·using·the·<a·title="a·way·to·impose·resource·limits"·href="___Pre__Run__Script.html">PreRunScript</a>·option·of·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">runExternalM2</a>.</p>
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28 b\x8be\x8es\x8st\x8t·w\x8wa\x8ay\x8y·t\x8to\x8o·s\x8se\x8et\x8t·the·limits·on·a·specific·process·is·to·run·something·like28 b\x8be\x8es\x8st\x8t·w\x8wa\x8ay\x8y·t\x8to\x8o·s\x8se\x8et\x8t·the·limits·on·a·specific·process·is·to·run·something·like
29 (ulimit·-S·-m·123456·-S·-t·5;·M2)29 (ulimit·-S·-m·123456·-S·-t·5;·M2)
30 which·will·run·M2·with·these·limits·while·leaving·the·shell's·ulimits30 which·will·run·M2·with·these·limits·while·leaving·the·shell's·ulimits
31 unchanged.31 unchanged.
32 F\x8Fr\x8ro\x8om\x8m·w\x8wi\x8it\x8th\x8hi\x8in\x8n·M\x8Ma\x8ac\x8ca\x8au\x8ul\x8la\x8ay\x8y2\x82,·you·can·v\x8vi\x8ie\x8ew\x8w·the·ulimits·currently·available·to·the32 F\x8Fr\x8ro\x8om\x8m·w\x8wi\x8it\x8th\x8hi\x8in\x8n·M\x8Ma\x8ac\x8ca\x8au\x8ul\x8la\x8ay\x8y2\x82,·you·can·v\x8vi\x8ie\x8ew\x8w·the·ulimits·currently·available·to·the
33 Macaulay2·process·by·using·the·_\x8r_\x8u_\x8n·command:33 Macaulay2·process·by·using·the·_\x8r_\x8u_\x8n·command:
34 i1·:·run("ulimit·-a")34 i1·:·run("ulimit·-a")
35 time(seconds)········700 
36 file(blocks)·········unlimited 
37 data(kbytes)·········unlimited 
38 stack(kbytes)········8192 
39 coredump(blocks)·····0 
40 memory(kbytes)·······850000 
41 locked·memory(kbytes)·3077552 
42 process··············96079 
43 nofiles··············512 
44 vmemory(kbytes)······unlimited35 real-time·non-blocking·time··(microseconds,·-R)·unlimited
 36 core·file·size··············(blocks,·-c)·0
 37 data·seg·size···············(kbytes,·-d)·unlimited
 38 scheduling·priority·················(-e)·0
 39 file·size···················(blocks,·-f)·unlimited
 40 pending·signals·····················(-i)·95981
 41 max·locked·memory···········(kbytes,·-l)·3076784
 42 max·memory·size·············(kbytes,·-m)·850000
 43 open·files··························(-n)·512
 44 pipe·size················(512·bytes,·-p)·8
 45 POSIX·message·queues·········(bytes,·-q)·819200
 46 real-time·priority··················(-r)·0
 47 stack·size··················(kbytes,·-s)·8192
 48 cpu·time···················(seconds,·-t)·700
 49 max·user·processes··················(-u)·95981
 50 virtual·memory··············(kbytes,·-v)·unlimited
45 locks················unlimited51 file·locks··························(-x)·unlimited
46 rtprio···············0 
  
47 o1·=·052 o1·=·0
48 This·starts·a·new·shell·and·executes·the·command·given,·which·in·this·case53 This·starts·a·new·shell·and·executes·the·command·given,·which·in·this·case
49 provides·the·list·of·ulimits·of·the·shell.·Since·ulimits·are·inherited,·this54 provides·the·list·of·ulimits·of·the·shell.·Since·ulimits·are·inherited,·this
50 should·be·the·same·as·the·ulimits·of·Macaulay2·itself.55 should·be·the·same·as·the·ulimits·of·Macaulay2·itself.
51 F\x8Fr\x8ro\x8om\x8m·w\x8wi\x8it\x8th\x8hi\x8in\x8n·M\x8Ma\x8ac\x8ca\x8au\x8ul\x8la\x8ay\x8y2\x82,·it·is·not·possible·to·set·ulimits·on·the·current56 F\x8Fr\x8ro\x8om\x8m·w\x8wi\x8it\x8th\x8hi\x8in\x8n·M\x8Ma\x8ac\x8ca\x8au\x8ul\x8la\x8ay\x8y2\x82,·it·is·not·possible·to·set·ulimits·on·the·current
52 Macaulay2·process·because·Macaulay2·does·not·provide·access·to·the·setrlimit57 Macaulay2·process·because·Macaulay2·does·not·provide·access·to·the·setrlimit
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91 ··········<p>The·hash·table·<span·class="tt">h</span>·stores·the·<span·class="tt">exit·code</span>·of·the·created·Macaulay2·process,·the·<span·class="tt">return·code</span>·of·the·created·Macaulay2·process·(see·<a·title="run·an·external·command"·href="../../Macaulay2Doc/html/_run.html">run</a>·for·details;·this·is·usually·256·times·the·exit·code,·plus·information·about·any·signals·received·by·the·child),·the·wall-clock·<span·class="tt">time·used</span>·(as·opposed·to·the·CPU·time),·the·name·of·the·<span·class="tt">output·file</span>·(unless·it·was·deleted),·the·name·of·the·<span·class="tt">answer·file</span>·(unless·it·was·deleted),·any·<span·class="tt">statistics</span>·recorded·about·the·resource·usage,·and·the·<span·class="tt">value</span>·returned·by·the·function·<span·class="tt">func</span>.·If·the·child·process·terminates·abnormally,·then·usually·the·<span·class="tt">exit·code</span>·is·nonzero·and·the·<span·class="tt">value</span>·returned·is·<a·title="the·unique·member·of·the·empty·class"·href="../../Macaulay2Doc/html/_null.html">null</a>.</p>91 ··········<p>The·hash·table·<span·class="tt">h</span>·stores·the·<span·class="tt">exit·code</span>·of·the·created·Macaulay2·process,·the·<span·class="tt">return·code</span>·of·the·created·Macaulay2·process·(see·<a·title="run·an·external·command"·href="../../Macaulay2Doc/html/_run.html">run</a>·for·details;·this·is·usually·256·times·the·exit·code,·plus·information·about·any·signals·received·by·the·child),·the·wall-clock·<span·class="tt">time·used</span>·(as·opposed·to·the·CPU·time),·the·name·of·the·<span·class="tt">output·file</span>·(unless·it·was·deleted),·the·name·of·the·<span·class="tt">answer·file</span>·(unless·it·was·deleted),·any·<span·class="tt">statistics</span>·recorded·about·the·resource·usage,·and·the·<span·class="tt">value</span>·returned·by·the·function·<span·class="tt">func</span>.·If·the·child·process·terminates·abnormally,·then·usually·the·<span·class="tt">exit·code</span>·is·nonzero·and·the·<span·class="tt">value</span>·returned·is·<a·title="the·unique·member·of·the·empty·class"·href="../../Macaulay2Doc/html/_null.html">null</a>.</p>
92 ··········<p>For·example,·we·can·write·a·few·functions·to·a·temporary·file:</p>92 ··········<p>For·example,·we·can·write·a·few·functions·to·a·temporary·file:</p>
93 ········</div>93 ········</div>
94 ········<table·class="examples">94 ········<table·class="examples">
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i1·:·fn=temporaryFileName()|&quot;.m2&quot;96 <td>··············<pre><code·class="language-macaulay2">i1·:·fn=temporaryFileName()|&quot;.m2&quot;
  
97 o1·=·/tmp/M2-3830577-0/0.m2</code></pre>97 o1·=·/tmp/M2-2801784-0/0.m2</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i2·:·fn&lt;&lt;///·square·=·(x)·->·(stderr&lt;&lt;&quot;Running&quot;&lt;&lt;endl;·sleep(1);·x^2);·///&lt;&lt;endl;</code></pre>100 <td>··············<pre><code·class="language-macaulay2">i2·:·fn&lt;&lt;///·square·=·(x)·->·(stderr&lt;&lt;&quot;Running&quot;&lt;&lt;endl;·sleep(1);·x^2);·///&lt;&lt;endl;</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i3·:·fn&lt;&lt;///·justexit·=·()·->·(·exit(27);·);·///&lt;&lt;endl;</code></pre>103 <td>··············<pre><code·class="language-macaulay2">i3·:·fn&lt;&lt;///·justexit·=·()·->·(·exit(27);·);·///&lt;&lt;endl;</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
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112 ········</table>112 ········</table>
113 ········<div>113 ········<div>
114 ··········<p>and·then·call·them:</p>114 ··········<p>and·then·call·them:</p>
115 ········</div>115 ········</div>
116 ········<table·class="examples">116 ········<table·class="examples">
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i6·:·h=runExternalM2(fn,&quot;square&quot;,(4));118 <td>··············<pre><code·class="language-macaulay2">i6·:·h=runExternalM2(fn,&quot;square&quot;,(4));
119 Running·(true·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-3830577-0/1.m2&quot;·>&quot;/tmp/M2-3830577-0/1.out&quot;·2>&amp;1·))119 Running·(true·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-2801784-0/1.m2&quot;·>&quot;/tmp/M2-2801784-0/1.out&quot;·2>&amp;1·))
120 Finished·running.</code></pre>120 Finished·running.</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i7·:·h123 <td>··············<pre><code·class="language-macaulay2">i7·:·h
  
124 o7·=·HashTable{&quot;answer·file&quot;·=>·null}124 o7·=·HashTable{&quot;answer·file&quot;·=>·null}
125 ···············&quot;exit·code&quot;·=>·0125 ···············&quot;exit·code&quot;·=>·0
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146 ········<div>146 ········<div>
147 ··········<p></p>147 ··········<p></p>
148 ··········<p>An·abnormal·program·exit·will·have·a·nonzero·<span·class="tt">exit·code</span>;·also,·the·<span·class="tt">value</span>·will·be·null,·the·<span·class="tt">output·file</span>·should·exist,·but·the·<span·class="tt">answer·file</span>·may·not·exist·unless·the·routine·finished·successfully.</p>148 ··········<p>An·abnormal·program·exit·will·have·a·nonzero·<span·class="tt">exit·code</span>;·also,·the·<span·class="tt">value</span>·will·be·null,·the·<span·class="tt">output·file</span>·should·exist,·but·the·<span·class="tt">answer·file</span>·may·not·exist·unless·the·routine·finished·successfully.</p>
149 ········</div>149 ········</div>
150 ········<table·class="examples">150 ········<table·class="examples">
151 ··········<tr>151 ··········<tr>
152 <td>··············<pre><code·class="language-macaulay2">i10·:·h=runExternalM2(fn,&quot;justexit&quot;,());152 <td>··············<pre><code·class="language-macaulay2">i10·:·h=runExternalM2(fn,&quot;justexit&quot;,());
153 Running·(true·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-3830577-0/2.m2&quot;·>&quot;/tmp/M2-3830577-0/2.out&quot;·2>&amp;1·))153 Running·(true·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-2801784-0/2.m2&quot;·>&quot;/tmp/M2-2801784-0/2.out&quot;·2>&amp;1·))
154 Finished·running.154 Finished·running.
155 RunExternalM2:·expected·answer·file·does·not·exist</code></pre>155 RunExternalM2:·expected·answer·file·does·not·exist</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
157 ··········<tr>157 ··········<tr>
158 <td>··············<pre><code·class="language-macaulay2">i11·:·h158 <td>··············<pre><code·class="language-macaulay2">i11·:·h
  
159 o11·=·HashTable{&quot;answer·file&quot;·=>·/tmp/M2-3830577-0/2.ans}159 o11·=·HashTable{&quot;answer·file&quot;·=>·/tmp/M2-2801784-0/2.ans}
160 ················&quot;exit·code&quot;·=>·27160 ················&quot;exit·code&quot;·=>·27
161 ················&quot;output·file&quot;·=>·/tmp/M2-3830577-0/2.out161 ················&quot;output·file&quot;·=>·/tmp/M2-2801784-0/2.out
162 ················&quot;return·code&quot;·=>·6912162 ················&quot;return·code&quot;·=>·6912
163 ················&quot;statistics&quot;·=>·null163 ················&quot;statistics&quot;·=>·null
164 ················&quot;time·used&quot;·=>·2164 ················&quot;time·used&quot;·=>·1
165 ················value·=>·null165 ················value·=>·null
  
166 o11·:·HashTable</code></pre>166 o11·:·HashTable</code></pre>
167 </td>··········</tr>167 </td>··········</tr>
168 ··········<tr>168 ··········<tr>
169 <td>··············<pre><code·class="language-macaulay2">i12·:·fileExists(h#&quot;output·file&quot;)169 <td>··············<pre><code·class="language-macaulay2">i12·:·fileExists(h#&quot;output·file&quot;)
  
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180 ········</table>180 ········</table>
181 ········<div>181 ········<div>
182 ··········<p>Here,·we·use·<a·title="how·to·use·resource·limits·with·Macaulay2"·href="_resource_splimits.html">resource·limits</a>·to·limit·the·routine·to·2·seconds·of·computational·time,·while·the·system·is·asked·to·use·10·seconds·of·computational·time:</p>182 ··········<p>Here,·we·use·<a·title="how·to·use·resource·limits·with·Macaulay2"·href="_resource_splimits.html">resource·limits</a>·to·limit·the·routine·to·2·seconds·of·computational·time,·while·the·system·is·asked·to·use·10·seconds·of·computational·time:</p>
183 ········</div>183 ········</div>
184 ········<table·class="examples">184 ········<table·class="examples">
185 ··········<tr>185 ··········<tr>
186 <td>··············<pre><code·class="language-macaulay2">i14·:·h=runExternalM2(fn,&quot;spin&quot;,10,PreRunScript=>&quot;ulimit·-t·2&quot;);186 <td>··············<pre><code·class="language-macaulay2">i14·:·h=runExternalM2(fn,&quot;spin&quot;,10,PreRunScript=>&quot;ulimit·-t·2&quot;);
187 Running·(ulimit·-t·2·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-3830577-0/3.m2&quot;·>&quot;/tmp/M2-3830577-0/3.out&quot;·2>&amp;1·))187 Running·(ulimit·-t·2·&amp;&amp;·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-2801784-0/3.m2&quot;·>&quot;/tmp/M2-2801784-0/3.out&quot;·2>&amp;1·))
188 Killed 
189 Finished·running.188 Finished·running.
190 RunExternalM2:·expected·answer·file·does·not·exist</code></pre>189 RunExternalM2:·expected·answer·file·does·not·exist</code></pre>
191 </td>··········</tr>190 </td>··········</tr>
192 ··········<tr>191 ··········<tr>
193 <td>··············<pre><code·class="language-macaulay2">i15·:·h192 <td>··············<pre><code·class="language-macaulay2">i15·:·h
  
194 o15·=·HashTable{&quot;answer·file&quot;·=>·/tmp/M2-3830577-0/3.ans}193 o15·=·HashTable{&quot;answer·file&quot;·=>·/tmp/M2-2801784-0/3.ans}
195 ················&quot;exit·code&quot;·=>·0194 ················&quot;exit·code&quot;·=>·0
196 ················&quot;output·file&quot;·=>·/tmp/M2-3830577-0/3.out195 ················&quot;output·file&quot;·=>·/tmp/M2-2801784-0/3.out
197 ················&quot;return·code&quot;·=>·9196 ················&quot;return·code&quot;·=>·9
198 ················&quot;statistics&quot;·=>·null197 ················&quot;statistics&quot;·=>·null
199 ················&quot;time·used&quot;·=>·4198 ················&quot;time·used&quot;·=>·3
200 ················value·=>·null199 ················value·=>·null
  
201 o15·:·HashTable</code></pre>200 o15·:·HashTable</code></pre>
202 </td>··········</tr>201 </td>··········</tr>
203 ··········<tr>202 ··········<tr>
204 <td>··············<pre><code·class="language-macaulay2">i16·:·if·h#&quot;output·file&quot;·=!=·null·and·fileExists(h#&quot;output·file&quot;)·then·get(h#&quot;output·file&quot;)203 <td>··············<pre><code·class="language-macaulay2">i16·:·if·h#&quot;output·file&quot;·=!=·null·and·fileExists(h#&quot;output·file&quot;)·then·get(h#&quot;output·file&quot;)
  
205 o16·=·204 o16·=·
206 ······i1·:·--·Script·/tmp/M2-3830577-0/3.m2·automatically·generated·by·RunExternalM2205 ······i1·:·--·Script·/tmp/M2-2801784-0/3.m2·automatically·generated·by·RunExternalM2
207 ···········needsPackage(&quot;RunExternalM2&quot;,Configuration=>{&quot;isChild&quot;=>true});206 ···········needsPackage(&quot;RunExternalM2&quot;,Configuration=>{&quot;isChild&quot;=>true});
  
208 ······i2·:·load·&quot;/tmp/M2-3830577-0/0.m2&quot;;207 ······i2·:·load·&quot;/tmp/M2-2801784-0/0.m2&quot;;
  
209 ······i3·:·runExternalM2ReturnAnswer(&quot;/tmp/M2-3830577-0/3.ans&quot;,spin·(10));208 ······i3·:·runExternalM2ReturnAnswer(&quot;/tmp/M2-2801784-0/3.ans&quot;,spin·(10));
210 ······Spinning!!</code></pre>209 ······Spinning!!</code></pre>
211 </td>··········</tr>210 </td>··········</tr>
212 ··········<tr>211 ··········<tr>
213 <td>··············<pre><code·class="language-macaulay2">i17·:·if·h#&quot;answer·file&quot;·=!=·null·and·fileExists(h#&quot;answer·file&quot;)·then·get(h#&quot;answer·file&quot;)</code></pre>212 <td>··············<pre><code·class="language-macaulay2">i17·:·if·h#&quot;answer·file&quot;·=!=·null·and·fileExists(h#&quot;answer·file&quot;)·then·get(h#&quot;answer·file&quot;)</code></pre>
214 </td>··········</tr>213 </td>··········</tr>
215 ········</table>214 ········</table>
216 ········<div>215 ········<div>
217 ··········<p></p>216 ··········<p></p>
218 ··········<p>We·can·get·quite·a·lot·of·detail·on·the·resources·used·with·the·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">KeepStatistics</a>·command:</p>217 ··········<p>We·can·get·quite·a·lot·of·detail·on·the·resources·used·with·the·<a·title="run·a·Macaulay2·function·in·a·new·Macaulay2·process"·href="_run__External__M2.html">KeepStatistics</a>·command:</p>
219 ········</div>218 ········</div>
220 ········<table·class="examples">219 ········<table·class="examples">
221 ··········<tr>220 ··········<tr>
222 <td>··············<pre><code·class="language-macaulay2">i18·:·h=runExternalM2(fn,&quot;spin&quot;,3,KeepStatistics=>true);221 <td>··············<pre><code·class="language-macaulay2">i18·:·h=runExternalM2(fn,&quot;spin&quot;,3,KeepStatistics=>true);
223 Running·(true·&amp;&amp;·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-3830577-0/4.m2&quot;·>&quot;/tmp/M2-3830577-0/4.out&quot;·2>&amp;1')·>&quot;/tmp/M2-3830577-0/4.stat&quot;·2>&amp;1·))222 Running·(true·&amp;&amp;·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-2801784-0/4.m2&quot;·>&quot;/tmp/M2-2801784-0/4.out&quot;·2>&amp;1')·>&quot;/tmp/M2-2801784-0/4.stat&quot;·2>&amp;1·))
224 Finished·running.</code></pre>223 Finished·running.</code></pre>
225 </td>··········</tr>224 </td>··········</tr>
226 ··········<tr>225 ··········<tr>
227 <td>··············<pre><code·class="language-macaulay2">i19·:·h#&quot;statistics&quot;226 <td>··············<pre><code·class="language-macaulay2">i19·:·h#&quot;statistics&quot;
  
228 o19·=·········Command·being·timed:·&quot;sh·-c·/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-3830577-0/4.m2&quot;·>&quot;/tmp/M2-3830577-0/4.out&quot;·2>&amp;1&quot;227 o19·=·········Command·being·timed:·&quot;sh·-c·/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··&lt;&quot;/tmp/M2-2801784-0/4.m2&quot;·>&quot;/tmp/M2-2801784-0/4.out&quot;·2>&amp;1&quot;
229 ··············User·time·(seconds):·4.25228 ··············User·time·(seconds):·4.84
230 ··············System·time·(seconds):·0.04229 ··············System·time·(seconds):·0.33
231 ··············Percent·of·CPU·this·job·got:·53%230 ··············Percent·of·CPU·this·job·got:·88%
232 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:08.01231 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:05.86
233 ··············Average·shared·text·size·(kbytes):·0232 ··············Average·shared·text·size·(kbytes):·0
234 ··············Average·unshared·data·size·(kbytes):·0233 ··············Average·unshared·data·size·(kbytes):·0
235 ··············Average·stack·size·(kbytes):·0234 ··············Average·stack·size·(kbytes):·0
236 ··············Average·total·size·(kbytes):·0235 ··············Average·total·size·(kbytes):·0
237 ··············Maximum·resident·set·size·(kbytes):·300512236 ··············Maximum·resident·set·size·(kbytes):·296888
238 ··············Average·resident·set·size·(kbytes):·0237 ··············Average·resident·set·size·(kbytes):·0
239 ··············Major·(requiring·I/O)·page·faults:·0238 ··············Major·(requiring·I/O)·page·faults:·0
240 ··············Minor·(reclaiming·a·frame)·page·faults:·10179239 ··············Minor·(reclaiming·a·frame)·page·faults:·71400
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Offset 46, 25 lines modifiedOffset 46, 25 lines modified
46 the·output·file·(unless·it·was·deleted),·the·name·of·the·answer·file·(unless·it46 the·output·file·(unless·it·was·deleted),·the·name·of·the·answer·file·(unless·it
47 was·deleted),·any·statistics·recorded·about·the·resource·usage,·and·the·value47 was·deleted),·any·statistics·recorded·about·the·resource·usage,·and·the·value
48 returned·by·the·function·func.·If·the·child·process·terminates·abnormally,·then48 returned·by·the·function·func.·If·the·child·process·terminates·abnormally,·then
49 usually·the·exit·code·is·nonzero·and·the·value·returned·is·_\x8n_\x8u_\x8l_\x8l.49 usually·the·exit·code·is·nonzero·and·the·value·returned·is·_\x8n_\x8u_\x8l_\x8l.
50 For·example,·we·can·write·a·few·functions·to·a·temporary·file:50 For·example,·we·can·write·a·few·functions·to·a·temporary·file:
51 i1·:·fn=temporaryFileName()|".m2"51 i1·:·fn=temporaryFileName()|".m2"
  
52 o1·=·/tmp/M2-3830577-0/0.m252 o1·=·/tmp/M2-2801784-0/0.m2
53 i2·:·fn<<///·square·=·(x)·->·(stderr<<"Running"<<endl;·sleep(1);·x^2);·///53 i2·:·fn<<///·square·=·(x)·->·(stderr<<"Running"<<endl;·sleep(1);·x^2);·///
54 <<endl;54 <<endl;
55 i3·:·fn<<///·justexit·=·()·->·(·exit(27);·);·///<<endl;55 i3·:·fn<<///·justexit·=·()·->·(·exit(27);·);·///<<endl;
56 i4·:·fn<<///·spin·=·(x)·->·(stderr<<"Spinning!!"<<endl;·startTime:=cpuTime();56 i4·:·fn<<///·spin·=·(x)·->·(stderr<<"Spinning!!"<<endl;·startTime:=cpuTime();
57 while(cpuTime()-startTime<x)·do·for·i·to·10000000·do·i;·return(x););·///<<endl;57 while(cpuTime()-startTime<x)·do·for·i·to·10000000·do·i;·return(x););·///<<endl;
58 i5·:·fn<<flush;58 i5·:·fn<<flush;
59 and·then·call·them:59 and·then·call·them:
60 i6·:·h=runExternalM2(fn,"square",(4));60 i6·:·h=runExternalM2(fn,"square",(4));
61 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/61 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/
62 M2-3830577-0/1.m2"·>"/tmp/M2-3830577-0/1.out"·2>&1·))62 M2-2801784-0/1.m2"·>"/tmp/M2-2801784-0/1.out"·2>&1·))
63 Finished·running.63 Finished·running.
64 i7·:·h64 i7·:·h
  
65 o7·=·HashTable{"answer·file"·=>·null}65 o7·=·HashTable{"answer·file"·=>·null}
66 ···············"exit·code"·=>·066 ···············"exit·code"·=>·0
67 ···············"output·file"·=>·null67 ···············"output·file"·=>·null
68 ···············"return·code"·=>·068 ···············"return·code"·=>·0
Offset 80, 167 lines modifiedOffset 80, 166 lines modified
  
80 o9·=·true80 o9·=·true
81 An·abnormal·program·exit·will·have·a·nonzero·exit·code;·also,·the·value·will·be81 An·abnormal·program·exit·will·have·a·nonzero·exit·code;·also,·the·value·will·be
82 null,·the·output·file·should·exist,·but·the·answer·file·may·not·exist·unless82 null,·the·output·file·should·exist,·but·the·answer·file·may·not·exist·unless
83 the·routine·finished·successfully.83 the·routine·finished·successfully.
84 i10·:·h=runExternalM2(fn,"justexit",());84 i10·:·h=runExternalM2(fn,"justexit",());
85 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/85 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/
86 M2-3830577-0/2.m2"·>"/tmp/M2-3830577-0/2.out"·2>&1·))86 M2-2801784-0/2.m2"·>"/tmp/M2-2801784-0/2.out"·2>&1·))
87 Finished·running.87 Finished·running.
88 RunExternalM2:·expected·answer·file·does·not·exist88 RunExternalM2:·expected·answer·file·does·not·exist
89 i11·:·h89 i11·:·h
  
90 o11·=·HashTable{"answer·file"·=>·/tmp/M2-3830577-0/2.ans}90 o11·=·HashTable{"answer·file"·=>·/tmp/M2-2801784-0/2.ans}
91 ················"exit·code"·=>·2791 ················"exit·code"·=>·27
92 ················"output·file"·=>·/tmp/M2-3830577-0/2.out92 ················"output·file"·=>·/tmp/M2-2801784-0/2.out
93 ················"return·code"·=>·691293 ················"return·code"·=>·6912
94 ················"statistics"·=>·null94 ················"statistics"·=>·null
95 ················"time·used"·=>·295 ················"time·used"·=>·1
96 ················value·=>·null96 ················value·=>·null
  
97 o11·:·HashTable97 o11·:·HashTable
98 i12·:·fileExists(h#"output·file")98 i12·:·fileExists(h#"output·file")
  
99 o12·=·true99 o12·=·true
100 i13·:·fileExists(h#"answer·file")100 i13·:·fileExists(h#"answer·file")
  
101 o13·=·false101 o13·=·false
102 Here,·we·use·_\x8r_\x8e_\x8s_\x8o_\x8u_\x8r_\x8c_\x8e_\x8·_\x8l_\x8i_\x8m_\x8i_\x8t_\x8s·to·limit·the·routine·to·2·seconds·of·computational102 Here,·we·use·_\x8r_\x8e_\x8s_\x8o_\x8u_\x8r_\x8c_\x8e_\x8·_\x8l_\x8i_\x8m_\x8i_\x8t_\x8s·to·limit·the·routine·to·2·seconds·of·computational
103 time,·while·the·system·is·asked·to·use·10·seconds·of·computational·time:103 time,·while·the·system·is·asked·to·use·10·seconds·of·computational·time:
104 i14·:·h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit·-t·2");104 i14·:·h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit·-t·2");
105 Running·(ulimit·-t·2·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-105 Running·(ulimit·-t·2·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-
106 q··<"/tmp/M2-3830577-0/3.m2"·>"/tmp/M2-3830577-0/3.out"·2>&1·))106 q··<"/tmp/M2-2801784-0/3.m2"·>"/tmp/M2-2801784-0/3.out"·2>&1·))
107 Killed 
108 Finished·running.107 Finished·running.
109 RunExternalM2:·expected·answer·file·does·not·exist108 RunExternalM2:·expected·answer·file·does·not·exist
110 i15·:·h109 i15·:·h
  
111 o15·=·HashTable{"answer·file"·=>·/tmp/M2-3830577-0/3.ans}110 o15·=·HashTable{"answer·file"·=>·/tmp/M2-2801784-0/3.ans}
112 ················"exit·code"·=>·0111 ················"exit·code"·=>·0
113 ················"output·file"·=>·/tmp/M2-3830577-0/3.out112 ················"output·file"·=>·/tmp/M2-2801784-0/3.out
114 ················"return·code"·=>·9113 ················"return·code"·=>·9
115 ················"statistics"·=>·null114 ················"statistics"·=>·null
116 ················"time·used"·=>·4115 ················"time·used"·=>·3
117 ················value·=>·null116 ················value·=>·null
  
118 o15·:·HashTable117 o15·:·HashTable
119 i16·:·if·h#"output·file"·=!=·null·and·fileExists(h#"output·file")·then·get118 i16·:·if·h#"output·file"·=!=·null·and·fileExists(h#"output·file")·then·get
120 (h#"output·file")119 (h#"output·file")
  
121 o16·=120 o16·=
122 ······i1·:·--·Script·/tmp/M2-3830577-0/3.m2·automatically·generated·by121 ······i1·:·--·Script·/tmp/M2-2801784-0/3.m2·automatically·generated·by
123 RunExternalM2122 RunExternalM2
124 ···········needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});123 ···········needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
  
125 ······i2·:·load·"/tmp/M2-3830577-0/0.m2";124 ······i2·:·load·"/tmp/M2-2801784-0/0.m2";
  
126 ······i3·:·runExternalM2ReturnAnswer("/tmp/M2-3830577-0/3.ans",spin·(10));125 ······i3·:·runExternalM2ReturnAnswer("/tmp/M2-2801784-0/3.ans",spin·(10));
127 ······Spinning!!126 ······Spinning!!
128 i17·:·if·h#"answer·file"·=!=·null·and·fileExists(h#"answer·file")·then·get127 i17·:·if·h#"answer·file"·=!=·null·and·fileExists(h#"answer·file")·then·get
129 (h#"answer·file")128 (h#"answer·file")
130 We·can·get·quite·a·lot·of·detail·on·the·resources·used·with·the·_\x8K_\x8e_\x8e_\x8p_\x8S_\x8t_\x8a_\x8t_\x8i_\x8s_\x8t_\x8i_\x8c_\x8s129 We·can·get·quite·a·lot·of·detail·on·the·resources·used·with·the·_\x8K_\x8e_\x8e_\x8p_\x8S_\x8t_\x8a_\x8t_\x8i_\x8s_\x8t_\x8i_\x8c_\x8s
131 command:130 command:
132 i18·:·h=runExternalM2(fn,"spin",3,KeepStatistics=>true);131 i18·:·h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
133 Running·(true·&&·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·-132 Running·(true·&&·(·(/usr/bin/time·--verbose·sh·-c·'/usr/bin/M2-binary··--stop·-
134 -no-debug·--silent··-q··<"/tmp/M2-3830577-0/4.m2"·>"/tmp/M2-3830577-0/4.out"133 -no-debug·--silent··-q··<"/tmp/M2-2801784-0/4.m2"·>"/tmp/M2-2801784-0/4.out"
135 2>&1')·>"/tmp/M2-3830577-0/4.stat"·2>&1·))134 2>&1')·>"/tmp/M2-2801784-0/4.stat"·2>&1·))
136 Finished·running.135 Finished·running.
137 i19·:·h#"statistics"136 i19·:·h#"statistics"
  
138 o19·=·········Command·being·timed:·"sh·-c·/usr/bin/M2-binary··--stop·--no-debug137 o19·=·········Command·being·timed:·"sh·-c·/usr/bin/M2-binary··--stop·--no-debug
139 --silent··-q··<"/tmp/M2-3830577-0/4.m2"·>"/tmp/M2-3830577-0/4.out"·2>&1"138 --silent··-q··<"/tmp/M2-2801784-0/4.m2"·>"/tmp/M2-2801784-0/4.out"·2>&1"
140 ··············User·time·(seconds):·4.25139 ··············User·time·(seconds):·4.84
141 ··············System·time·(seconds):·0.04140 ··············System·time·(seconds):·0.33
142 ··············Percent·of·CPU·this·job·got:·53%141 ··············Percent·of·CPU·this·job·got:·88%
143 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:08.01142 ··············Elapsed·(wall·clock)·time·(h:mm:ss·or·m:ss):·0:05.86
144 ··············Average·shared·text·size·(kbytes):·0143 ··············Average·shared·text·size·(kbytes):·0
145 ··············Average·unshared·data·size·(kbytes):·0144 ··············Average·unshared·data·size·(kbytes):·0
146 ··············Average·stack·size·(kbytes):·0145 ··············Average·stack·size·(kbytes):·0
147 ··············Average·total·size·(kbytes):·0146 ··············Average·total·size·(kbytes):·0
148 ··············Maximum·resident·set·size·(kbytes):·300512147 ··············Maximum·resident·set·size·(kbytes):·296888
149 ··············Average·resident·set·size·(kbytes):·0148 ··············Average·resident·set·size·(kbytes):·0
150 ··············Major·(requiring·I/O)·page·faults:·0149 ··············Major·(requiring·I/O)·page·faults:·0
151 ··············Minor·(reclaiming·a·frame)·page·faults:·10179150 ··············Minor·(reclaiming·a·frame)·page·faults:·71400
152 ··············Voluntary·context·switches:·2593151 ··············Voluntary·context·switches:·6148
153 ··············Involuntary·context·switches:·720152 ··············Involuntary·context·switches:·2185
154 ··············Swaps:·0153 ··············Swaps:·0
155 ··············File·system·inputs:·0154 ··············File·system·inputs:·0
156 ··············File·system·outputs:·16155 ··············File·system·outputs:·24
157 ··············Socket·messages·sent:·0156 ··············Socket·messages·sent:·0
158 ··············Socket·messages·received:·0157 ··············Socket·messages·received:·0
159 ··············Signals·delivered:·0158 ··············Signals·delivered:·0
160 ··············Page·size·(bytes):·4096159 ··············Page·size·(bytes):·4096
161 ··············Exit·status:·0160 ··············Exit·status:·0
162 We·can·handle·most·kinds·of·objects·as·return·values,·although·_\x8M_\x8u_\x8t_\x8a_\x8b_\x8l_\x8e_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x161 We·can·handle·most·kinds·of·objects·as·return·values,·although·_\x8M_\x8u_\x8t_\x8a_\x8b_\x8l_\x8e_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x
163 does·not·work.·Here,·we·use·the·built-in·_\x8i_\x8d_\x8e_\x8n_\x8t_\x8i_\x8t_\x8y·function:162 does·not·work.·Here,·we·use·the·built-in·_\x8i_\x8d_\x8e_\x8n_\x8t_\x8i_\x8t_\x8y·function:
164 i20·:·v=///·A·complicated·string^%&C@#CERQVASDFQ#BQBSDH"'·ewrjwklsf///;163 i20·:·v=///·A·complicated·string^%&C@#CERQVASDFQ#BQBSDH"'·ewrjwklsf///;
165 i21·:·(runExternalM2(fn,identity,v))#value===v164 i21·:·(runExternalM2(fn,identity,v))#value===v
166 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/165 Running·(true·&&·(/usr/bin/M2-binary··--stop·--no-debug·--silent··-q··<"/tmp/
167 M2-3830577-0/6.m2"·>"/tmp/M2-3830577-0/6.out"·2>&1·))166 M2-2801784-0/6.m2"·>"/tmp/M2-2801784-0/6.out"·2>&1·))
168 Finished·running.167 Finished·running.
  
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739 B
./usr/share/doc/Macaulay2/SCMAlgebras/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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7 #:len=227 #:len=22
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9 #:len=2669 #:len=266
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11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj
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735 B
./usr/share/doc/Macaulay2/SCSCP/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=297 #:len=29
8 UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=8 UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=
9 #:len=2179 #:len=217
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB010 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB0
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741 B
./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=217 #:len=21
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10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTQ5LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTQ5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z
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1.14 KB
./usr/share/doc/Macaulay2/SLPexpressions/example-output/___S__L__Pexpressions.out
    
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30 ············································)30 ············································)
  
31 ··························"variable·positions"·=>·{-1}31 ··························"variable·positions"·=>·{-1}
  
32 o5·:·InterpretedSLProgram32 o5·:·InterpretedSLProgram
  
33 i6·:·time·A·=·evaluate(slp,matrix{{1}});33 i6·:·time·A·=·evaluate(slp,matrix{{1}});
34 ·--·used·0.000995909s·(cpu);·0.000148504s·(thread);·0s·(gc)34 ·--·used·0.00103368s·(cpu);·0.000244217s·(thread);·0s·(gc)
  
35 ··············1·······135 ··············1·······1
36 o6·:·Matrix·ZZ··<--·ZZ36 o6·:·Matrix·ZZ··<--·ZZ
  
37 i7·:·ZZ[y];37 i7·:·ZZ[y];
  
38 i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})38 i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})
39 ·--·used·3.08366s·(cpu);·2.52611s·(thread);·0s·(gc)39 ·--·used·3.71934s·(cpu);·2.91805s·(thread);·0s·(gc)
  
40 o8·=·10443888814131525066917527107166243825799642490473837803842334832839539040 o8·=·104438888141315250669175271071662438257996424904738378038423348328395390
41 ·····79715574568488268119349975583408901067144392628379875734381857936072632341 ·····797155745684882681193499755834089010671443926283798757343818579360726323
42 ·····60878513652779459569765437099983403615901343837183144280700118559462263742 ·····608785136527794595697654370999834036159013438371831442807001185594622637
43 ·····63188393977127456723346843445866174968079087058037040712840487401186091143 ·····631883939771274567233468434458661749680790870580370407128404874011860911
44 ·····44679777835980290066869389768817877859469056301902609405995794534328234644 ·····446797778359802900668693897688178778594690563019026094059957945343282346
45 ·····93030266964430590250159723998677142155416938355598852914863182379144344945 ·····930302669644305902501597239986771421554169383555988529148631823791443449
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2.43 KB
./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
    
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86 ··························&quot;variable·positions&quot;·=>·{-1}86 ··························&quot;variable·positions&quot;·=>·{-1}
  
87 o5·:·InterpretedSLProgram</code></pre>87 o5·:·InterpretedSLProgram</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i6·:·time·A·=·evaluate(slp,matrix{{1}});90 <td>··············<pre><code·class="language-macaulay2">i6·:·time·A·=·evaluate(slp,matrix{{1}});
91 ·--·used·0.000995909s·(cpu);·0.000148504s·(thread);·0s·(gc)91 ·--·used·0.00103368s·(cpu);·0.000244217s·(thread);·0s·(gc)
  
92 ··············1·······192 ··············1·······1
93 o6·:·Matrix·ZZ··&lt;--·ZZ</code></pre>93 o6·:·Matrix·ZZ··&lt;--·ZZ</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i7·:·ZZ[y];</code></pre>96 <td>··············<pre><code·class="language-macaulay2">i7·:·ZZ[y];</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})99 <td>··············<pre><code·class="language-macaulay2">i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})
100 ·--·used·3.08366s·(cpu);·2.52611s·(thread);·0s·(gc)100 ·--·used·3.71934s·(cpu);·2.91805s·(thread);·0s·(gc)
  
101 o8·=·104438888141315250669175271071662438257996424904738378038423348328395390101 o8·=·104438888141315250669175271071662438257996424904738378038423348328395390
102 ·····797155745684882681193499755834089010671443926283798757343818579360726323102 ·····797155745684882681193499755834089010671443926283798757343818579360726323
103 ·····608785136527794595697654370999834036159013438371831442807001185594622637103 ·····608785136527794595697654370999834036159013438371831442807001185594622637
104 ·····631883939771274567233468434458661749680790870580370407128404874011860911104 ·····631883939771274567233468434458661749680790870580370407128404874011860911
105 ·····446797778359802900668693897688178778594690563019026094059957945343282346105 ·····446797778359802900668693897688178778594690563019026094059957945343282346
106 ·····930302669644305902501597239986771421554169383555988529148631823791443449106 ·····930302669644305902501597239986771421554169383555988529148631823791443449
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1.06 KB
html2text {}
    
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38 ············································output·nodes:·138 ············································output·nodes:·1
39 ············································)39 ············································)
  
40 ··························"variable·positions"·=>·{-1}40 ··························"variable·positions"·=>·{-1}
  
41 o5·:·InterpretedSLProgram41 o5·:·InterpretedSLProgram
42 i6·:·time·A·=·evaluate(slp,matrix{{1}});42 i6·:·time·A·=·evaluate(slp,matrix{{1}});
43 ·--·used·0.000995909s·(cpu);·0.000148504s·(thread);·0s·(gc)43 ·--·used·0.00103368s·(cpu);·0.000244217s·(thread);·0s·(gc)
  
44 ··············1·······144 ··············1·······1
45 o6·:·Matrix·ZZ··<--·ZZ45 o6·:·Matrix·ZZ··<--·ZZ
46 i7·:·ZZ[y];46 i7·:·ZZ[y];
47 i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})47 i8·:·time·B·=·sub((y+1)^(2^n),{y=>1})
48 ·--·used·3.08366s·(cpu);·2.52611s·(thread);·0s·(gc)48 ·--·used·3.71934s·(cpu);·2.91805s·(thread);·0s·(gc)
  
49 o8·=·10443888814131525066917527107166243825799642490473837803842334832839539049 o8·=·104438888141315250669175271071662438257996424904738378038423348328395390
50 ·····79715574568488268119349975583408901067144392628379875734381857936072632350 ·····797155745684882681193499755834089010671443926283798757343818579360726323
51 ·····60878513652779459569765437099983403615901343837183144280700118559462263751 ·····608785136527794595697654370999834036159013438371831442807001185594622637
52 ·····63188393977127456723346843445866174968079087058037040712840487401186091152 ·····631883939771274567233468434458661749680790870580370407128404874011860911
53 ·····44679777835980290066869389768817877859469056301902609405995794534328234653 ·····446797778359802900668693897688178778594690563019026094059957945343282346
54 ·····93030266964430590250159723998677142155416938355598852914863182379144344954 ·····930302669644305902501597239986771421554169383555988529148631823791443449
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766 B
./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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8 c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==8 c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==
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758 B
./usr/share/doc/Macaulay2/SRdeformations/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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9 #:len=31749 #:len=3174
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11 dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy11 dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy
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761 B
./usr/share/doc/Macaulay2/SRdeformations/example-output/___Co__Complex.out
    
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
  
56 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)56 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
57 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·157 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
  
58 i8·:·cC=complement·C58 i8·:·cC=complement·C
  
59 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··59 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··
60 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·060 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
61 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)61 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
62 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·162 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
  
63 i9·:·dualize·cC63 i9·:·dualize·cC
  
64 o9·=·2:·v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··64 o9·=·2:·v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··
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696 B
./usr/share/doc/Macaulay2/SRdeformations/example-output/___Complex.out
    
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
  
56 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)56 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
57 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·157 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
  
58 i8·:·complement·C58 i8·:·complement·C
  
59 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··59 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··
60 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·060 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
61 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)61 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
62 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·162 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
  
63 i9·:·R=QQ[x_0..x_5]63 i9·:·R=QQ[x_0..x_5]
  
64 o9·=·R64 o9·=·R
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./usr/share/doc/Macaulay2/SRdeformations/html/___Co__Complex.html
    
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136 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)136 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
137 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>137 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>
138 </td>··········</tr>138 </td>··········</tr>
139 ··········<tr>139 ··········<tr>
140 <td>··············<pre><code·class="language-macaulay2">i8·:·cC=complement·C140 <td>··············<pre><code·class="language-macaulay2">i8·:·cC=complement·C
  
141 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··141 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··
142 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·0142 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
143 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)143 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
144 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>144 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i9·:·dualize·cC147 <td>··············<pre><code·class="language-macaulay2">i9·:·dualize·cC
  
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82 ·········0·1·2··0·1·4··0·3·4··1·2·3··1·2·5··1·4·5··2·3·4··3·4·582 ·········0·1·2··0·1·4··0·3·4··1·2·3··1·2·5··1·4·5··2·3·4··3·4·5
  
83 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)83 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
84 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·184 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
85 i8·:·cC=complement·C85 i8·:·cC=complement·C
  
86 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x86 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x
87 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·087 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
88 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)88 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
89 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·189 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
90 i9·:·dualize·cC90 i9·:·dualize·cC
  
91 o9·=·2:·v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v91 o9·=·2:·v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v··v·v·v
92 ·········3·4·5··2·3·5··1·2·5··0·4·5··0·3·4··0·2·3··0·1·5··0·1·292 ·········3·4·5··2·3·5··1·2·5··0·4·5··0·3·4··0·2·3··0·1·5··0·1·2
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./usr/share/doc/Macaulay2/SRdeformations/html/___Complex.html
    
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146 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)146 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
147 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>147 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>
148 </td>··········</tr>148 </td>··········</tr>
149 ··········<tr>149 ··········<tr>
150 <td>··············<pre><code·class="language-macaulay2">i8·:·complement·C150 <td>··············<pre><code·class="language-macaulay2">i8·:·complement·C
  
151 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··151 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··
152 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·0152 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
153 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)153 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
154 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>154 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1</code></pre>
155 </td>··········</tr>155 </td>··········</tr>
156 ········</table>156 ········</table>
157 ········<div>157 ········<div>
158 ··········<p></p>158 ··········<p></p>
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95 ·········0·1·2··0·1·4··0·3·4··1·2·3··1·2·5··1·4·5··2·3·4··3·4·595 ·········0·1·2··0·1·4··0·3·4··1·2·3··1·2·5··1·4·5··2·3·4··3·4·5
  
96 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)96 o7·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
97 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·197 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
98 i8·:·complement·C98 i8·:·complement·C
  
99 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x99 o8·=·2:·x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x··x·x·x
100 ·········3·4·5··1·4·5··2·1·5··2·3·4··3·0·4··1·0·4··2·3·1··2·1·0100 ·········4·5·3··1·4·5··1·5·2··4·2·3··0·4·3··1·0·4··1·2·3··1·0·2
  
101 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)101 o8·:·co-complex·of·dim·2·embedded·in·dim·5·(printing·facets)
102 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1102 ·····equidimensional,·simplicial,·F-vector·{0,·0,·0,·8,·12,·6,·1},·Euler·=·1
103 i9·:·R=QQ[x_0..x_5]103 i9·:·R=QQ[x_0..x_5]
  
104 o9·=·R104 o9·=·R
  
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./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=107 #:len=10
8 U1ZEQ29tcGxleA==8 U1ZEQ29tcGxleA==
9 #:len=31359 #:len=3135
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSB0aGUgU1ZEIGRlY29tcG9z10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSB0aGUgU1ZEIGRlY29tcG9z
11 aXRpb24gb2YgYSBjaGFpbkNvbXBsZXggb3ZlciBSUiIsICJsaW5lbnVtIiA9PiAxMTAyLCBJbnB111 aXRpb24gb2YgYSBjaGFpbkNvbXBsZXggb3ZlciBSUiIsICJsaW5lbnVtIiA9PiAxMTAyLCBJbnB1
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./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Complex.out
    
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15 i3·:·r={5,11,3,2}15 i3·:·r={5,11,3,2}
  
16 o3·=·{5,·11,·3,·2}16 o3·=·{5,·11,·3,·2}
  
17 o3·:·List17 o3·:·List
  
18 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)18 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)
19 ·--·.00578819s·elapsed19 ·--·.00581555s·elapsed
  
20 ·······6·······19·······19·······7·······320 ·······6·······19·······19·······7·······3
21 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ··<--·ZZ21 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ··<--·ZZ
22 ········································22 ········································
23 ·····0·······1········2········3·······423 ·····0·······1········2········3·······4
  
24 o4·:·ChainComplex24 o4·:·ChainComplex
Offset 51, 15 lines modifiedOffset 51, 15 lines modified
51 ·······53········53·········53·········53········5351 ·······53········53·········53·········53········53
52 ················································52 ················································
53 ·····-1········0··········1··········2·········353 ·····-1········0··········1··········2·········3
  
54 o6·:·ChainComplex54 o6·:·ChainComplex
  
55 i7·:·elapsedTime·(h,U)=SVDComplex·CR;55 i7·:·elapsedTime·(h,U)=SVDComplex·CR;
56 ·--·.00198659s·elapsed56 ·--·.00192744s·elapsed
  
57 i8·:·h57 i8·:·h
  
58 o8·=·HashTable{-1·=>·1}58 o8·=·HashTable{-1·=>·1}
59 ···············0·=>·359 ···············0·=>·3
60 ···············1·=>·560 ···············1·=>·5
61 ···············2·=>·261 ···············2·=>·2
Offset 95, 15 lines modifiedOffset 95, 15 lines modified
95 i12·:·maximalEntry·chainComplex·errors95 i12·:·maximalEntry·chainComplex·errors
  
96 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}96 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}
  
97 o12·:·List97 o12·:·List
  
98 i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);98 i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);
99 ·--·.00379819s·elapsed99 ·--·.00393356s·elapsed
  
100 i14·:·hL·===·h100 i14·:·hL·===·h
  
101 o14·=·true101 o14·=·true
  
102 i15·:·SigmaL·=source·U;102 i15·:·SigmaL·=source·U;
  
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1.41 KB
./usr/share/doc/Macaulay2/SVDComplexes/example-output/___S__V__D__Homology.out
    
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15 i3·:·r={4,3,3}15 i3·:·r={4,3,3}
  
16 o3·=·{4,·3,·3}16 o3·=·{4,·3,·3}
  
17 o3·:·List17 o3·:·List
  
18 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)18 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
19 ·--·.0023834s·elapsed19 ·--·.00338394s·elapsed
  
20 ·······5·······10·······11·······520 ·······5·······10·······11·······5
21 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ21 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
22 ································22 ································
23 ·····0·······1········2········323 ·····0·······1········2········3
  
24 o4·:·ChainComplex24 o4·:·ChainComplex
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47 ·······53········53·········53·········5347 ·······53········53·········53·········53
48 ······································48 ······································
49 ·····0·········1··········2··········349 ·····0·········1··········2··········3
  
50 o6·:·ChainComplex50 o6·:·ChainComplex
  
51 i7·:·elapsedTime·(h,h1)=SVDHomology·CR51 i7·:·elapsedTime·(h,h1)=SVDHomology·CR
52 ·--·.000486324s·elapsed52 ·--·.00076275s·elapsed
  
53 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})53 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})
54 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)54 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)
55 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)55 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)
56 ················3·=>·256 ················3·=>·2
  
57 o7·:·Sequence57 o7·:·Sequence
  
58 i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)58 i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
59 ·--·.0174319s·elapsed59 ·--·.00165508s·elapsed
  
60 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})60 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})
61 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)61 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)
62 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)62 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)
63 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)63 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)
  
64 o8·:·Sequence64 o8·:·Sequence
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./usr/share/doc/Macaulay2/SVDComplexes/example-output/_common__Entries.out
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 i4·:·r={4,3,5}17 i4·:·r={4,3,5}
  
18 o4·=·{4,·3,·5}18 o4·=·{4,·3,·5}
  
19 o4·:·List19 o4·:·List
  
20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
21 ·--·.00324918s·elapsed21 ·--·.00325484s·elapsed
  
22 ·······6·······10·······13·······822 ·······6·······10·······13·······8
23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
24 ································24 ································
25 ·····0·······1········2········325 ·····0·······1········2········3
  
26 o5·:·ChainComplex26 o5·:·ChainComplex
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./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 i4·:·r={4,3,3}17 i4·:·r={4,3,3}
  
18 o4·=·{4,·3,·3}18 o4·=·{4,·3,·3}
  
19 o4·:·List19 o4·:·List
  
20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
21 ·--·.00226766s·elapsed21 ·--·.00268037s·elapsed
  
22 ·······6·······10·······11·······522 ·······6·······10·······11·······5
23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
24 ································24 ································
25 ·····0·······1········2········325 ·····0·······1········2········3
  
26 o5·:·ChainComplex26 o5·:·ChainComplex
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./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 i4·:·r={4,3,3}17 i4·:·r={4,3,3}
  
18 o4·=·{4,·3,·3}18 o4·=·{4,·3,·3}
  
19 o4·:·List19 o4·:·List
  
20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)20 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
21 ·--·.00633112s·elapsed21 ·--·.00254071s·elapsed
  
22 ·······6·······10·······11·······522 ·······6·······10·······11·······5
23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ23 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
24 ································24 ································
25 ·····0·······1········2········325 ·····0·······1········2········3
  
26 o5·:·ChainComplex26 o5·:·ChainComplex
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Offset 103, 15 lines modifiedOffset 103, 15 lines modified
  
103 o3·=·{5,·11,·3,·2}103 o3·=·{5,·11,·3,·2}
  
104 o3·:·List</code></pre>104 o3·:·List</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)107 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)
108 ·--·.00578819s·elapsed108 ·--·.00581555s·elapsed
  
109 ·······6·······19·······19·······7·······3109 ·······6·······19·······19·······7·······3
110 o4·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ··&lt;--·ZZ110 o4·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ··&lt;--·ZZ
111 ········································111 ········································
112 ·····0·······1········2········3·······4112 ·····0·······1········2········3·······4
  
113 o4·:·ChainComplex</code></pre>113 o4·:·ChainComplex</code></pre>
Offset 142, 15 lines modifiedOffset 142, 15 lines modified
142 ················································142 ················································
143 ·····-1········0··········1··········2·········3143 ·····-1········0··········1··········2·········3
  
144 o6·:·ChainComplex</code></pre>144 o6·:·ChainComplex</code></pre>
145 </td>··········</tr>145 </td>··········</tr>
146 ··········<tr>146 ··········<tr>
147 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·(h,U)=SVDComplex·CR;147 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·(h,U)=SVDComplex·CR;
148 ·--·.00198659s·elapsed</code></pre>148 ·--·.00192744s·elapsed</code></pre>
149 </td>··········</tr>149 </td>··········</tr>
150 ··········<tr>150 ··········<tr>
151 <td>··············<pre><code·class="language-macaulay2">i8·:·h151 <td>··············<pre><code·class="language-macaulay2">i8·:·h
  
152 o8·=·HashTable{-1·=>·1}152 o8·=·HashTable{-1·=>·1}
153 ···············0·=>·3153 ···············0·=>·3
154 ···············1·=>·5154 ···············1·=>·5
Offset 192, 15 lines modifiedOffset 192, 15 lines modified
  
192 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}192 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}
  
193 o12·:·List</code></pre>193 o12·:·List</code></pre>
194 </td>··········</tr>194 </td>··········</tr>
195 ··········<tr>195 ··········<tr>
196 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);196 <td>··············<pre><code·class="language-macaulay2">i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);
197 ·--·.00379819s·elapsed</code></pre>197 ·--·.00393356s·elapsed</code></pre>
198 </td>··········</tr>198 </td>··········</tr>
199 ··········<tr>199 ··········<tr>
200 <td>··············<pre><code·class="language-macaulay2">i14·:·hL·===·h200 <td>··············<pre><code·class="language-macaulay2">i14·:·hL·===·h
  
201 o14·=·true</code></pre>201 o14·=·true</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ··········<tr>203 ··········<tr>
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Offset 38, 15 lines modifiedOffset 38, 15 lines modified
38 o2·:·List38 o2·:·List
39 i3·:·r={5,11,3,2}39 i3·:·r={5,11,3,2}
  
40 o3·=·{5,·11,·3,·2}40 o3·=·{5,·11,·3,·2}
  
41 o3·:·List41 o3·:·List
42 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)42 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>4)
43 ·--·.00578819s·elapsed43 ·--·.00581555s·elapsed
  
44 ·······6·······19·······19·······7·······344 ·······6·······19·······19·······7·······3
45 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ··<--·ZZ45 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ··<--·ZZ
  
46 ·····0·······1········2········3·······446 ·····0·······1········2········3·······4
  
47 o4·:·ChainComplex47 o4·:·ChainComplex
Offset 71, 15 lines modifiedOffset 71, 15 lines modified
71 o6·=·RR····<--·RR·····<--·RR·····<--·RR····<--·RR71 o6·=·RR····<--·RR·····<--·RR·····<--·RR····<--·RR
72 ·······53········53·········53·········53········5372 ·······53········53·········53·········53········53
  
73 ·····-1········0··········1··········2·········373 ·····-1········0··········1··········2·········3
  
74 o6·:·ChainComplex74 o6·:·ChainComplex
75 i7·:·elapsedTime·(h,U)=SVDComplex·CR;75 i7·:·elapsedTime·(h,U)=SVDComplex·CR;
76 ·--·.00198659s·elapsed76 ·--·.00192744s·elapsed
77 i8·:·h77 i8·:·h
  
78 o8·=·HashTable{-1·=>·1}78 o8·=·HashTable{-1·=>·1}
79 ···············0·=>·379 ···············0·=>·3
80 ···············1·=>·580 ···············1·=>·5
81 ···············2·=>·281 ···············2·=>·2
82 ···············3·=>·182 ···············3·=>·1
Offset 110, 15 lines modifiedOffset 110, 15 lines modified
110 1)*Sigma.dd_ell*transpose·U_ell);110 1)*Sigma.dd_ell*transpose·U_ell);
111 i12·:·maximalEntry·chainComplex·errors111 i12·:·maximalEntry·chainComplex·errors
  
112 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}112 o12·=·{8.43769e-15,·6.39488e-14,·1.06581e-13,·9.76996e-15}
  
113 o12·:·List113 o12·:·List
114 i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);114 i13·:·elapsedTime·(hL,U)=SVDComplex(CR,Strategy=>Laplacian);
115 ·--·.00379819s·elapsed115 ·--·.00393356s·elapsed
116 i14·:·hL·===·h116 i14·:·hL·===·h
  
117 o14·=·true117 o14·=·true
118 i15·:·SigmaL·=source·U;118 i15·:·SigmaL·=source·U;
119 i16·:·for·i·from·min·CR+1·to·max·CR·list·maximalEntry(SigmaL.dd_i·-Sigma.dd_i)119 i16·:·for·i·from·min·CR+1·to·max·CR·list·maximalEntry(SigmaL.dd_i·-Sigma.dd_i)
  
120 o16·=·{1.77636e-14,·6.39488e-14,·8.52651e-14,·3.55271e-15}120 o16·=·{1.77636e-14,·6.39488e-14,·8.52651e-14,·3.55271e-15}
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Offset 105, 15 lines modifiedOffset 105, 15 lines modified
  
105 o3·=·{4,·3,·3}105 o3·=·{4,·3,·3}
  
106 o3·:·List</code></pre>106 o3·:·List</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)109 <td>··············<pre><code·class="language-macaulay2">i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
110 ·--·.0023834s·elapsed110 ·--·.00338394s·elapsed
  
111 ·······5·······10·······11·······5111 ·······5·······10·······11·······5
112 o4·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ112 o4·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ
113 ································113 ································
114 ·····0·······1········2········3114 ·····0·······1········2········3
  
115 o4·:·ChainComplex</code></pre>115 o4·:·ChainComplex</code></pre>
Offset 140, 26 lines modifiedOffset 140, 26 lines modified
140 ······································140 ······································
141 ·····0·········1··········2··········3141 ·····0·········1··········2··········3
  
142 o6·:·ChainComplex</code></pre>142 o6·:·ChainComplex</code></pre>
143 </td>··········</tr>143 </td>··········</tr>
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·(h,h1)=SVDHomology·CR145 <td>··············<pre><code·class="language-macaulay2">i7·:·elapsedTime·(h,h1)=SVDHomology·CR
146 ·--·.000486324s·elapsed146 ·--·.00076275s·elapsed
  
147 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})147 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})
148 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)148 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)
149 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)149 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)
150 ················3·=>·2150 ················3·=>·2
  
151 o7·:·Sequence</code></pre>151 o7·:·Sequence</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)154 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
155 ·--·.0174319s·elapsed155 ·--·.00165508s·elapsed
  
156 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})156 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})
157 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)157 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)
158 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)158 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)
159 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)159 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)
  
160 o8·:·Sequence</code></pre>160 o8·:·Sequence</code></pre>
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41 o2·:·List41 o2·:·List
42 i3·:·r={4,3,3}42 i3·:·r={4,3,3}
  
43 o3·=·{4,·3,·3}43 o3·=·{4,·3,·3}
  
44 o3·:·List44 o3·:·List
45 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)45 i4·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
46 ·--·.0023834s·elapsed46 ·--·.00338394s·elapsed
  
47 ·······5·······10·······11·······547 ·······5·······10·······11·······5
48 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ48 o4·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
  
49 ·····0·······1········2········349 ·····0·······1········2········3
  
50 o4·:·ChainComplex50 o4·:·ChainComplex
Offset 70, 24 lines modifiedOffset 70, 24 lines modified
70 o6·=·RR····<--·RR·····<--·RR·····<--·RR70 o6·=·RR····<--·RR·····<--·RR·····<--·RR
71 ·······53········53·········53·········5371 ·······53········53·········53·········53
  
72 ·····0·········1··········2··········372 ·····0·········1··········2··········3
  
73 o6·:·ChainComplex73 o6·:·ChainComplex
74 i7·:·elapsedTime·(h,h1)=SVDHomology·CR74 i7·:·elapsedTime·(h,h1)=SVDHomology·CR
75 ·--·.000486324s·elapsed75 ·--·.00076275s·elapsed
  
76 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})76 o7·=·(HashTable{0·=>·1},·HashTable{1·=>·(7.87842,·1.31052,·)···········})
77 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)77 ················1·=>·3·············2·=>·(37.9214,·30.3707,·1.61954e-14)
78 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)78 ················2·=>·5·············3·=>·(14.972,·8.57847,·3.90646e-15)
79 ················3·=>·279 ················3·=>·2
  
80 o7·:·Sequence80 o7·:·Sequence
81 i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)81 i8·:·elapsedTime·(hL,hL1)=SVDHomology(CR,Strategy=>Laplacian)
82 ·--·.0174319s·elapsed82 ·--·.00165508s·elapsed
  
83 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})83 o8·=·(HashTable{0·=>·1},·HashTable{0·=>·(,·1.71747,·-1.72291e-14)······})
84 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)84 ················1·=>·3·············1·=>·(1.71747,·922.381,·2.51496e-13)
85 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)85 ················2·=>·5·············2·=>·(922.381,·73.5901,·1.81323e-13)
86 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)86 ················3·=>·2·············3·=>·(73.5901,·,·2.82914e-13)
  
87 o8·:·Sequence87 o8·:·Sequence
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104 o4·=·{4,·3,·5}104 o4·=·{4,·3,·5}
  
105 o4·:·List</code></pre>105 o4·:·List</code></pre>
106 </td>··········</tr>106 </td>··········</tr>
107 ··········<tr>107 ··········<tr>
108 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)108 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
109 ·--·.00324918s·elapsed109 ·--·.00325484s·elapsed
  
110 ·······6·······10·······13·······8110 ·······6·······10·······13·······8
111 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ111 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ
112 ································112 ································
113 ·····0·······1········2········3113 ·····0·······1········2········3
  
114 o5·:·ChainComplex</code></pre>114 o5·:·ChainComplex</code></pre>
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34 o3·:·List34 o3·:·List
35 i4·:·r={4,3,5}35 i4·:·r={4,3,5}
  
36 o4·=·{4,·3,·5}36 o4·=·{4,·3,·5}
  
37 o4·:·List37 o4·:·List
38 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)38 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
39 ·--·.00324918s·elapsed39 ·--·.00325484s·elapsed
  
40 ·······6·······10·······13·······840 ·······6·······10·······13·······8
41 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ41 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
  
42 ·····0·······1········2········342 ·····0·······1········2········3
  
43 o5·:·ChainComplex43 o5·:·ChainComplex
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95 o4·=·{4,·3,·3}95 o4·=·{4,·3,·3}
  
96 o4·:·List</code></pre>96 o4·:·List</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)99 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
100 ·--·.00226766s·elapsed100 ·--·.00268037s·elapsed
  
101 ·······6·······10·······11·······5101 ·······6·······10·······11·······5
102 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ102 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ
103 ································103 ································
104 ·····0·······1········2········3104 ·····0·······1········2········3
  
105 o5·:·ChainComplex</code></pre>105 o5·:·ChainComplex</code></pre>
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29 o3·:·List29 o3·:·List
30 i4·:·r={4,3,3}30 i4·:·r={4,3,3}
  
31 o4·=·{4,·3,·3}31 o4·=·{4,·3,·3}
  
32 o4·:·List32 o4·:·List
33 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)33 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
34 ·--·.00226766s·elapsed34 ·--·.00268037s·elapsed
  
35 ·······6·······10·······11·······535 ·······6·······10·······11·······5
36 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ36 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
  
37 ·····0·······1········2········337 ·····0·······1········2········3
  
38 o5·:·ChainComplex38 o5·:·ChainComplex
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Offset 95, 15 lines modifiedOffset 95, 15 lines modified
  
95 o4·=·{4,·3,·3}95 o4·=·{4,·3,·3}
  
96 o4·:·List</code></pre>96 o4·:·List</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)99 <td>··············<pre><code·class="language-macaulay2">i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
100 ·--·.00633112s·elapsed100 ·--·.00254071s·elapsed
  
101 ·······6·······10·······11·······5101 ·······6·······10·······11·······5
102 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ102 o5·=·ZZ··&lt;--·ZZ···&lt;--·ZZ···&lt;--·ZZ
103 ································103 ································
104 ·····0·······1········2········3104 ·····0·······1········2········3
  
105 o5·:·ChainComplex</code></pre>105 o5·:·ChainComplex</code></pre>
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Offset 29, 15 lines modifiedOffset 29, 15 lines modified
29 o3·:·List29 o3·:·List
30 i4·:·r={4,3,3}30 i4·:·r={4,3,3}
  
31 o4·=·{4,·3,·3}31 o4·=·{4,·3,·3}
  
32 o4·:·List32 o4·:·List
33 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)33 i5·:·elapsedTime·C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
34 ·--·.00633112s·elapsed34 ·--·.00254071s·elapsed
  
35 ·······6·······10·······11·······535 ·······6·······10·······11·······5
36 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ36 o5·=·ZZ··<--·ZZ···<--·ZZ···<--·ZZ
  
37 ·····0·······1········2········337 ·····0·······1········2········3
  
38 o5·:·ChainComplex38 o5·:·ChainComplex
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./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump
    
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733 B
./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
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8 c2F0dXJhdGUoSWRlYWwsTGlzdCk=8 c2F0dXJhdGUoSWRlYWwsTGlzdCk=
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11 ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2F0dXJhdGUs11 ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2F0dXJhdGUs
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1.11 KB
./usr/share/doc/Macaulay2/Saturation/example-output/_quotient_lp..._cm__Strategy_eq_gt..._rp.out
    
Offset 38, 33 lines modifiedOffset 38, 33 lines modified
38 o5·:·Ideal·of·S38 o5·:·Ideal·of·S
  
39 i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));39 i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));
  
40 o6·:·Ideal·of·S40 o6·:·Ideal·of·S
  
41 i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);41 i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);
42 ·--·used·0.241897s·(cpu);·0.241667s·(thread);·0s·(gc)42 ·--·used·0.366139s·(cpu);·0.366099s·(thread);·0s·(gc)
  
43 o7·:·Ideal·of·S43 o7·:·Ideal·of·S
  
44 i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);44 i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);
45 ·--·used·0.49439s·(cpu);·0.458884s·(thread);·0s·(gc)45 ·--·used·0.621376s·(cpu);·0.547772s·(thread);·0s·(gc)
  
46 o8·:·Ideal·of·S46 o8·:·Ideal·of·S
  
47 i9·:·S·=·ZZ/101[vars(0..4)];47 i9·:·S·=·ZZ/101[vars(0..4)];
  
48 i10·:·I·=·ideal·vars·S;48 i10·:·I·=·ideal·vars·S;
  
49 o10·:·Ideal·of·S49 o10·:·Ideal·of·S
  
50 i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);50 i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);
51 ·--·used·0.0159628s·(cpu);·0.017219s·(thread);·0s·(gc)51 ·--·used·0.0256564s·(cpu);·0.0262704s·(thread);·0s·(gc)
  
52 o11·:·Ideal·of·S52 o11·:·Ideal·of·S
  
53 i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);53 i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);
54 ·--·used·0.00580675s·(cpu);·0.00644167s·(thread);·0s·(gc)54 ·--·used·0.00644005s·(cpu);·0.00841738s·(thread);·0s·(gc)
  
55 o12·:·Ideal·of·S55 o12·:·Ideal·of·S
  
56 i13·:·56 i13·:·
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./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html
    
Offset 106, 21 lines modifiedOffset 106, 21 lines modified
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));107 <td>··············<pre><code·class="language-macaulay2">i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));
  
108 o6·:·Ideal·of·S</code></pre>108 o6·:·Ideal·of·S</code></pre>
109 </td>··········</tr>109 </td>··········</tr>
110 ··········<tr>110 ··········<tr>
111 <td>··············<pre><code·class="language-macaulay2">i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);111 <td>··············<pre><code·class="language-macaulay2">i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);
112 ·--·used·0.241897s·(cpu);·0.241667s·(thread);·0s·(gc)112 ·--·used·0.366139s·(cpu);·0.366099s·(thread);·0s·(gc)
  
113 o7·:·Ideal·of·S</code></pre>113 o7·:·Ideal·of·S</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);116 <td>··············<pre><code·class="language-macaulay2">i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);
117 ·--·used·0.49439s·(cpu);·0.458884s·(thread);·0s·(gc)117 ·--·used·0.621376s·(cpu);·0.547772s·(thread);·0s·(gc)
  
118 o8·:·Ideal·of·S</code></pre>118 o8·:·Ideal·of·S</code></pre>
119 </td>··········</tr>119 </td>··········</tr>
120 ········</table>120 ········</table>
121 ········<div>121 ········<div>
122 ··········<p><span·class="tt">Strategy·=>·Quotient</span>·is·faster·in·other·cases:</p>122 ··········<p><span·class="tt">Strategy·=>·Quotient</span>·is·faster·in·other·cases:</p>
123 ········</div>123 ········</div>
Offset 131, 21 lines modifiedOffset 131, 21 lines modified
131 ··········<tr>131 ··········<tr>
132 <td>··············<pre><code·class="language-macaulay2">i10·:·I·=·ideal·vars·S;132 <td>··············<pre><code·class="language-macaulay2">i10·:·I·=·ideal·vars·S;
  
133 o10·:·Ideal·of·S</code></pre>133 o10·:·Ideal·of·S</code></pre>
134 </td>··········</tr>134 </td>··········</tr>
135 ··········<tr>135 ··········<tr>
136 <td>··············<pre><code·class="language-macaulay2">i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);136 <td>··············<pre><code·class="language-macaulay2">i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);
137 ·--·used·0.0159628s·(cpu);·0.017219s·(thread);·0s·(gc)137 ·--·used·0.0256564s·(cpu);·0.0262704s·(thread);·0s·(gc)
  
138 o11·:·Ideal·of·S</code></pre>138 o11·:·Ideal·of·S</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ··········<tr>140 ··········<tr>
141 <td>··············<pre><code·class="language-macaulay2">i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);141 <td>··············<pre><code·class="language-macaulay2">i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);
142 ·--·used·0.00580675s·(cpu);·0.00644167s·(thread);·0s·(gc)142 ·--·used·0.00644005s·(cpu);·0.00841738s·(thread);·0s·(gc)
  
143 o12·:·Ideal·of·S</code></pre>143 o12·:·Ideal·of·S</code></pre>
144 </td>··········</tr>144 </td>··········</tr>
145 ········</table>145 ········</table>
146 ······</div>146 ······</div>
147 ······<div>147 ······<div>
148 ········<h2>Further·information</h2>148 ········<h2>Further·information</h2>
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1.32 KB
html2text {}
    
Offset 57, 32 lines modifiedOffset 57, 32 lines modified
57 i5·:·I·=·monomialCurveIdeal(S,·1..n-1);57 i5·:·I·=·monomialCurveIdeal(S,·1..n-1);
  
58 o5·:·Ideal·of·S58 o5·:·Ideal·of·S
59 i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));59 i6·:·J·=·ideal(map(S^1,·S^n,·(p,·q)·->·S_q^5));
  
60 o6·:·Ideal·of·S60 o6·:·Ideal·of·S
61 i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);61 i7·:·time·quotient(I^3,·J^2,·Strategy·=>·Iterate);
62 ·--·used·0.241897s·(cpu);·0.241667s·(thread);·0s·(gc)62 ·--·used·0.366139s·(cpu);·0.366099s·(thread);·0s·(gc)
  
63 o7·:·Ideal·of·S63 o7·:·Ideal·of·S
64 i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);64 i8·:·time·quotient(I^3,·J^2,·Strategy·=>·Quotient);
65 ·--·used·0.49439s·(cpu);·0.458884s·(thread);·0s·(gc)65 ·--·used·0.621376s·(cpu);·0.547772s·(thread);·0s·(gc)
  
66 o8·:·Ideal·of·S66 o8·:·Ideal·of·S
67 Strategy·=>·Quotient·is·faster·in·other·cases:67 Strategy·=>·Quotient·is·faster·in·other·cases:
68 i9·:·S·=·ZZ/101[vars(0..4)];68 i9·:·S·=·ZZ/101[vars(0..4)];
69 i10·:·I·=·ideal·vars·S;69 i10·:·I·=·ideal·vars·S;
  
70 o10·:·Ideal·of·S70 o10·:·Ideal·of·S
71 i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);71 i11·:·time·quotient(I^5,·I^3,·Strategy·=>·Iterate);
72 ·--·used·0.0159628s·(cpu);·0.017219s·(thread);·0s·(gc)72 ·--·used·0.0256564s·(cpu);·0.0262704s·(thread);·0s·(gc)
  
73 o11·:·Ideal·of·S73 o11·:·Ideal·of·S
74 i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);74 i12·:·time·quotient(I^5,·I^3,·Strategy·=>·Quotient);
75 ·--·used·0.00580675s·(cpu);·0.00644167s·(thread);·0s·(gc)75 ·--·used·0.00644005s·(cpu);·0.00841738s·(thread);·0s·(gc)
  
76 o12·:·Ideal·of·S76 o12·:·Ideal·of·S
77 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*77 *\x8**\x8**\x8**\x8**\x8*·F\x8Fu\x8ur\x8rt\x8th\x8he\x8er\x8r·i\x8in\x8nf\x8fo\x8or\x8rm\x8ma\x8at\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
78 ····*·Default·value:·null78 ····*·Default·value:·null
79 ····*·Function:·_\x8q_\x8u_\x8o_\x8t_\x8i_\x8e_\x8n_\x8t·--·quotient·or·division79 ····*·Function:·_\x8q_\x8u_\x8o_\x8t_\x8i_\x8e_\x8n_\x8t·--·quotient·or·division
80 ····*·Option·key:·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y·--·an·optional·argument80 ····*·Option·key:·_\x8S_\x8t_\x8r_\x8a_\x8t_\x8e_\x8g_\x8y·--·an·optional·argument
81 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*81 *\x8**\x8**\x8**\x8**\x8*·R\x8Re\x8ef\x8fe\x8er\x8re\x8en\x8nc\x8ce\x8es\x8s·*\x8**\x8**\x8**\x8**\x8*
⊟
747 B
./usr/share/doc/Macaulay2/Schubert2/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:38·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=327 #:len=32
8 aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik=8 aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik=
9 #:len=2609 #:len=260
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1NSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1NSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x111 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x1
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1.48 KB
./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out
    
Offset 40, 23 lines modifiedOffset 40, 23 lines modified
40 ··········)40 ··········)
  
41 o6·=·f41 o6·=·f
  
42 o6·:·FunctionClosure42 o6·:·FunctionClosure
  
43 i7·:·for·n·from·2·to·10·list·time·f·n43 i7·:·for·n·from·2·to·10·list·time·f·n
44 ·--·used·0.00772742s·(cpu);·0.00437985s·(thread);·0s·(gc) 
45 ·--·used·0.00361408s·(cpu);·0.00569648s·(thread);·0s·(gc)44 ·--·used·0.00636584s·(cpu);·0.00512841s·(thread);·0s·(gc)
46 ·--·used·0.00803871s·(cpu);·0.00819785s·(thread);·0s·(gc)45 ·--·used·0.00498346s·(cpu);·0.00654066s·(thread);·0s·(gc)
 46 ·--·used·0.00948554s·(cpu);·0.010502s·(thread);·0s·(gc)
47 ·--·used·0.0134933s·(cpu);·0.0157863s·(thread);·0s·(gc)47 ·--·used·0.0132901s·(cpu);·0.0158032s·(thread);·0s·(gc)
48 ·--·used·0.0216741s·(cpu);·0.0248108s·(thread);·0s·(gc)48 ·--·used·0.0214054s·(cpu);·0.0252183s·(thread);·0s·(gc)
49 ·--·used·0.0418132s·(cpu);·0.0432409s·(thread);·0s·(gc)49 ·--·used·0.0518857s·(cpu);·0.0524637s·(thread);·0s·(gc)
 50 ·--·used·0.0901787s·(cpu);·0.0910968s·(thread);·0s·(gc)
50 ·--·used·0.0699501s·(cpu);·0.0718607s·(thread);·0s·(gc)51 ·--·used·0.146991s·(cpu);·0.150788s·(thread);·0s·(gc)
51 ·--·used·0.107763s·(cpu);·0.110456s·(thread);·0s·(gc) 
52 ·--·used·0.24516s·(cpu);·0.206552s·(thread);·0s·(gc)52 ·--·used·0.316999s·(cpu);·0.27087s·(thread);·0s·(gc)
  
53 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,53 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
54 ·····------------------------------------------------------------------------54 ·····------------------------------------------------------------------------
55 ·····289139638632755625,·520764738758073845321}55 ·····289139638632755625,·520764738758073845321}
  
56 o7·:·List56 o7·:·List
  
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3.07 KB
./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html
    
Offset 106, 23 lines modifiedOffset 106, 23 lines modified
  
106 o6·=·f106 o6·=·f
  
107 o6·:·FunctionClosure</code></pre>107 o6·:·FunctionClosure</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i7·:·for·n·from·2·to·10·list·time·f·n110 <td>··············<pre><code·class="language-macaulay2">i7·:·for·n·from·2·to·10·list·time·f·n
111 ·--·used·0.00772742s·(cpu);·0.00437985s·(thread);·0s·(gc) 
112 ·--·used·0.00361408s·(cpu);·0.00569648s·(thread);·0s·(gc)111 ·--·used·0.00636584s·(cpu);·0.00512841s·(thread);·0s·(gc)
113 ·--·used·0.00803871s·(cpu);·0.00819785s·(thread);·0s·(gc)112 ·--·used·0.00498346s·(cpu);·0.00654066s·(thread);·0s·(gc)
 113 ·--·used·0.00948554s·(cpu);·0.010502s·(thread);·0s·(gc)
114 ·--·used·0.0134933s·(cpu);·0.0157863s·(thread);·0s·(gc)114 ·--·used·0.0132901s·(cpu);·0.0158032s·(thread);·0s·(gc)
115 ·--·used·0.0216741s·(cpu);·0.0248108s·(thread);·0s·(gc)115 ·--·used·0.0214054s·(cpu);·0.0252183s·(thread);·0s·(gc)
116 ·--·used·0.0418132s·(cpu);·0.0432409s·(thread);·0s·(gc)116 ·--·used·0.0518857s·(cpu);·0.0524637s·(thread);·0s·(gc)
 117 ·--·used·0.0901787s·(cpu);·0.0910968s·(thread);·0s·(gc)
117 ·--·used·0.0699501s·(cpu);·0.0718607s·(thread);·0s·(gc)118 ·--·used·0.146991s·(cpu);·0.150788s·(thread);·0s·(gc)
118 ·--·used·0.107763s·(cpu);·0.110456s·(thread);·0s·(gc) 
119 ·--·used·0.24516s·(cpu);·0.206552s·(thread);·0s·(gc)119 ·--·used·0.316999s·(cpu);·0.27087s·(thread);·0s·(gc)
  
120 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,120 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
121 ·····------------------------------------------------------------------------121 ·····------------------------------------------------------------------------
122 ·····289139638632755625,·520764738758073845321}122 ·····289139638632755625,·520764738758073845321}
  
123 o7·:·List</code></pre>123 o7·:·List</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
⊟
1.48 KB
html2text {}
    
Offset 56, 23 lines modifiedOffset 56, 23 lines modified
56 ··········integral·chern·symmetricPower_(2*n-3)·last·bundles·G56 ··········integral·chern·symmetricPower_(2*n-3)·last·bundles·G
57 ··········)57 ··········)
  
58 o6·=·f58 o6·=·f
  
59 o6·:·FunctionClosure59 o6·:·FunctionClosure
60 i7·:·for·n·from·2·to·10·list·time·f·n60 i7·:·for·n·from·2·to·10·list·time·f·n
61 ·--·used·0.00772742s·(cpu);·0.00437985s·(thread);·0s·(gc) 
62 ·--·used·0.00361408s·(cpu);·0.00569648s·(thread);·0s·(gc)61 ·--·used·0.00636584s·(cpu);·0.00512841s·(thread);·0s·(gc)
63 ·--·used·0.00803871s·(cpu);·0.00819785s·(thread);·0s·(gc)62 ·--·used·0.00498346s·(cpu);·0.00654066s·(thread);·0s·(gc)
 63 ·--·used·0.00948554s·(cpu);·0.010502s·(thread);·0s·(gc)
64 ·--·used·0.0134933s·(cpu);·0.0157863s·(thread);·0s·(gc)64 ·--·used·0.0132901s·(cpu);·0.0158032s·(thread);·0s·(gc)
65 ·--·used·0.0216741s·(cpu);·0.0248108s·(thread);·0s·(gc)65 ·--·used·0.0214054s·(cpu);·0.0252183s·(thread);·0s·(gc)
66 ·--·used·0.0418132s·(cpu);·0.0432409s·(thread);·0s·(gc)66 ·--·used·0.0518857s·(cpu);·0.0524637s·(thread);·0s·(gc)
 67 ·--·used·0.0901787s·(cpu);·0.0910968s·(thread);·0s·(gc)
67 ·--·used·0.0699501s·(cpu);·0.0718607s·(thread);·0s·(gc)68 ·--·used·0.146991s·(cpu);·0.150788s·(thread);·0s·(gc)
68 ·--·used·0.107763s·(cpu);·0.110456s·(thread);·0s·(gc) 
69 ·--·used·0.24516s·(cpu);·0.206552s·(thread);·0s·(gc)69 ·--·used·0.316999s·(cpu);·0.27087s·(thread);·0s·(gc)
  
70 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,70 o7·=·{1,·27,·2875,·698005,·305093061,·210480374951,·210776836330775,
71 ·····------------------------------------------------------------------------71 ·····------------------------------------------------------------------------
72 ·····289139638632755625,·520764738758073845321}72 ·····289139638632755625,·520764738758073845321}
  
73 o7·:·List73 o7·:·List
74 Note:·in·characteristic·zero,·using·Bertini's·theorem,·the·numbers·computed·can74 Note:·in·characteristic·zero,·using·Bertini's·theorem,·the·numbers·computed·can
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730 B
./usr/share/doc/Macaulay2/SchurComplexes/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
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11 Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU1MiwgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi11 Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU1MiwgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi
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731 B
./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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11 b2x5bm9taWFsIGFzIGEgc3VtIG9mIFNjaHVyIGZ1bmN0aW9ucyIsICJsaW5lbnVtIiA9PiA0MDEs11 b2x5bm9taWFsIGFzIGEgc3VtIG9mIFNjaHVyIGZ1bmN0aW9ucyIsICJsaW5lbnVtIiA9PiA0MDEs
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749 B
./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=337 #:len=33
8 c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp8 c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp
9 #:len=2869 #:len=286
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjQ0OSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjQ0OSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp
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732 B
./usr/share/doc/Macaulay2/SchurVeronese/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=147 #:len=14
8 bWFrZUJldHRpVGFsbHk=8 bWFrZUJldHRpVGFsbHk=
9 #:len=13349 #:len=1334
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl
11 cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g11 cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g
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719 B
./usr/share/doc/Macaulay2/SectionRing/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=87 #:len=8
8 bVJlZ3VsYXI=8 bVJlZ3VsYXI=
9 #:len=11369 #:len=1136
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB010 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB0
11 aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp11 aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp
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745 B
./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=267 #:len=26
8 bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk=8 bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk=
9 #:len=2779 #:len=277
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr
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1000 B
./usr/share/doc/Macaulay2/SegreClasses/example-output/_is__Component__Contained.out
    
Offset 53, 15 lines modifiedOffset 53, 15 lines modified
53 o9·:·Ideal·of·R53 o9·:·Ideal·of·R
  
54 i10·:·X=((W)*ideal(y)+ideal(f));54 i10·:·X=((W)*ideal(y)+ideal(f));
  
55 o10·:·Ideal·of·R55 o10·:·Ideal·of·R
  
56 i11·:·time·isComponentContained(X,Y)56 i11·:·time·isComponentContained(X,Y)
57 ·--·used·36.7865s·(cpu);·6.48724s·(thread);·0s·(gc)57 ·--·used·34.0123s·(cpu);·6.39819s·(thread);·0s·(gc)
  
58 o11·=·true58 o11·=·true
  
59 i12·:·print·"we·could·confirm·this·with·the·computation:"59 i12·:·print·"we·could·confirm·this·with·the·computation:"
60 we·could·confirm·this·with·the·computation:60 we·could·confirm·this·with·the·computation:
  
61 i13·:·B=ideal(x)*ideal(y)*ideal(z)61 i13·:·B=ideal(x)*ideal(y)*ideal(z)
Offset 71, 12 lines modifiedOffset 71, 12 lines modified
71 ······b*d*g,·b*d*h,·b*d*i,·b*e*g,·b*e*h,·b*e*i,·b*f*g,·b*f*h,·b*f*i,·c*d*g,71 ······b*d*g,·b*d*h,·b*d*i,·b*e*g,·b*e*h,·b*e*i,·b*f*g,·b*f*h,·b*f*i,·c*d*g,
72 ······-----------------------------------------------------------------------72 ······-----------------------------------------------------------------------
73 ······c*d*h,·c*d*i,·c*e*g,·c*e*h,·c*e*i,·c*f*g,·c*f*h,·c*f*i)73 ······c*d*h,·c*d*i,·c*e*g,·c*e*h,·c*e*i,·c*f*g,·c*f*h,·c*f*i)
  
74 o13·:·Ideal·of·R74 o13·:·Ideal·of·R
  
75 i14·:·time·isSubset(saturate(Y,B),saturate(X,B))75 i14·:·time·isSubset(saturate(Y,B),saturate(X,B))
76 ·--·used·83.065s·(cpu);·52.4295s·(thread);·0s·(gc)76 ·--·used·85.3014s·(cpu);·58.4834s·(thread);·0s·(gc)
  
77 o14·=·true77 o14·=·true
  
78 i15·:·78 i15·:·
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743 B
./usr/share/doc/Macaulay2/SegreClasses/example-output/_segre__Dim__X.out
    
Offset 23, 24 lines modifiedOffset 23, 24 lines modified
23 i5·:·A·=·makeChowRing(R)23 i5·:·A·=·makeChowRing(R)
  
24 o5·=·A24 o5·=·A
  
25 o5·:·QuotientRing25 o5·:·QuotientRing
  
26 i6·:·time·s·=·segreDimX(X,Y,A)26 i6·:·time·s·=·segreDimX(X,Y,A)
27 ·--·used·5.68461s·(cpu);·0.787334s·(thread);·0s·(gc)27 ·--·used·4.88784s·(cpu);·0.689362s·(thread);·0s·(gc)
  
28 ·······2·············228 ·······2·············2
29 o6·=·2H··+·4H·H··+·2H29 o6·=·2H··+·4H·H··+·2H
30 ·······1·····1·2·····230 ·······1·····1·2·····2
  
31 o6·:·A31 o6·:·A
  
32 i7·:·time·segre(X,Y,A)32 i7·:·time·segre(X,Y,A)
33 ·--·used·0.133503s·(cpu);·0.0875373s·(thread);·0s·(gc)33 ·--·used·0.511186s·(cpu);·0.139065s·(thread);·0s·(gc)
  
34 ········2·2·····2·········2·····2·············234 ········2·2·····2·········2·····2·············2
35 o7·=·12H·H··-·6H·H··-·6H·H··+·2H··+·4H·H··+·2H35 o7·=·12H·H··-·6H·H··-·6H·H··+·2H··+·4H·H··+·2H
36 ········1·2·····1·2·····1·2·····1·····1·2·····236 ········1·2·····1·2·····1·2·····1·····1·2·····2
  
37 o7·:·A37 o7·:·A
  
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2.82 KB
./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html
    
Offset 144, 15 lines modifiedOffset 144, 15 lines modified
144 ··········<tr>144 ··········<tr>
145 <td>··············<pre><code·class="language-macaulay2">i10·:·X=((W)*ideal(y)+ideal(f));145 <td>··············<pre><code·class="language-macaulay2">i10·:·X=((W)*ideal(y)+ideal(f));
  
146 o10·:·Ideal·of·R</code></pre>146 o10·:·Ideal·of·R</code></pre>
147 </td>··········</tr>147 </td>··········</tr>
148 ··········<tr>148 ··········<tr>
149 <td>··············<pre><code·class="language-macaulay2">i11·:·time·isComponentContained(X,Y)149 <td>··············<pre><code·class="language-macaulay2">i11·:·time·isComponentContained(X,Y)
150 ·--·used·36.7865s·(cpu);·6.48724s·(thread);·0s·(gc)150 ·--·used·34.0123s·(cpu);·6.39819s·(thread);·0s·(gc)
  
151 o11·=·true</code></pre>151 o11·=·true</code></pre>
152 </td>··········</tr>152 </td>··········</tr>
153 ··········<tr>153 ··········<tr>
154 <td>··············<pre><code·class="language-macaulay2">i12·:·print·&quot;we·could·confirm·this·with·the·computation:&quot;154 <td>··············<pre><code·class="language-macaulay2">i12·:·print·&quot;we·could·confirm·this·with·the·computation:&quot;
155 we·could·confirm·this·with·the·computation:</code></pre>155 we·could·confirm·this·with·the·computation:</code></pre>
156 </td>··········</tr>156 </td>··········</tr>
Offset 165, 15 lines modifiedOffset 165, 15 lines modified
165 ······-----------------------------------------------------------------------165 ······-----------------------------------------------------------------------
166 ······c*d*h,·c*d*i,·c*e*g,·c*e*h,·c*e*i,·c*f*g,·c*f*h,·c*f*i)166 ······c*d*h,·c*d*i,·c*e*g,·c*e*h,·c*e*i,·c*f*g,·c*f*h,·c*f*i)
  
167 o13·:·Ideal·of·R</code></pre>167 o13·:·Ideal·of·R</code></pre>
168 </td>··········</tr>168 </td>··········</tr>
169 ··········<tr>169 ··········<tr>
170 <td>··············<pre><code·class="language-macaulay2">i14·:·time·isSubset(saturate(Y,B),saturate(X,B))170 <td>··············<pre><code·class="language-macaulay2">i14·:·time·isSubset(saturate(Y,B),saturate(X,B))
171 ·--·used·83.065s·(cpu);·52.4295s·(thread);·0s·(gc)171 ·--·used·85.3014s·(cpu);·58.4834s·(thread);·0s·(gc)
  
172 o14·=·true</code></pre>172 o14·=·true</code></pre>
173 </td>··········</tr>173 </td>··········</tr>
174 ········</table>174 ········</table>
175 ······</div>175 ······</div>
176 ······<div·class="waystouse">176 ······<div·class="waystouse">
177 ········<h2>Ways·to·use·<span·class="tt">isComponentContained</span>:</h2>177 ········<h2>Ways·to·use·<span·class="tt">isComponentContained</span>:</h2>
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1.41 KB
html2text {}
    
Offset 69, 29 lines modifiedOffset 69, 29 lines modified
69 i9·:·Y=ideal·(z_0*W_0-z_1*W_1)+ideal(f);69 i9·:·Y=ideal·(z_0*W_0-z_1*W_1)+ideal(f);
  
70 o9·:·Ideal·of·R70 o9·:·Ideal·of·R
71 i10·:·X=((W)*ideal(y)+ideal(f));71 i10·:·X=((W)*ideal(y)+ideal(f));
  
72 o10·:·Ideal·of·R72 o10·:·Ideal·of·R
73 i11·:·time·isComponentContained(X,Y)73 i11·:·time·isComponentContained(X,Y)
74 ·--·used·36.7865s·(cpu);·6.48724s·(thread);·0s·(gc)74 ·--·used·34.0123s·(cpu);·6.39819s·(thread);·0s·(gc)
  
75 o11·=·true75 o11·=·true
76 i12·:·print·"we·could·confirm·this·with·the·computation:"76 i12·:·print·"we·could·confirm·this·with·the·computation:"
77 we·could·confirm·this·with·the·computation:77 we·could·confirm·this·with·the·computation:
78 i13·:·B=ideal(x)*ideal(y)*ideal(z)78 i13·:·B=ideal(x)*ideal(y)*ideal(z)
  
79 o13·=·ideal·(a*d*g,·a*d*h,·a*d*i,·a*e*g,·a*e*h,·a*e*i,·a*f*g,·a*f*h,·a*f*i,79 o13·=·ideal·(a*d*g,·a*d*h,·a*d*i,·a*e*g,·a*e*h,·a*e*i,·a*f*g,·a*f*h,·a*f*i,
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84 o13·:·Ideal·of·R84 o13·:·Ideal·of·R
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111 o5·=·A111 o5·=·A
  
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117 ·······2·············2117 ·······2·············2
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./usr/share/doc/Macaulay2/SimplicialDecomposability/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/SimplicialPosets/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/SlackIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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./usr/share/doc/Macaulay2/SlackIdeals/example-output/_rehomogenize__Polynomial.out
    
Offset 9, 14 lines modifiedOffset 9, 14 lines modified
  
9 i3·:·(Y,·T)·=·setOnesForest·X;9 i3·:·(Y,·T)·=·setOnesForest·X;
  
10 i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};10 i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};
  
11 i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)11 i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)
  
12 ······2·2·2·2··········2·2·2·2··················2·2 
13 o5·=·x·x·x·x·x··x···-·x·x·x·x·x··x···+·x·x·x·x·x·x·x·x12 ······2·2·····2··2·················2··2·····2·2·····2·2
 13 o5·=·x·x·x·x·x··x···+·x·x·x·x·x·x·x··x···-·x·x·x·x·x·x
14 ······1·4·6·7·10·11····2·3·5·8·10·11····1·2·3·4·6·7·9·1214 ······1·4·6·7·10·11····1·2·3·4·5·8·10·11····2·3·6·7·9·12
  
15 o5·:·R15 o5·:·R
  
16 i6·:·16 i6·:·
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./usr/share/doc/Macaulay2/SlackIdeals/html/_rehomogenize__Polynomial.html
    
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93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};</code></pre>95 <td>··············<pre><code·class="language-macaulay2">i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)98 <td>··············<pre><code·class="language-macaulay2">i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)
  
99 ······2·2·2·2··········2·2·2·2··················2·2 
100 o5·=·x·x·x·x·x··x···-·x·x·x·x·x··x···+·x·x·x·x·x·x·x·x99 ······2·2·····2··2·················2··2·····2·2·····2·2
 100 o5·=·x·x·x·x·x··x···+·x·x·x·x·x·x·x··x···-·x·x·x·x·x·x
101 ······1·4·6·7·10·11····2·3·5·8·10·11····1·2·3·4·6·7·9·12101 ······1·4·6·7·10·11····1·2·3·4·5·8·10·11····2·3·6·7·9·12
  
102 o5·:·R</code></pre>102 o5·:·R</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ········</table>104 ········</table>
105 ······</div>105 ······</div>
106 ······<div>106 ······<div>
107 ········<h2>See·also</h2>107 ········<h2>See·also</h2>
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html2text {}
    
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32 ·············6······532 ·············6······5
33 o2·:·Matrix·R··<--·R33 o2·:·Matrix·R··<--·R
34 i3·:·(Y,·T)·=·setOnesForest·X;34 i3·:·(Y,·T)·=·setOnesForest·X;
35 i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};35 i4·:·remVars·:=·flatten·entries·Y·-·set{0_(ring·Y),·1_(ring·Y)};
36 i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)36 i5·:·h·=·rehomogenizePolynomial(X,·Y,·T,·remVars_0^2+remVars_0*remVars_1-1)
  
37 ······2·2·2·2··········2·2·2·2··················2·2 
38 o5·=·x·x·x·x·x··x···-·x·x·x·x·x··x···+·x·x·x·x·x·x·x·x37 ······2·2·····2··2·················2··2·····2·2·····2·2
 38 o5·=·x·x·x·x·x··x···+·x·x·x·x·x·x·x··x···-·x·x·x·x·x·x
39 ······1·4·6·7·10·11····2·3·5·8·10·11····1·2·3·4·6·7·9·1239 ······1·4·6·7·10·11····1·2·3·4·5·8·10·11····2·3·6·7·9·12
  
40 o5·:·R40 o5·:·R
41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*41 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
42 ····*·_\x8s_\x8e_\x8t_\x8O_\x8n_\x8e_\x8s_\x8F_\x8o_\x8r_\x8e_\x8s_\x8t·--·sets·to·1·variables·in·a·symbolic·slack·matrix·which42 ····*·_\x8s_\x8e_\x8t_\x8O_\x8n_\x8e_\x8s_\x8F_\x8o_\x8r_\x8e_\x8s_\x8t·--·sets·to·1·variables·in·a·symbolic·slack·matrix·which
43 ······corresponding·to·edges·of·a·spanning·forest43 ······corresponding·to·edges·of·a·spanning·forest
44 ····*·_\x8s_\x8l_\x8a_\x8c_\x8k_\x8I_\x8d_\x8e_\x8a_\x8l·--·computes·the·slack·ideal44 ····*·_\x8s_\x8l_\x8a_\x8c_\x8k_\x8I_\x8d_\x8e_\x8a_\x8l·--·computes·the·slack·ideal
45 ····*·_\x8s_\x8y_\x8m_\x8b_\x8o_\x8l_\x8i_\x8c_\x8S_\x8l_\x8a_\x8c_\x8k_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·computes·the·symbolic·slack·matrix45 ····*·_\x8s_\x8y_\x8m_\x8b_\x8o_\x8l_\x8i_\x8c_\x8S_\x8l_\x8a_\x8c_\x8k_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·computes·the·symbolic·slack·matrix
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731 B
./usr/share/doc/Macaulay2/SpaceCurves/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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782 B
./usr/share/doc/Macaulay2/SparseResultants/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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./usr/share/doc/Macaulay2/SparseResultants/example-output/_degree__Determinant.out
    
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3 i1·:·n·=·{2,3,2}3 i1·:·n·=·{2,3,2}
  
4 o1·=·{2,·3,·2}4 o1·=·{2,·3,·2}
  
5 o1·:·List5 o1·:·List
  
6 i2·:·time·degreeDeterminant·n6 i2·:·time·degreeDeterminant·n
7 ·--·used·0.000862569s·(cpu);·5.5804e-05s·(thread);·0s·(gc)7 ·--·used·0.00287196s·(cpu);·8.0771e-05s·(thread);·0s·(gc)
  
8 o2·=·68 o2·=·6
  
9 i3·:·M·=·genericMultidimensionalMatrix·n;9 i3·:·M·=·genericMultidimensionalMatrix·n;
  
10 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]10 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]
11 ······················································0,0,0···1,2,111 ······················································0,0,0···1,2,1
  
12 i4·:·time·degree·determinant·M12 i4·:·time·degree·determinant·M
13 ·--·used·0.521489s·(cpu);·0.128662s·(thread);·0s·(gc)13 ·--·used·0.515928s·(cpu);·0.139254s·(thread);·0s·(gc)
  
14 o4·=·{6}14 o4·=·{6}
  
15 o4·:·List15 o4·:·List
  
16 i5·:·16 i5·:·
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678 B
./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Discriminant.out
    
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1 --·-*-·M2-comint·-*-·hash:·171303219021082231781 --·-*-·M2-comint·-*-·hash:·17130321902108223178
  
2 i1·:·(d,n)·:=·(2,3);2 i1·:·(d,n)·:=·(2,3);
  
3 i2·:·time·Disc·=·denseDiscriminant(d,n)3 i2·:·time·Disc·=·denseDiscriminant(d,n)
4 ·--·used·0.224336s·(cpu);·0.159631s·(thread);·0s·(gc)4 ·--·used·0.317507s·(cpu);·0.230065s·(thread);·0s·(gc)
  
5 o2·=·Disc5 o2·=·Disc
  
6 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·1·2·|)6 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·1·2·|)
7 ···························································|·0·0·0·1·1·2·0·0·1·0·|7 ···························································|·0·0·0·1·1·2·0·0·1·0·|
8 ···························································|·0·1·2·0·1·0·0·1·0·0·|8 ···························································|·0·1·2·0·1·0·0·1·0·0·|
  
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./usr/share/doc/Macaulay2/SparseResultants/example-output/_dense__Resultant.out
    
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9 ·················29 ·················2
10 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)10 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
11 ······4·1·2····2·2····3·1····1·2····011 ······4·1·2····2·2····3·1····1·2····0
  
12 o1·:·Sequence12 o1·:·Sequence
  
13 i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula13 i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula
14 ·--·used·0.0638138s·(cpu);·0.0647453s·(thread);·0s·(gc)14 ·--·used·0.0725585s·(cpu);·0.0760792s·(thread);·0s·(gc)
  
15 i3·:·time·denseResultant(f0,f1,f2,Algorithm=>"Macaulay");·--·using·Macaulay·formula15 i3·:·time·denseResultant(f0,f1,f2,Algorithm=>"Macaulay");·--·using·Macaulay·formula
16 ·--·used·0.204187s·(cpu);·0.180209s·(thread);·0s·(gc)16 ·--·used·0.332s·(cpu);·0.279115s·(thread);·0s·(gc)
  
17 i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant17 i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant
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19 i5·:·assert(o2·==·o3·and·o3·==·o4)19 i5·:·assert(o2·==·o3·and·o3·==·o4)
  
20 i6·:·20 i6·:·
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1.31 KB
./usr/share/doc/Macaulay2/SparseResultants/example-output/_determinant_lp__Multidimensional__Matrix_rp.out
    
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5 o1·=·{{{{8,·1},·{3,·7}},·{{8,·3},·{3,·7}}},·{{{8,·8},·{5,·7}},·{{8,·5},·{2,5 o1·=·{{{{8,·1},·{3,·7}},·{{8,·3},·{3,·7}}},·{{{8,·8},·{5,·7}},·{{8,·5},·{2,
6 ·····------------------------------------------------------------------------6 ·····------------------------------------------------------------------------
7 ·····3}}}}7 ·····3}}}}
  
8 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ8 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ
  
9 i2·:·time·det·M9 i2·:·time·det·M
10 ·--·used·0.992258s·(cpu);·0.157985s·(thread);·0s·(gc)10 ·--·used·0.870332s·(cpu);·0.147197s·(thread);·0s·(gc)
  
11 o2·=·969833799042151219211 o2·=·9698337990421512192
  
12 i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)12 i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)
  
13 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,13 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,
14 ·····------------------------------------------------------------------------14 ·····------------------------------------------------------------------------
Offset 24, 13 lines modifiedOffset 24, 13 lines modified
24 ·····7,·4,·5}}},·{{{4,·0,·1,·4,·4},·{2,·6,·1,·1,·4}},·{{5,·4,·9,·7,·4},·{6,24 ·····7,·4,·5}}},·{{{4,·0,·1,·4,·4},·{2,·6,·1,·1,·4}},·{{5,·4,·9,·7,·4},·{6,
25 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
26 ·····4,·8,·4,·2}}}}}26 ·····4,·8,·4,·2}}}}}
  
27 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ27 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ
  
28 i4·:·time·det·M28 i4·:·time·det·M
29 ·--·used·0.323404s·(cpu);·0.290089s·(thread);·0s·(gc)29 ·--·used·0.438499s·(cpu);·0.408216s·(thread);·0s·(gc)
  
30 o4·=·91298449999693898047944772788564453075318452578698694073740730127880628730 o4·=·912984499996938980479447727885644530753184525786986940737407301278806287
31 ·····925713949392658640018792781388831 ·····9257139493926586400187927813888
  
32 i5·:·32 i5·:·
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Offset 11, 15 lines modifiedOffset 11, 15 lines modified
11 ·····a·····x·y·z··+·a·····x·y·z··+·a·····x·y·z11 ·····a·····x·y·z··+·a·····x·y·z··+·a·····x·y·z
12 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·112 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·1
  
13 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]13 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]
14 ·········0,0,0···1,2,1···0···1···0···2···0···114 ·········0,0,0···1,2,1···0···1···0···2···0···1
  
15 i2·:·time·sparseDiscriminant·f15 i2·:·time·sparseDiscriminant·f
16 ·--·used·2.26531s·(cpu);·2.07853s·(thread);·0s·(gc)16 ·--·used·2.49466s·(cpu);·2.26393s·(thread);·0s·(gc)
  
17 ···················································2························17 ···················································2························
18 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-18 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-
19 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,0··19 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,0··
20 ·····------------------------------------------------------------------------20 ·····------------------------------------------------------------------------
21 ············2·····2································2············21 ············2·····2································2············
22 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-22 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-
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./usr/share/doc/Macaulay2/SparseResultants/example-output/_sparse__Resultant.out
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 --·-*-·M2-comint·-*-·hash:·162283638219457300641 --·-*-·M2-comint·-*-·hash:·16228363821945730064
  
2 i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})2 i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
3 ·--·used·0.577685s·(cpu);·0.373009s·(thread);·0s·(gc)3 ·--·used·0.768478s·(cpu);·0.544176s·(thread);·0s·(gc)
  
4 o1·=·Res4 o1·=·Res
  
5 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·1·2·2·|,·|·0·0·1·1·|})5 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·1·2·2·|,·|·0·0·1·1·|})
6 ····························································|·0·0·1·1·|··|·1·0·1·2·|··|·0·1·0·1·|6 ····························································|·0·0·1·1·|··|·1·0·1·2·|··|·0·1·0·1·|
  
7 i2·:·QQ[c_(1,1)..c_(3,4)][x,y];7 i2·:·QQ[c_(1,1)..c_(3,4)][x,y];
Offset 18, 15 lines modifiedOffset 18, 15 lines modified
18 ·····------------------------------------------------------------------------18 ·····------------------------------------------------------------------------
19 ·····c···x*y·+·c···x·+·c···y·+·c···)19 ·····c···x*y·+·c···x·+·c···y·+·c···)
20 ······3,3·······3,4·····3,2·····3,120 ······3,3·······3,4·····3,2·····3,1
  
21 o3·:·Sequence21 o3·:·Sequence
  
22 i4·:·time·Res(f,g,h)22 i4·:·time·Res(f,g,h)
23 ·--·used·0.00614827s·(cpu);·0.00731995s·(thread);·0s·(gc)23 ·--·used·0.00799013s·(cpu);·0.0087872s·(thread);·0s·(gc)
  
24 ········2·······················4······2···2···············4····24 ········2·······················4······2···2···············4····
25 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+25 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+
26 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,1··26 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,1··
27 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
28 ······3·······2·······3···············2···················3········28 ······3·······2·······3···············2···················3········
29 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+29 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+
Offset 730, 30 lines modifiedOffset 730, 30 lines modified
  
730 o4·:·QQ[c···..c···]730 o4·:·QQ[c···..c···]
731 ·········1,1···3,4731 ·········1,1···3,4
  
732 i5·:·assert(Res(f,g,h)·==·sparseResultant(f,g,h))732 i5·:·assert(Res(f,g,h)·==·sparseResultant(f,g,h))
  
733 i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);733 i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
734 ·--·used·0.771274s·(cpu);·0.13058s·(thread);·0s·(gc)734 ·--·used·0.656266s·(cpu);·0.0858027s·(thread);·0s·(gc)
  
735 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over·ZZ/3331)735 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over·ZZ/3331)
736 ·····························································|·0·1·0·1·|736 ·····························································|·0·1·0·1·|
  
737 i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];737 i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
  
738 i8·:·(f,g,h)·=·(a_0·+·a_1*x·+·a_2*y·+·a_3*x*y,·b_0·+·b_1*x·+·b_2*y·+·b_3*x*y,·c_0·+·c_1*x·+·c_2*y·+·c_3*x*y)738 i8·:·(f,g,h)·=·(a_0·+·a_1*x·+·a_2*y·+·a_3*x*y,·b_0·+·b_1*x·+·b_2*y·+·b_3*x*y,·c_0·+·c_1*x·+·c_2*y·+·c_3*x*y)
  
739 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)739 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)
740 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0740 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0
  
741 o8·:·Sequence741 o8·:·Sequence
  
742 i9·:·time·Res(f,g,h)742 i9·:·time·Res(f,g,h)
743 ·--·used·0.00400011s·(cpu);·0.0025948s·(thread);·0s·(gc)743 ·--·used·0.00370975s·(cpu);·0.00315709s·(thread);·0s·(gc)
  
744 ······2·····2············2············2········2·2····2··········744 ······2·····2············2············2········2·2····2··········
745 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-745 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-
746 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1··746 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1··
747 ·····------------------------------------------------------------------------747 ·····------------------------------------------------------------------------
748 ·························2·······················2·························748 ·························2·······················2·························
749 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+749 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+
Offset 822, 15 lines modifiedOffset 822, 15 lines modified
822 ··················2822 ··················2
823 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)823 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
824 ·······4·1·2····2·2····3·1····1·2····0824 ·······4·1·2····2·2····3·1····1·2····0
  
825 o11·:·Sequence825 o11·:·Sequence
  
826 i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));826 i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
827 ·--·used·1.24143s·(cpu);·0.50713s·(thread);·0s·(gc)827 ·--·used·1.23788s·(cpu);·0.512561s·(thread);·0s·(gc)
  
828 i13·:·quotientRemainder(UnmixedRes,MixedRes)828 i13·:·quotientRemainder(UnmixedRes,MixedRes)
  
829 ········2·2···················2····2·······························2·2829 ········2·2···················2····2·······························2·2
830 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)830 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)
831 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5831 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5
  
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Offset 73, 27 lines modifiedOffset 73, 27 lines modified
  
73 o1·=·{2,·3,·2}73 o1·=·{2,·3,·2}
  
74 o1·:·List</code></pre>74 o1·:·List</code></pre>
75 </td>··········</tr>75 </td>··········</tr>
76 ··········<tr>76 ··········<tr>
77 <td>··············<pre><code·class="language-macaulay2">i2·:·time·degreeDeterminant·n77 <td>··············<pre><code·class="language-macaulay2">i2·:·time·degreeDeterminant·n
78 ·--·used·0.000862569s·(cpu);·5.5804e-05s·(thread);·0s·(gc)78 ·--·used·0.00287196s·(cpu);·8.0771e-05s·(thread);·0s·(gc)
  
79 o2·=·6</code></pre>79 o2·=·6</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·M·=·genericMultidimensionalMatrix·n;82 <td>··············<pre><code·class="language-macaulay2">i3·:·M·=·genericMultidimensionalMatrix·n;
  
83 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]83 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]
84 ······················································0,0,0···1,2,1</code></pre>84 ······················································0,0,0···1,2,1</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degree·determinant·M87 <td>··············<pre><code·class="language-macaulay2">i4·:·time·degree·determinant·M
88 ·--·used·0.521489s·(cpu);·0.128662s·(thread);·0s·(gc)88 ·--·used·0.515928s·(cpu);·0.139254s·(thread);·0s·(gc)
  
89 o4·=·{6}89 o4·=·{6}
  
90 o4·:·List</code></pre>90 o4·:·List</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ········</table>92 ········</table>
93 ······</div>93 ······</div>
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html2text {}
    
Offset 16, 23 lines modifiedOffset 16, 23 lines modified
16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*16 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
17 i1·:·n·=·{2,3,2}17 i1·:·n·=·{2,3,2}
  
18 o1·=·{2,·3,·2}18 o1·=·{2,·3,·2}
  
19 o1·:·List19 o1·:·List
20 i2·:·time·degreeDeterminant·n20 i2·:·time·degreeDeterminant·n
21 ·--·used·0.000862569s·(cpu);·5.5804e-05s·(thread);·0s·(gc)21 ·--·used·0.00287196s·(cpu);·8.0771e-05s·(thread);·0s·(gc)
  
22 o2·=·622 o2·=·6
23 i3·:·M·=·genericMultidimensionalMatrix·n;23 i3·:·M·=·genericMultidimensionalMatrix·n;
  
24 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]24 o3·:·3-dimensional·matrix·of·shape·2·x·3·x·2·over·ZZ[a·····..a·····]
25 ······················································0,0,0···1,2,125 ······················································0,0,0···1,2,1
26 i4·:·time·degree·determinant·M26 i4·:·time·degree·determinant·M
27 ·--·used·0.521489s·(cpu);·0.128662s·(thread);·0s·(gc)27 ·--·used·0.515928s·(cpu);·0.139254s·(thread);·0s·(gc)
  
28 o4·=·{6}28 o4·=·{6}
  
29 o4·:·List29 o4·:·List
30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*30 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
31 ····*·_\x8d_\x8e_\x8t_\x8e_\x8r_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·hyperdeterminant·of·a31 ····*·_\x8d_\x8e_\x8t_\x8e_\x8r_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8M_\x8u_\x8l_\x8t_\x8i_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x_\x8)·--·hyperdeterminant·of·a
32 ······multidimensional·matrix32 ······multidimensional·matrix
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./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html
    
Offset 82, 15 lines modifiedOffset 82, 15 lines modified
82 ········<h2>Description</h2>82 ········<h2>Description</h2>
83 ········<table·class="examples">83 ········<table·class="examples">
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i1·:·(d,n)·:=·(2,3);</code></pre>85 <td>··············<pre><code·class="language-macaulay2">i1·:·(d,n)·:=·(2,3);</code></pre>
86 </td>··········</tr>86 </td>··········</tr>
87 ··········<tr>87 ··········<tr>
88 <td>··············<pre><code·class="language-macaulay2">i2·:·time·Disc·=·denseDiscriminant(d,n)88 <td>··············<pre><code·class="language-macaulay2">i2·:·time·Disc·=·denseDiscriminant(d,n)
89 ·--·used·0.224336s·(cpu);·0.159631s·(thread);·0s·(gc)89 ·--·used·0.317507s·(cpu);·0.230065s·(thread);·0s·(gc)
  
90 o2·=·Disc90 o2·=·Disc
  
91 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·1·2·|)91 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·1·2·|)
92 ···························································|·0·0·0·1·1·2·0·0·1·0·|92 ···························································|·0·0·0·1·1·2·0·0·1·0·|
93 ···························································|·0·1·2·0·1·0·0·1·0·0·|</code></pre>93 ···························································|·0·1·2·0·1·0·0·1·0·0·|</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
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19 ····*·Outputs:19 ····*·Outputs:
20 ··········o·for·(d,n),·this·is·the·same·as·_\x8s_\x8p_\x8a_\x8r_\x8s_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·_\x8e_\x8x_\x8p_\x8o_\x8n_\x8e_\x8n_\x8t_\x8s_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x20 ··········o·for·(d,n),·this·is·the·same·as·_\x8s_\x8p_\x8a_\x8r_\x8s_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·_\x8e_\x8x_\x8p_\x8o_\x8n_\x8e_\x8n_\x8t_\x8s_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x
21 ············"\x8"g\x8ge\x8en\x8ne\x8er\x8ri\x8ic\x8c·p\x8po\x8ol\x8ly\x8yn\x8no\x8om\x8mi\x8ia\x8al\x8l·o\x8of\x8f·d\x8de\x8eg\x8gr\x8re\x8ee\x8e·d\x8d·i\x8in\x8n·n\x8n·v\x8va\x8ar\x8ri\x8ia\x8ab\x8bl\x8le\x8es\x8s"\x8";\x8;21 ············"\x8"g\x8ge\x8en\x8ne\x8er\x8ri\x8ic\x8c·p\x8po\x8ol\x8ly\x8yn\x8no\x8om\x8mi\x8ia\x8al\x8l·o\x8of\x8f·d\x8de\x8eg\x8gr\x8re\x8ee\x8e·d\x8d·i\x8in\x8n·n\x8n·v\x8va\x8ar\x8ri\x8ia\x8ab\x8bl\x8le\x8es\x8s"\x8";\x8;
22 ··········o·for·f,·this·is·the·same·as·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t(f).22 ··········o·for·f,·this·is·the·same·as·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t(f).
23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*23 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
24 i1·:·(d,n)·:=·(2,3);24 i1·:·(d,n)·:=·(2,3);
25 i2·:·time·Disc·=·denseDiscriminant(d,n)25 i2·:·time·Disc·=·denseDiscriminant(d,n)
26 ·--·used·0.224336s·(cpu);·0.159631s·(thread);·0s·(gc)26 ·--·used·0.317507s·(cpu);·0.230065s·(thread);·0s·(gc)
  
27 o2·=·Disc27 o2·=·Disc
  
28 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·128 o2·:·SparseDiscriminant·(sparse·discriminant·associated·to·|·0·0·0·0·0·0·1·1·1
29 2·|)29 2·|)
30 ···························································|·0·0·0·1·1·2·0·0·130 ···························································|·0·0·0·1·1·2·0·0·1
31 0·|31 0·|
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Offset 92, 23 lines modifiedOffset 92, 23 lines modified
92 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)92 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
93 ······4·1·2····2·2····3·1····1·2····093 ······4·1·2····2·2····3·1····1·2····0
  
94 o1·:·Sequence</code></pre>94 o1·:·Sequence</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula97 <td>··············<pre><code·class="language-macaulay2">i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula
98 ·--·used·0.0638138s·(cpu);·0.0647453s·(thread);·0s·(gc)</code></pre>98 ·--·used·0.0725585s·(cpu);·0.0760792s·(thread);·0s·(gc)</code></pre>
99 </td>··········</tr>99 </td>··········</tr>
100 ··········<tr>100 ··········<tr>
101 <td>··············<pre><code·class="language-macaulay2">i3·:·time·denseResultant(f0,f1,f2,Algorithm=>&quot;Macaulay&quot;);·--·using·Macaulay·formula101 <td>··············<pre><code·class="language-macaulay2">i3·:·time·denseResultant(f0,f1,f2,Algorithm=>&quot;Macaulay&quot;);·--·using·Macaulay·formula
102 ·--·used·0.204187s·(cpu);·0.180209s·(thread);·0s·(gc)</code></pre>102 ·--·used·0.332s·(cpu);·0.279115s·(thread);·0s·(gc)</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant
106 ·--·used·0.490012s·(cpu);·0.255368s·(thread);·0s·(gc)</code></pre>106 ·--·used·0.45966s·(cpu);·0.283291s·(thread);·0s·(gc)</code></pre>
107 </td>··········</tr>107 </td>··········</tr>
108 ··········<tr>108 ··········<tr>
109 <td>··············<pre><code·class="language-macaulay2">i5·:·assert(o2·==·o3·and·o3·==·o4)</code></pre>109 <td>··············<pre><code·class="language-macaulay2">i5·:·assert(o2·==·o3·and·o3·==·o4)</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ········</table>111 ········</table>
112 ······</div>112 ······</div>
113 ······<div>113 ······<div>
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Offset 29, 20 lines modifiedOffset 29, 20 lines modified
29 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
30 ·················230 ·················2
31 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)31 ·····c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
32 ······4·1·2····2·2····3·1····1·2····032 ······4·1·2····2·2····3·1····1·2····0
  
33 o1·:·Sequence33 o1·:·Sequence
34 i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula34 i2·:·time·denseResultant(f0,f1,f2);·--·using·Poisson·formula
35 ·--·used·0.0638138s·(cpu);·0.0647453s·(thread);·0s·(gc)35 ·--·used·0.0725585s·(cpu);·0.0760792s·(thread);·0s·(gc)
36 i3·:·time·denseResultant(f0,f1,f2,Algorithm=>"Macaulay");·--·using·Macaulay36 i3·:·time·denseResultant(f0,f1,f2,Algorithm=>"Macaulay");·--·using·Macaulay
37 formula37 formula
38 ·--·used·0.204187s·(cpu);·0.180209s·(thread);·0s·(gc)38 ·--·used·0.332s·(cpu);·0.279115s·(thread);·0s·(gc)
39 i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant39 i4·:·time·(denseResultant(1,2,2))·(f0,f1,f2);·--·using·sparseResultant
40 ·--·used·0.490012s·(cpu);·0.255368s·(thread);·0s·(gc)40 ·--·used·0.45966s·(cpu);·0.283291s·(thread);·0s·(gc)
41 i5·:·assert(o2·==·o3·and·o3·==·o4)41 i5·:·assert(o2·==·o3·and·o3·==·o4)
42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*42 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
43 ····*·_\x8s_\x8p_\x8a_\x8r_\x8s_\x8e_\x8R_\x8e_\x8s_\x8u_\x8l_\x8t_\x8a_\x8n_\x8t·--·sparse·resultant·(A-resultant)43 ····*·_\x8s_\x8p_\x8a_\x8r_\x8s_\x8e_\x8R_\x8e_\x8s_\x8u_\x8l_\x8t_\x8a_\x8n_\x8t·--·sparse·resultant·(A-resultant)
44 ····*·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8R_\x8e_\x8s_\x8u_\x8l_\x8t_\x8a_\x8n_\x8t·--·affine·resultant44 ····*·_\x8a_\x8f_\x8f_\x8i_\x8n_\x8e_\x8R_\x8e_\x8s_\x8u_\x8l_\x8t_\x8a_\x8n_\x8t·--·affine·resultant
45 ····*·_\x8d_\x8e_\x8n_\x8s_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·dense·discriminant·(classical·discriminant)45 ····*·_\x8d_\x8e_\x8n_\x8s_\x8e_\x8D_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·dense·discriminant·(classical·discriminant)
46 ····*·_\x8e_\x8x_\x8p_\x8o_\x8n_\x8e_\x8n_\x8t_\x8s_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·exponents·in·one·or·more·polynomials46 ····*·_\x8e_\x8x_\x8p_\x8o_\x8n_\x8e_\x8n_\x8t_\x8s_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·exponents·in·one·or·more·polynomials
47 ····*·_\x8g_\x8e_\x8n_\x8e_\x8r_\x8i_\x8c_\x8L_\x8a_\x8u_\x8r_\x8e_\x8n_\x8t_\x8P_\x8o_\x8l_\x8y_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8s·--·generic·(Laurent)·polynomials47 ····*·_\x8g_\x8e_\x8n_\x8e_\x8r_\x8i_\x8c_\x8L_\x8a_\x8u_\x8r_\x8e_\x8n_\x8t_\x8P_\x8o_\x8l_\x8y_\x8n_\x8o_\x8m_\x8i_\x8a_\x8l_\x8s·--·generic·(Laurent)·polynomials
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Offset 84, 15 lines modifiedOffset 84, 15 lines modified
84 ·····------------------------------------------------------------------------84 ·····------------------------------------------------------------------------
85 ·····3}}}}85 ·····3}}}}
  
86 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ</code></pre>86 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i2·:·time·det·M89 <td>··············<pre><code·class="language-macaulay2">i2·:·time·det·M
90 ·--·used·0.992258s·(cpu);·0.157985s·(thread);·0s·(gc)90 ·--·used·0.870332s·(cpu);·0.147197s·(thread);·0s·(gc)
  
91 o2·=·9698337990421512192</code></pre>91 o2·=·9698337990421512192</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)94 <td>··············<pre><code·class="language-macaulay2">i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)
  
95 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,95 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,
Offset 105, 15 lines modifiedOffset 105, 15 lines modified
105 ·····------------------------------------------------------------------------105 ·····------------------------------------------------------------------------
106 ·····4,·8,·4,·2}}}}}106 ·····4,·8,·4,·2}}}}}
  
107 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ</code></pre>107 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i4·:·time·det·M110 <td>··············<pre><code·class="language-macaulay2">i4·:·time·det·M
111 ·--·used·0.323404s·(cpu);·0.290089s·(thread);·0s·(gc)111 ·--·used·0.438499s·(cpu);·0.408216s·(thread);·0s·(gc)
  
112 o4·=·912984499996938980479447727885644530753184525786986940737407301278806287112 o4·=·912984499996938980479447727885644530753184525786986940737407301278806287
113 ·····9257139493926586400187927813888</code></pre>113 ·····9257139493926586400187927813888</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ········</table>115 ········</table>
116 ······</div>116 ······</div>
117 ······<div>117 ······<div>
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26 o1·=·{{{{8,·1},·{3,·7}},·{{8,·3},·{3,·7}}},·{{{8,·8},·{5,·7}},·{{8,·5},·{2,26 o1·=·{{{{8,·1},·{3,·7}},·{{8,·3},·{3,·7}}},·{{{8,·8},·{5,·7}},·{{8,·5},·{2,
27 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
28 ·····3}}}}28 ·····3}}}}
  
29 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ29 o1·:·4-dimensional·matrix·of·shape·2·x·2·x·2·x·2·over·ZZ
30 i2·:·time·det·M30 i2·:·time·det·M
31 ·--·used·0.992258s·(cpu);·0.157985s·(thread);·0s·(gc)31 ·--·used·0.870332s·(cpu);·0.147197s·(thread);·0s·(gc)
  
32 o2·=·969833799042151219232 o2·=·9698337990421512192
33 i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)33 i3·:·M·=·randomMultidimensionalMatrix(2,2,2,2,5)
  
34 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,34 o3·=·{{{{{6,·3,·6,·8,·6},·{9,·3,·7,·6,·9}},·{{6,·2,·6,·0,·2},·{6,·9,·3,·5,
35 ·····------------------------------------------------------------------------35 ·····------------------------------------------------------------------------
36 ·····6}}},·{{{3,·5,·7,·7,·9},·{4,·5,·0,·4,·3}},·{{1,·8,·9,·1,·2},·{9,·6,·6,36 ·····6}}},·{{{3,·5,·7,·7,·9},·{4,·5,·0,·4,·3}},·{{1,·8,·9,·1,·2},·{9,·6,·6,
Offset 43, 15 lines modifiedOffset 43, 15 lines modified
43 ·····------------------------------------------------------------------------43 ·····------------------------------------------------------------------------
44 ·····7,·4,·5}}},·{{{4,·0,·1,·4,·4},·{2,·6,·1,·1,·4}},·{{5,·4,·9,·7,·4},·{6,44 ·····7,·4,·5}}},·{{{4,·0,·1,·4,·4},·{2,·6,·1,·1,·4}},·{{5,·4,·9,·7,·4},·{6,
45 ·····------------------------------------------------------------------------45 ·····------------------------------------------------------------------------
46 ·····4,·8,·4,·2}}}}}46 ·····4,·8,·4,·2}}}}}
  
47 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ47 o3·:·5-dimensional·matrix·of·shape·2·x·2·x·2·x·2·x·5·over·ZZ
48 i4·:·time·det·M48 i4·:·time·det·M
49 ·--·used·0.323404s·(cpu);·0.290089s·(thread);·0s·(gc)49 ·--·used·0.438499s·(cpu);·0.408216s·(thread);·0s·(gc)
  
50 o4·=·91298449999693898047944772788564453075318452578698694073740730127880628750 o4·=·912984499996938980479447727885644530753184525786986940737407301278806287
51 ·····925713949392658640018792781388851 ·····9257139493926586400187927813888
52 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*52 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
53 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·the·class·of·all·multidimensional·matrices53 ····*·_\x8M_\x8u_\x8l_\x8t_\x8i_\x8d_\x8i_\x8m_\x8e_\x8n_\x8s_\x8i_\x8o_\x8n_\x8a_\x8l_\x8M_\x8a_\x8t_\x8r_\x8i_\x8x·--·the·class·of·all·multidimensional·matrices
54 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8D_\x8e_\x8t_\x8e_\x8r_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·degree·of·the·hyperdeterminant·of·a·generic54 ····*·_\x8d_\x8e_\x8g_\x8r_\x8e_\x8e_\x8D_\x8e_\x8t_\x8e_\x8r_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t·--·degree·of·the·hyperdeterminant·of·a·generic
55 ······multidimensional·matrix55 ······multidimensional·matrix
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Offset 89, 15 lines modifiedOffset 89, 15 lines modified
89 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·189 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·1
  
90 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]90 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]
91 ·········0,0,0···1,2,1···0···1···0···2···0···1</code></pre>91 ·········0,0,0···1,2,1···0···1···0···2···0···1</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·time·sparseDiscriminant·f94 <td>··············<pre><code·class="language-macaulay2">i2·:·time·sparseDiscriminant·f
95 ·--·used·2.26531s·(cpu);·2.07853s·(thread);·0s·(gc)95 ·--·used·2.49466s·(cpu);·2.26393s·(thread);·0s·(gc)
  
96 ···················································2························96 ···················································2························
97 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-97 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-
98 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,0··98 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,0··
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ············2·····2································2············100 ············2·····2································2············
101 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-101 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-
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38 ·····------------------------------------------------------------------------38 ·····------------------------------------------------------------------------
39 ·····a·····x·y·z··+·a·····x·y·z··+·a·····x·y·z39 ·····a·····x·y·z··+·a·····x·y·z··+·a·····x·y·z
40 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·140 ······1,1,1·1·1·1····1,2,0·1·2·0····1,2,1·1·2·1
  
41 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]41 o1·:·ZZ[a·····..a·····][x·..x·,·y·..y·,·z·..z·]
42 ·········0,0,0···1,2,1···0···1···0···2···0···142 ·········0,0,0···1,2,1···0···1···0···2···0···1
43 i2·:·time·sparseDiscriminant·f43 i2·:·time·sparseDiscriminant·f
44 ·--·used·2.26531s·(cpu);·2.07853s·(thread);·0s·(gc)44 ·--·used·2.49466s·(cpu);·2.26393s·(thread);·0s·(gc)
  
45 ···················································245 ···················································2
46 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-46 o2·=·a·····a·····a·····a·····a·····a······-·a·····a·····a·····a·····a······-
47 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,047 ······0,1,1·0,2,0·0,2,1·1,0,0·1,0,1·1,1,0····0,1,0·0,2,1·1,0,0·1,0,1·1,1,0
48 ·····------------------------------------------------------------------------48 ·····------------------------------------------------------------------------
49 ············2·····2································249 ············2·····2································2
50 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-50 ·····a·····a·····a·····a······+·a·····a·····a·····a·····a······-
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76 ······<div>76 ······<div>
77 ········<h2>Description</h2>77 ········<h2>Description</h2>
78 ········<p>Alternatively,·one·can·apply·the·method·directly·to·the·list·of·Laurent·polynomials·$f_0,\ldots,f_n$.·In·this·case,·the·matrices·$A_0,\ldots,A_n$·are·automatically·determined·by·<a·title="exponents·in·one·or·more·polynomials"·href="_exponents__Matrix.html">exponentsMatrix</a>.·If·you·want·require·that·$A_0=\cdots=A_n$,·then·use·the·option·<span·class="tt">Unmixed=>true</span>·(this·could·be·faster).·Below·we·consider·some·examples.</p>78 ········<p>Alternatively,·one·can·apply·the·method·directly·to·the·list·of·Laurent·polynomials·$f_0,\ldots,f_n$.·In·this·case,·the·matrices·$A_0,\ldots,A_n$·are·automatically·determined·by·<a·title="exponents·in·one·or·more·polynomials"·href="_exponents__Matrix.html">exponentsMatrix</a>.·If·you·want·require·that·$A_0=\cdots=A_n$,·then·use·the·option·<span·class="tt">Unmixed=>true</span>·(this·could·be·faster).·Below·we·consider·some·examples.</p>
79 ········<p>In·the·first·example,·we·calculate·the·sparse·(mixed)·resultant·associated·to·the·three·sets·of·monomials·$(1,x·y,x^2·y,x),(y,x^2·y^2,x^2·y,x),(1,y,x·y,x)$.·Then·we·evaluate·it·at·the·three·polynomials·$f·=·c_{(1,1)}+c_{(1,2)}·x·y+c_{(1,3)}·x^2·y+c_{(1,4)}·x,·g·=·c_{(2,1)}·y+c_{(2,2)}·x^2·y^2+c_{(2,3)}·x^2·y+c_{(2,4)}·x,·h·=·c_{(3,1)}+c_{(3,2)}·y+c_{(3,3)}·x·y+c_{(3,4)}·x$.</p>79 ········<p>In·the·first·example,·we·calculate·the·sparse·(mixed)·resultant·associated·to·the·three·sets·of·monomials·$(1,x·y,x^2·y,x),(y,x^2·y^2,x^2·y,x),(1,y,x·y,x)$.·Then·we·evaluate·it·at·the·three·polynomials·$f·=·c_{(1,1)}+c_{(1,2)}·x·y+c_{(1,3)}·x^2·y+c_{(1,4)}·x,·g·=·c_{(2,1)}·y+c_{(2,2)}·x^2·y^2+c_{(2,3)}·x^2·y+c_{(2,4)}·x,·h·=·c_{(3,1)}+c_{(3,2)}·y+c_{(3,3)}·x·y+c_{(3,4)}·x$.</p>
80 ········<table·class="examples">80 ········<table·class="examples">
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})82 <td>··············<pre><code·class="language-macaulay2">i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
83 ·--·used·0.577685s·(cpu);·0.373009s·(thread);·0s·(gc)83 ·--·used·0.768478s·(cpu);·0.544176s·(thread);·0s·(gc)
  
84 o1·=·Res84 o1·=·Res
  
85 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·1·2·2·|,·|·0·0·1·1·|})85 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·1·2·2·|,·|·0·0·1·1·|})
86 ····························································|·0·0·1·1·|··|·1·0·1·2·|··|·0·1·0·1·|</code></pre>86 ····························································|·0·0·1·1·|··|·1·0·1·2·|··|·0·1·0·1·|</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
Offset 100, 15 lines modifiedOffset 100, 15 lines modified
100 ·····c···x*y·+·c···x·+·c···y·+·c···)100 ·····c···x*y·+·c···x·+·c···y·+·c···)
101 ······3,3·······3,4·····3,2·····3,1101 ······3,3·······3,4·····3,2·····3,1
  
102 o3·:·Sequence</code></pre>102 o3·:·Sequence</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ··········<tr>104 ··········<tr>
105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Res(f,g,h)105 <td>··············<pre><code·class="language-macaulay2">i4·:·time·Res(f,g,h)
106 ·--·used·0.00614827s·(cpu);·0.00731995s·(thread);·0s·(gc)106 ·--·used·0.00799013s·(cpu);·0.0087872s·(thread);·0s·(gc)
  
107 ········2·······················4······2···2···············4····107 ········2·······················4······2···2···············4····
108 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+108 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+
109 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,1··109 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,1··
110 ·····------------------------------------------------------------------------110 ·····------------------------------------------------------------------------
111 ······3·······2·······3···············2···················3········111 ······3·······2·······3···············2···················3········
112 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+112 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+
Offset 817, 15 lines modifiedOffset 817, 15 lines modified
817 <td>··············<pre><code·class="language-macaulay2">i5·:·assert(Res(f,g,h)·==·sparseResultant(f,g,h))</code></pre>817 <td>··············<pre><code·class="language-macaulay2">i5·:·assert(Res(f,g,h)·==·sparseResultant(f,g,h))</code></pre>
818 </td>··········</tr>818 </td>··········</tr>
819 ········</table>819 ········</table>
820 ········<p>In·the·second·example,·we·calculate·the·sparse·unmixed·resultant·associated·to·the·set·of·monomials·$(1,x,y,xy)$.·Then·we·evaluate·it·at·the·three·polynomials·$f·=·a_0·+·a_1·x·+·a_2·y·+·a_3·x·y,·g·=·b_0·+·b_1·x·+·b_2·y·+·b_3·x·y,·h·=·c_0·+·c_1·x·+·c_2·y·+·c_3·x·y$.·Moreover,·we·perform·all·the·computation·over·$\mathbb{Z}/3331$.</p>820 ········<p>In·the·second·example,·we·calculate·the·sparse·unmixed·resultant·associated·to·the·set·of·monomials·$(1,x,y,xy)$.·Then·we·evaluate·it·at·the·three·polynomials·$f·=·a_0·+·a_1·x·+·a_2·y·+·a_3·x·y,·g·=·b_0·+·b_1·x·+·b_2·y·+·b_3·x·y,·h·=·c_0·+·c_1·x·+·c_2·y·+·c_3·x·y$.·Moreover,·we·perform·all·the·computation·over·$\mathbb{Z}/3331$.</p>
821 ········<table·class="examples">821 ········<table·class="examples">
822 ··········<tr>822 ··········<tr>
823 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);823 <td>··············<pre><code·class="language-macaulay2">i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
824 ·--·used·0.771274s·(cpu);·0.13058s·(thread);·0s·(gc)824 ·--·used·0.656266s·(cpu);·0.0858027s·(thread);·0s·(gc)
  
825 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over·ZZ/3331)825 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over·ZZ/3331)
826 ·····························································|·0·1·0·1·|</code></pre>826 ·····························································|·0·1·0·1·|</code></pre>
827 </td>··········</tr>827 </td>··········</tr>
828 ··········<tr>828 ··········<tr>
829 <td>··············<pre><code·class="language-macaulay2">i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];</code></pre>829 <td>··············<pre><code·class="language-macaulay2">i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];</code></pre>
830 </td>··········</tr>830 </td>··········</tr>
Offset 835, 15 lines modifiedOffset 835, 15 lines modified
835 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)835 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)
836 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0836 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0
  
837 o8·:·Sequence</code></pre>837 o8·:·Sequence</code></pre>
838 </td>··········</tr>838 </td>··········</tr>
839 ··········<tr>839 ··········<tr>
840 <td>··············<pre><code·class="language-macaulay2">i9·:·time·Res(f,g,h)840 <td>··············<pre><code·class="language-macaulay2">i9·:·time·Res(f,g,h)
841 ·--·used·0.00400011s·(cpu);·0.0025948s·(thread);·0s·(gc)841 ·--·used·0.00370975s·(cpu);·0.00315709s·(thread);·0s·(gc)
  
842 ······2·····2············2············2········2·2····2··········842 ······2·····2············2············2········2·2····2··········
843 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-843 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-
844 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1··844 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1··
845 ·····------------------------------------------------------------------------845 ·····------------------------------------------------------------------------
846 ·························2·······················2·························846 ·························2·······················2·························
847 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+847 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+
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918 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)918 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
919 ·······4·1·2····2·2····3·1····1·2····0919 ·······4·1·2····2·2····3·1····1·2····0
  
920 o11·:·Sequence</code></pre>920 o11·:·Sequence</code></pre>
921 </td>··········</tr>921 </td>··········</tr>
922 ··········<tr>922 ··········<tr>
923 <td>··············<pre><code·class="language-macaulay2">i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));923 <td>··············<pre><code·class="language-macaulay2">i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
924 ·--·used·1.24143s·(cpu);·0.50713s·(thread);·0s·(gc)</code></pre>924 ·--·used·1.23788s·(cpu);·0.512561s·(thread);·0s·(gc)</code></pre>
925 </td>··········</tr>925 </td>··········</tr>
926 ··········<tr>926 ··········<tr>
927 <td>··············<pre><code·class="language-macaulay2">i13·:·quotientRemainder(UnmixedRes,MixedRes)927 <td>··············<pre><code·class="language-macaulay2">i13·:·quotientRemainder(UnmixedRes,MixedRes)
  
928 ········2·2···················2····2·······························2·2928 ········2·2···················2····2·······························2·2
929 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)929 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)
930 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5930 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5
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35 In·the·first·example,·we·calculate·the·sparse·(mixed)·resultant·associated·to35 In·the·first·example,·we·calculate·the·sparse·(mixed)·resultant·associated·to
36 the·three·sets·of·monomials·$(1,x·y,x^2·y,x),(y,x^2·y^2,x^2·y,x),(1,y,x·y,x)$.36 the·three·sets·of·monomials·$(1,x·y,x^2·y,x),(y,x^2·y^2,x^2·y,x),(1,y,x·y,x)$.
37 Then·we·evaluate·it·at·the·three·polynomials·$f·=·c_{(1,1)}+c_{(1,2)}·x·y+c_{37 Then·we·evaluate·it·at·the·three·polynomials·$f·=·c_{(1,1)}+c_{(1,2)}·x·y+c_{
38 (1,3)}·x^2·y+c_{(1,4)}·x,·g·=·c_{(2,1)}·y+c_{(2,2)}·x^2·y^2+c_{(2,3)}·x^2·y+c_{38 (1,3)}·x^2·y+c_{(1,4)}·x,·g·=·c_{(2,1)}·y+c_{(2,2)}·x^2·y^2+c_{(2,3)}·x^2·y+c_{
39 (2,4)}·x,·h·=·c_{(3,1)}+c_{(3,2)}·y+c_{(3,3)}·x·y+c_{(3,4)}·x$.39 (2,4)}·x,·h·=·c_{(3,1)}+c_{(3,2)}·y+c_{(3,3)}·x·y+c_{(3,4)}·x$.
40 i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},40 i1·:·time·Res·=·sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},
41 {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})41 {1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
42 ·--·used·0.577685s·(cpu);·0.373009s·(thread);·0s·(gc)42 ·--·used·0.768478s·(cpu);·0.544176s·(thread);·0s·(gc)
  
43 o1·=·Res43 o1·=·Res
  
44 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·144 o1·:·SparseResultant·(sparse·mixed·resultant·associated·to·{|·0·1·1·2·|,·|·0·1
45 2·2·|,·|·0·0·1·1·|})45 2·2·|,·|·0·0·1·1·|})
46 ····························································|·0·0·1·1·|··|·1·046 ····························································|·0·0·1·1·|··|·1·0
47 1·2·|··|·0·1·0·1·|47 1·2·|··|·0·1·0·1·|
Offset 56, 15 lines modifiedOffset 56, 15 lines modified
56 ·······1,3·······1,2·······1,4·····1,1···2,2········2,3·······2,4·····2,156 ·······1,3·······1,2·······1,4·····1,1···2,2········2,3·······2,4·····2,1
57 ·····------------------------------------------------------------------------57 ·····------------------------------------------------------------------------
58 ·····c···x*y·+·c···x·+·c···y·+·c···)58 ·····c···x*y·+·c···x·+·c···y·+·c···)
59 ······3,3·······3,4·····3,2·····3,159 ······3,3·······3,4·····3,2·····3,1
  
60 o3·:·Sequence60 o3·:·Sequence
61 i4·:·time·Res(f,g,h)61 i4·:·time·Res(f,g,h)
62 ·--·used·0.00614827s·(cpu);·0.00731995s·(thread);·0s·(gc)62 ·--·used·0.00799013s·(cpu);·0.0087872s·(thread);·0s·(gc)
  
63 ········2·······················4······2···2···············463 ········2·······················4······2···2···············4
64 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+64 o4·=·-·c···c···c···c···c···c···c····+·c···c···c···c···c···c····+
65 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,165 ········1,2·1,3·1,4·2,1·2,2·2,3·3,1····1,2·1,3·2,1·2,2·2,4·3,1
66 ·····------------------------------------------------------------------------66 ·····------------------------------------------------------------------------
67 ······3·······2·······3···············2···················367 ······3·······2·······3···············2···················3
68 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+68 ·····c···c···c···c···c···c····-·2c···c···c···c···c···c···c···c····+
Offset 772, 29 lines modifiedOffset 772, 29 lines modified
772 In·the·second·example,·we·calculate·the·sparse·unmixed·resultant·associated·to772 In·the·second·example,·we·calculate·the·sparse·unmixed·resultant·associated·to
773 the·set·of·monomials·$(1,x,y,xy)$.·Then·we·evaluate·it·at·the·three·polynomials773 the·set·of·monomials·$(1,x,y,xy)$.·Then·we·evaluate·it·at·the·three·polynomials
774 $f·=·a_0·+·a_1·x·+·a_2·y·+·a_3·x·y,·g·=·b_0·+·b_1·x·+·b_2·y·+·b_3·x·y,·h·=·c_0774 $f·=·a_0·+·a_1·x·+·a_2·y·+·a_3·x·y,·g·=·b_0·+·b_1·x·+·b_2·y·+·b_3·x·y,·h·=·c_0
775 +·c_1·x·+·c_2·y·+·c_3·x·y$.·Moreover,·we·perform·all·the·computation·over775 +·c_1·x·+·c_2·y·+·c_3·x·y$.·Moreover,·we·perform·all·the·computation·over
776 $\mathbb{Z}/3331$.776 $\mathbb{Z}/3331$.
777 i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},777 i6·:·time·Res·=·sparseResultant(matrix{{0,0,1,1},
778 {0,1,0,1}},CoefficientRing=>ZZ/3331);778 {0,1,0,1}},CoefficientRing=>ZZ/3331);
779 ·--·used·0.771274s·(cpu);·0.13058s·(thread);·0s·(gc)779 ·--·used·0.656266s·(cpu);·0.0858027s·(thread);·0s·(gc)
  
780 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over780 o6·:·SparseResultant·(sparse·unmixed·resultant·associated·to·|·0·0·1·1·|·over
781 ZZ/3331)781 ZZ/3331)
782 ·····························································|·0·1·0·1·|782 ·····························································|·0·1·0·1·|
783 i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];783 i7·:·ZZ/3331[a_0..a_3,b_0..b_3,c_0..c_3][x,y];
784 i8·:·(f,g,h)·=·(a_0·+·a_1*x·+·a_2*y·+·a_3*x*y,·b_0·+·b_1*x·+·b_2*y·+·b_3*x*y,784 i8·:·(f,g,h)·=·(a_0·+·a_1*x·+·a_2*y·+·a_3*x*y,·b_0·+·b_1*x·+·b_2*y·+·b_3*x*y,
785 c_0·+·c_1*x·+·c_2*y·+·c_3*x*y)785 c_0·+·c_1*x·+·c_2*y·+·c_3*x*y)
  
786 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)786 o8·=·(a·x*y·+·a·x·+·a·y·+·a·,·b·x*y·+·b·x·+·b·y·+·b·,·c·x*y·+·c·x·+·c·y·+·c·)
787 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0787 ·······3·······1·····2·····0···3·······1·····2·····0···3·······1·····2·····0
  
788 o8·:·Sequence788 o8·:·Sequence
789 i9·:·time·Res(f,g,h)789 i9·:·time·Res(f,g,h)
790 ·--·used·0.00400011s·(cpu);·0.0025948s·(thread);·0s·(gc)790 ·--·used·0.00370975s·(cpu);·0.00315709s·(thread);·0s·(gc)
  
791 ······2·····2············2············2········2·2····2791 ······2·····2············2············2········2·2····2
792 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-792 o9·=·a·b·b·c··-·a·a·b·b·c··-·a·a·b·b·c··+·a·a·b·c··-·a·b·b·c·c··-
793 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1793 ······3·1·2·0····2·3·1·3·0····1·3·2·3·0····1·2·3·0····3·0·2·0·1
794 ·····------------------------------------------------------------------------794 ·····------------------------------------------------------------------------
795 ·························2·······················2795 ·························2·······················2
796 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+796 ·····a·a·b·b·c·c··+·a·a·b·c·c··+·a·a·b·b·c·c··+·a·b·b·c·c··-·a·a·b·b·c·c··+
Offset 864, 15 lines modifiedOffset 864, 15 lines modified
864 ··················2864 ··················2
865 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)865 ······c·x·x··+·c·x··+·c·x··+·c·x··+·c·)
866 ·······4·1·2····2·2····3·1····1·2····0866 ·······4·1·2····2·2····3·1····1·2····0
  
867 o11·:·Sequence867 o11·:·Sequence
868 i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant868 i12·:·time·(MixedRes,UnmixedRes)·=·(sparseResultant(f,g,h),sparseResultant
869 (f,g,h,Unmixed=>true));869 (f,g,h,Unmixed=>true));
870 ·--·used·1.24143s·(cpu);·0.50713s·(thread);·0s·(gc)870 ·--·used·1.23788s·(cpu);·0.512561s·(thread);·0s·(gc)
871 i13·:·quotientRemainder(UnmixedRes,MixedRes)871 i13·:·quotientRemainder(UnmixedRes,MixedRes)
  
872 ········2·2···················2····2·······························2·2872 ········2·2···················2····2·······························2·2
873 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)873 o13·=·(b·c··-·b·b·c·c··+·b·b·c··+·b·c·c··-·2b·b·c·c··-·b·b·c·c··+·b·c·,·0)
874 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5874 ········5·2····4·5·2·4····2·5·4····4·2·5·····2·5·2·5····2·4·4·5····2·5
  
875 o13·:·Sequence875 o13·:·Sequence
⊟
761 B
./usr/share/doc/Macaulay2/SpechtModule/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=387 #:len=38
8 c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=8 c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=
9 #:len=2879 #:len=287
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s
⊟
2.44 KB
./usr/share/doc/Macaulay2/SpechtModule/example-output/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.out
    
Offset 25, 15 lines modifiedOffset 25, 15 lines modified
25 o4·=·|·0·1·|25 o4·=·|·0·1·|
26 ·····|·2·3·|26 ·····|·2·3·|
27 ·····|·4·|27 ·····|·4·|
  
28 o4·:·YoungTableau28 o4·:·YoungTableau
  
29 i5·:·time·higherSpechtPolynomial(S,T,R)29 i5·:·time·higherSpechtPolynomial(S,T,R)
30 ·--·used·0.0022521s·(cpu);·0.000926164s·(thread);·0s·(gc)30 ·--·used·0.00223019s·(cpu);·0.00109961s·(thread);·0s·(gc)
  
31 ······3·2··········2·3······3·····2········3·2····3···2······2···3····31 ······3·2··········2·3······3·····2········3·2····3···2······2···3····
32 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-32 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
33 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··33 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··
34 ·····------------------------------------------------------------------------34 ·····------------------------------------------------------------------------
35 ······3·2········3·2········2·3········2·3········3···2········3·2····35 ······3·2········3·2········2·3········2·3········3···2········3·2····
36 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-36 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
Offset 46, 15 lines modifiedOffset 46, 15 lines modified
46 ········2···3····2·····3····2·····3······2···3········2·3········2·346 ········2···3····2·····3····2·····3······2···3········2·3········2·3
47 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x47 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
48 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·448 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
49 o5·:·R49 o5·:·R
  
50 i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)50 i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)
51 ·--·used·5.286e-05s·(cpu);·0.000938202s·(thread);·0s·(gc)51 ·--·used·0.000462847s·(cpu);·0.00112547s·(thread);·0s·(gc)
  
52 ······3·2··········2·3······3·····2········3·2····3···2······2···3····52 ······3·2··········2·3······3·····2········3·2····3···2······2···3····
53 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-53 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
54 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··54 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ······3·2········3·2········2·3········2·3········3···2········3·2····56 ······3·2········3·2········2·3········2·3········3···2········3·2····
57 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-57 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
Offset 67, 15 lines modifiedOffset 67, 15 lines modified
67 ········2···3····2·····3····2·····3······2···3········2·3········2·367 ········2···3····2·····3····2·····3······2···3········2·3········2·3
68 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x68 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
69 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·469 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
70 o6·:·R70 o6·:·R
  
71 i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)71 i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)
72 ·--·used·0.000838432s·(cpu);·0.00144067s·(thread);·0s·(gc)72 ·--·used·0.0020504s·(cpu);·0.00217022s·(thread);·0s·(gc)
  
73 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)(x·)(x·))73 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)(x·)(x·))
74 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4···2···074 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4···2···0
  
75 o7·:·Expression·of·class·Product75 o7·:·Expression·of·class·Product
  
76 i8·:·76 i8·:·
⊟
751 B
./usr/share/doc/Macaulay2/SpechtModule/example-output/_representation__Multiplicity.out
    
Offset 25, 15 lines modifiedOffset 25, 15 lines modified
25 o2·:·List25 o2·:·List
  
26 i3·:·tal·:=·tally·apply·(H,h->conjugacyClass·h);26 i3·:·tal·:=·tally·apply·(H,h->conjugacyClass·h);
  
27 i4·:·partis·=·partitions·6;27 i4·:·partis·=·partitions·6;
  
28 i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity(tal,p))28 i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity(tal,p))
29 ·--·used·0.189444s·(cpu);·0.166948s·(thread);·0s·(gc)29 ·--·used·0.228763s·(cpu);·0.195076s·(thread);·0s·(gc)
  
30 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}30 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}
31 ···············Partition{2,·1,·1,·1,·1}·=>·031 ···············Partition{2,·1,·1,·1,·1}·=>·0
32 ···············Partition{2,·2,·1,·1}·=>·132 ···············Partition{2,·2,·1,·1}·=>·1
33 ···············Partition{2,·2,·2}·=>·133 ···············Partition{2,·2,·2}·=>·1
34 ···············Partition{3,·1,·1,·1}·=>·034 ···············Partition{3,·1,·1,·1}·=>·0
35 ···············Partition{3,·2,·1}·=>·035 ···············Partition{3,·2,·1}·=>·0
⊟
876 B
./usr/share/doc/Macaulay2/SpechtModule/example-output/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.out
    
Offset 9, 15 lines modifiedOffset 9, 15 lines modified
9 i2·:·genList·=·{{1,2,3,0,5,4},{0,4,2,5,1,3}}9 i2·:·genList·=·{{1,2,3,0,5,4},{0,4,2,5,1,3}}
  
10 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}10 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}
  
11 o2·:·List11 o2·:·List
  
12 i3·:·time·seco·=·secondaryInvariants(genList,R);12 i3·:·time·seco·=·secondaryInvariants(genList,R);
13 ·--·used·0.419118s·(cpu);·0.335759s·(thread);·0s·(gc)13 ·--·used·0.507903s·(cpu);·0.381046s·(thread);·0s·(gc)
14 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)14 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)
15 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)15 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)
16 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)16 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)
17 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)17 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
18 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)18 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)
19 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)19 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)
20 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)20 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
⊟
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122 ·····|·2·3·|122 ·····|·2·3·|
123 ·····|·4·|123 ·····|·4·|
  
124 o4·:·YoungTableau</code></pre>124 o4·:·YoungTableau</code></pre>
125 </td>··········</tr>125 </td>··········</tr>
126 ··········<tr>126 ··········<tr>
127 <td>··············<pre><code·class="language-macaulay2">i5·:·time·higherSpechtPolynomial(S,T,R)127 <td>··············<pre><code·class="language-macaulay2">i5·:·time·higherSpechtPolynomial(S,T,R)
128 ·--·used·0.0022521s·(cpu);·0.000926164s·(thread);·0s·(gc)128 ·--·used·0.00223019s·(cpu);·0.00109961s·(thread);·0s·(gc)
  
129 ······3·2··········2·3······3·····2········3·2····3···2······2···3····129 ······3·2··········2·3······3·····2········3·2····3···2······2···3····
130 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-130 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
131 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··131 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··
132 ·····------------------------------------------------------------------------132 ·····------------------------------------------------------------------------
133 ······3·2········3·2········2·3········2·3········3···2········3·2····133 ······3·2········3·2········2·3········2·3········3···2········3·2····
134 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-134 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
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144 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x144 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
145 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4145 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
146 o5·:·R</code></pre>146 o5·:·R</code></pre>
147 </td>··········</tr>147 </td>··········</tr>
148 ··········<tr>148 ··········<tr>
149 <td>··············<pre><code·class="language-macaulay2">i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)149 <td>··············<pre><code·class="language-macaulay2">i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)
150 ·--·used·5.286e-05s·(cpu);·0.000938202s·(thread);·0s·(gc)150 ·--·used·0.000462847s·(cpu);·0.00112547s·(thread);·0s·(gc)
  
151 ······3·2··········2·3······3·····2········3·2····3···2······2···3····151 ······3·2··········2·3······3·····2········3·2····3···2······2···3····
152 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-152 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
153 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··153 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4··
154 ·····------------------------------------------------------------------------154 ·····------------------------------------------------------------------------
155 ······3·2········3·2········2·3········2·3········3···2········3·2····155 ······3·2········3·2········2·3········2·3········3···2········3·2····
156 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-156 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
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166 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x166 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
167 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4167 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
168 o6·:·R</code></pre>168 o6·:·R</code></pre>
169 </td>··········</tr>169 </td>··········</tr>
170 ··········<tr>170 ··········<tr>
171 <td>··············<pre><code·class="language-macaulay2">i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)171 <td>··············<pre><code·class="language-macaulay2">i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)
172 ·--·used·0.000838432s·(cpu);·0.00144067s·(thread);·0s·(gc)172 ·--·used·0.0020504s·(cpu);·0.00217022s·(thread);·0s·(gc)
  
173 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)(x·)(x·))173 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)(x·)(x·))
174 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4···2···0174 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4···2···0
  
175 o7·:·Expression·of·class·Product</code></pre>175 o7·:·Expression·of·class·Product</code></pre>
176 </td>··········</tr>176 </td>··········</tr>
177 ········</table>177 ········</table>
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69 o4·=·|·0·1·|69 o4·=·|·0·1·|
70 ·····|·2·3·|70 ·····|·2·3·|
71 ·····|·4·|71 ·····|·4·|
  
72 o4·:·YoungTableau72 o4·:·YoungTableau
73 i5·:·time·higherSpechtPolynomial(S,T,R)73 i5·:·time·higherSpechtPolynomial(S,T,R)
74 ·--·used·0.0022521s·(cpu);·0.000926164s·(thread);·0s·(gc)74 ·--·used·0.00223019s·(cpu);·0.00109961s·(thread);·0s·(gc)
  
75 ······3·2··········2·3······3·····2········3·2····3···2······2···375 ······3·2··········2·3······3·····2········3·2····3···2······2···3
76 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-76 o5·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
77 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·477 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4
78 ·····------------------------------------------------------------------------78 ·····------------------------------------------------------------------------
79 ······3·2········3·2········2·3········2·3········3···2········3·279 ······3·2········3·2········2·3········2·3········3···2········3·2
80 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-80 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
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89 ·····------------------------------------------------------------------------89 ·····------------------------------------------------------------------------
90 ········2···3····2·····3····2·····3······2···3········2·3········2·390 ········2···3····2·····3····2·····3······2···3········2·3········2·3
91 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x91 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
92 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·492 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
93 o5·:·R93 o5·:·R
94 i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)94 i6·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false)
95 ·--·used·5.286e-05s·(cpu);·0.000938202s·(thread);·0s·(gc)95 ·--·used·0.000462847s·(cpu);·0.00112547s·(thread);·0s·(gc)
  
96 ······3·2··········2·3······3·····2········3·2····3···2······2···396 ······3·2··········2·3······3·····2········3·2····3···2······2···3
97 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-97 o6·=·x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
98 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·498 ······0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·3····0·1·2·4····0·1·2·4
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ······3·2········3·2········2·3········2·3········3···2········3·2100 ······3·2········3·2········2·3········2·3········3···2········3·2
101 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-101 ·····x·x·x·x··-·x·x·x·x··+·x·x·x·x··+·x·x·x·x··+·x·x·x·x··-·x·x·x·x··-
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109 ·····------------------------------------------------------------------------109 ·····------------------------------------------------------------------------
110 ········2···3····2·····3····2·····3······2···3········2·3········2·3110 ········2···3····2·····3····2·····3······2···3········2·3········2·3
111 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x111 ·····x·x·x·x··-·x·x·x·x··-·x·x·x·x··+·x·x·x·x··-·x·x·x·x··+·x·x·x·x
112 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4112 ······0·1·3·4····0·2·3·4····1·2·3·4····0·2·3·4····0·1·3·4····1·2·3·4
  
113 o6·:·R113 o6·:·R
114 i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)114 i7·:·time·higherSpechtPolynomial(S,T,R,·Robust·=>·false,·AsExpression·=>·true)
115 ·--·used·0.000838432s·(cpu);·0.00144067s·(thread);·0s·(gc)115 ·--·used·0.0020504s·(cpu);·0.00217022s·(thread);·0s·(gc)
  
116 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)116 o7·=·(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)(-·x··+·x·)((x··+·x··+·x·)(x·)(x·)·+·(x·)
117 (x·)(x·))117 (x·)(x·))
118 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4118 ·········0····2·····0····4·····2····4·····1····3····0····2····4···3···1······4
119 2···0119 2···0
  
120 o7·:·Expression·of·class·Product120 o7·:·Expression·of·class·Product
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119 ········</div>119 ········</div>
120 ········<table·class="examples">120 ········<table·class="examples">
121 ··········<tr>121 ··········<tr>
122 <td>··············<pre><code·class="language-macaulay2">i4·:·partis·=·partitions·6;</code></pre>122 <td>··············<pre><code·class="language-macaulay2">i4·:·partis·=·partitions·6;</code></pre>
123 </td>··········</tr>123 </td>··········</tr>
124 ··········<tr>124 ··········<tr>
125 <td>··············<pre><code·class="language-macaulay2">i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity(tal,p))125 <td>··············<pre><code·class="language-macaulay2">i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity(tal,p))
126 ·--·used·0.189444s·(cpu);·0.166948s·(thread);·0s·(gc)126 ·--·used·0.228763s·(cpu);·0.195076s·(thread);·0s·(gc)
  
127 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}127 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}
128 ···············Partition{2,·1,·1,·1,·1}·=>·0128 ···············Partition{2,·1,·1,·1,·1}·=>·0
129 ···············Partition{2,·2,·1,·1}·=>·1129 ···············Partition{2,·2,·1,·1}·=>·1
130 ···············Partition{2,·2,·2}·=>·1130 ···············Partition{2,·2,·2}·=>·1
131 ···············Partition{3,·1,·1,·1}·=>·0131 ···············Partition{3,·1,·1,·1}·=>·0
132 ···············Partition{3,·2,·1}·=>·0132 ···············Partition{3,·2,·1}·=>·0
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64 representations·of·$H$·in·each·irreducible·representation·of·$S_6$.·We·take64 representations·of·$H$·in·each·irreducible·representation·of·$S_6$.·We·take
65 into·account·that·there·are·multiple·copies·of·each·representation·by65 into·account·that·there·are·multiple·copies·of·each·representation·by
66 multiplying·the·values·with·the·number·of·copies·which·is·given·by·the66 multiplying·the·values·with·the·number·of·copies·which·is·given·by·the
67 hookLengthFormula.67 hookLengthFormula.
68 i4·:·partis·=·partitions·6;68 i4·:·partis·=·partitions·6;
69 i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity69 i5·:·time·multi·=·hashTable·apply·(partis,·p->·p=>·representationMultiplicity
70 (tal,p))70 (tal,p))
71 ·--·used·0.189444s·(cpu);·0.166948s·(thread);·0s·(gc)71 ·--·used·0.228763s·(cpu);·0.195076s·(thread);·0s·(gc)
  
72 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}72 o5·=·HashTable{Partition{1,·1,·1,·1,·1,·1}·=>·1}
73 ···············Partition{2,·1,·1,·1,·1}·=>·073 ···············Partition{2,·1,·1,·1,·1}·=>·0
74 ···············Partition{2,·2,·1,·1}·=>·174 ···············Partition{2,·2,·1,·1}·=>·1
75 ···············Partition{2,·2,·2}·=>·175 ···············Partition{2,·2,·2}·=>·1
76 ···············Partition{3,·1,·1,·1}·=>·076 ···············Partition{3,·1,·1,·1}·=>·0
77 ···············Partition{3,·2,·1}·=>·077 ···············Partition{3,·2,·1}·=>·0
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Offset 98, 15 lines modifiedOffset 98, 15 lines modified
  
98 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}98 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}
  
99 o2·:·List</code></pre>99 o2·:·List</code></pre>
100 </td>··········</tr>100 </td>··········</tr>
101 ··········<tr>101 ··········<tr>
102 <td>··············<pre><code·class="language-macaulay2">i3·:·time·seco·=·secondaryInvariants(genList,R);102 <td>··············<pre><code·class="language-macaulay2">i3·:·time·seco·=·secondaryInvariants(genList,R);
103 ·--·used·0.419118s·(cpu);·0.335759s·(thread);·0s·(gc)103 ·--·used·0.507903s·(cpu);·0.381046s·(thread);·0s·(gc)
104 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)104 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)
105 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)105 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)
106 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)106 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)
107 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)107 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
108 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)108 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)
109 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)109 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)
110 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)110 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
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46 o1·:·PolynomialRing46 o1·:·PolynomialRing
47 i2·:·genList·=·{{1,2,3,0,5,4},{0,4,2,5,1,3}}47 i2·:·genList·=·{{1,2,3,0,5,4},{0,4,2,5,1,3}}
  
48 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}48 o2·=·{{1,·2,·3,·0,·5,·4},·{0,·4,·2,·5,·1,·3}}
  
49 o2·:·List49 o2·:·List
50 i3·:·time·seco·=·secondaryInvariants(genList,R);50 i3·:·time·seco·=·secondaryInvariants(genList,R);
51 ·--·used·0.419118s·(cpu);·0.335759s·(thread);·0s·(gc)51 ·--·used·0.507903s·(cpu);·0.381046s·(thread);·0s·(gc)
52 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)52 (Partition{6},·Ambient_Dimension,·1,·Rank,·1)
53 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)53 (Partition{5,·1},·Ambient_Dimension,·5,·Rank,·0)
54 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)54 (Partition{4,·2},·Ambient_Dimension,·9,·Rank,·1)
55 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)55 (Partition{4,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
56 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)56 (Partition{3,·3},·Ambient_Dimension,·5,·Rank,·1)
57 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)57 (Partition{3,·2,·1},·Ambient_Dimension,·16,·Rank,·0)
58 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)58 (Partition{3,·1,·1,·1},·Ambient_Dimension,·10,·Rank,·0)
⊟
769 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=317 #:len=31
8 SW50ZXJzZWN0aW9uT2ZUaHJlZVF1YWRyaWNzSW5QNw==8 SW50ZXJzZWN0aW9uT2ZUaHJlZVF1YWRyaWNzSW5QNw==
9 #:len=6259 #:len=625
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBzcGVjaWFs10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBzcGVjaWFs
11 IGludGVyc2VjdGlvbiBvZiB0aHJlZSBxdWFkcmljcyBpbiBQXjciLCAibGluZW51bSIgPT4gMzY111 IGludGVyc2VjdGlvbiBvZiB0aHJlZSBxdWFkcmljcyBpbiBQXjciLCAibGluZW51bSIgPT4gMzY1
⊟
725 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
    
Offset 10, 15 lines modifiedOffset 10, 15 lines modified
10 ·····of·discriminant·31·=·det|·8·1·|10 ·····of·discriminant·31·=·det|·8·1·|
11 ·····························|·1·4·|11 ·····························|·1·4·|
12 ·····containing·a·surface·of·degree·1·and·sectional·genus·012 ·····containing·a·surface·of·degree·1·and·sectional·genus·0
13 ·····cut·out·by·5·hypersurfaces·of·degree·113 ·····cut·out·by·5·hypersurfaces·of·degree·1
14 ·····(This·is·a·classical·example·of·rational·fourfold)14 ·····(This·is·a·classical·example·of·rational·fourfold)
  
15 i3·:·time·U'·=·associatedCastelnuovoSurface·X;15 i3·:·time·U'·=·associatedCastelnuovoSurface·X;
16 ·--·used·3.59693s·(cpu);·1.06332s·(thread);·0s·(gc)16 ·--·used·4.40484s·(cpu);·1.04889s·(thread);·0s·(gc)
  
17 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X17 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X
  
18 i4·:·(mu,U,C,f)·=·building·U';18 i4·:·(mu,U,C,f)·=·building·U';
  
19 i5·:·?·mu19 i5·:·?·mu
  
⊟
690 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Cubic__Fourfold_rp.out
    
Offset 7, 15 lines modifiedOffset 7, 15 lines modified
7 i2·:·describe·X7 i2·:·describe·X
  
8 o2·=·Special·cubic·fourfold·of·discriminant·148 o2·=·Special·cubic·fourfold·of·discriminant·14
9 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·09 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·0
10 ·····cut·out·by·6·hypersurfaces·of·degree·210 ·····cut·out·by·6·hypersurfaces·of·degree·2
  
11 i3·:·time·U'·=·associatedK3surface·X;11 i3·:·time·U'·=·associatedK3surface·X;
12 ·--·used·3.57308s·(cpu);·1.10736s·(thread);·0s·(gc)12 ·--·used·3.99832s·(cpu);·1.12526s·(thread);·0s·(gc)
  
13 o3·:·ProjectiveVariety,·K3·surface·associated·to·X13 o3·:·ProjectiveVariety,·K3·surface·associated·to·X
  
14 i4·:·(mu,U,C,f)·=·building·U';14 i4·:·(mu,U,C,f)·=·building·U';
  
15 i5·:·?·mu15 i5·:·?·mu
  
⊟
774 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Special__Gushel__Mukai__Fourfold_rp.out
    
Offset 10, 15 lines modifiedOffset 10, 15 lines modified
10 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·010 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0
11 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)11 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
12 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)12 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
13 ·····Type:·ordinary13 ·····Type:·ordinary
14 ·····(case·1·of·Table·1·in·arXiv:2002.07026)14 ·····(case·1·of·Table·1·in·arXiv:2002.07026)
  
15 i3·:·time·U'·=·associatedK3surface·X;15 i3·:·time·U'·=·associatedK3surface·X;
16 ·--·used·7.9956s·(cpu);·3.7328s·(thread);·0s·(gc)16 ·--·used·10.0322s·(cpu);·4.13645s·(thread);·0s·(gc)
  
17 o3·:·ProjectiveVariety,·K3·surface·associated·to·X17 o3·:·ProjectiveVariety,·K3·surface·associated·to·X
  
18 i4·:·(mu,U,C,f)·=·building·U';18 i4·:·(mu,U,C,f)·=·building·U';
  
19 i5·:·?·mu19 i5·:·?·mu
  
⊟
1.29 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Cubic__Fourfold_cm__Z__Z_rp.out
    
Offset 8, 28 lines modifiedOffset 8, 28 lines modified
8 i2·:·describe·X8 i2·:·describe·X
  
9 o2·=·Special·cubic·fourfold·of·discriminant·269 o2·=·Special·cubic·fourfold·of·discriminant·26
10 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·010 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
11 ·····cut·out·by·13·hypersurfaces·of·degree·311 ·····cut·out·by·13·hypersurfaces·of·degree·3
  
12 i3·:·time·f·=·detectCongruence(X,Verbose=>true);12 i3·:·time·f·=·detectCongruence(X,Verbose=>true);
13 ·--·used·2.96759s·(cpu);·1.73627s·(thread);·0s·(gc)13 ·--·used·3.66454s·(cpu);·1.59606s·(thread);·0s·(gc)
14 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a·general·point:·814 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a·general·point:·8
15 number·2-secant·lines·=·715 number·2-secant·lines·=·7
16 number·5-secant·conics·=·116 number·5-secant·conics·=·1
  
17 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^517 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^5
  
18 i4·:·p·:=·point·ambient·X·--·random·point·on·P^518 i4·:·p·:=·point·ambient·X·--·random·point·on·P^5
  
19 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]19 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]
  
20 o4·:·ProjectiveVariety,·a·point·in·PP^520 o4·:·ProjectiveVariety,·a·point·in·PP^5
  
21 i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface21 i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface
22 ·--·used·0.372948s·(cpu);·0.237981s·(thread);·0s·(gc)22 ·--·used·0.386309s·(cpu);·0.226969s·(thread);·0s·(gc)
  
23 o5·:·ProjectiveVariety,·curve·in·PP^523 o5·:·ProjectiveVariety,·curve·in·PP^5
  
24 i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree(C·*·surface·X)·==·5·and·isSubset(p,·C))24 i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree(C·*·surface·X)·==·5·and·isSubset(p,·C))
  
25 i7·:·25 i7·:·
⊟
1.48 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Special__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
    
Offset 11, 15 lines modifiedOffset 11, 15 lines modified
11 ·····containing·a·surface·in·PP^8·of·degree·9·and·sectional·genus·211 ·····containing·a·surface·in·PP^8·of·degree·9·and·sectional·genus·2
12 ·····cut·out·by·19·hypersurfaces·of·degree·212 ·····cut·out·by·19·hypersurfaces·of·degree·2
13 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)13 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)
14 ·····Type:·ordinary14 ·····Type:·ordinary
15 ·····(case·17·of·Table·1·in·arXiv:2002.07026)15 ·····(case·17·of·Table·1·in·arXiv:2002.07026)
  
16 i3·:·time·f·=·detectCongruence(X,Verbose=>true);16 i3·:·time·f·=·detectCongruence(X,Verbose=>true);
17 ·--·used·11.0735s·(cpu);·5.31108s·(thread);·0s·(gc)17 ·--·used·14.1479s·(cpu);·5.55996s·(thread);·0s·(gc)
18 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a·general·point:·718 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a·general·point:·7
19 number·1-secant·lines·=·619 number·1-secant·lines·=·6
20 number·3-secant·conics·=·120 number·3-secant·conics·=·1
  
21 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^821 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^8
  
22 i4·:·Y·=·ambientFivefold·X;·--·del·Pezzo·fivefold·containing·X22 i4·:·Y·=·ambientFivefold·X;·--·del·Pezzo·fivefold·containing·X
Offset 29, 15 lines modifiedOffset 29, 15 lines modified
29 i5·:·p·:=·point·Y·--·random·point·on·Y29 i5·:·p·:=·point·Y·--·random·point·on·Y
  
30 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,·13402,·1]30 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,·13402,·1]
  
31 o5·:·ProjectiveVariety,·a·point·in·PP^831 o5·:·ProjectiveVariety,·a·point·in·PP^8
  
32 i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface32 i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface
33 ·--·used·0.294189s·(cpu);·0.190741s·(thread);·0s·(gc)33 ·--·used·0.562943s·(cpu);·0.214401s·(thread);·0s·(gc)
  
34 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)34 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)
  
35 i7·:·S·=·surface·X;35 i7·:·S·=·surface·X;
  
36 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)36 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)
  
⊟
606 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Cubic__Fourfold_rp.out
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
1 --·-*-·M2-comint·-*-·hash:·17298908135795611111 --·-*-·M2-comint·-*-·hash:·1729890813579561111
  
2 i1·:·X·=·specialCubicFourfold·"quintic·del·Pezzo·surface";2 i1·:·X·=·specialCubicFourfold·"quintic·del·Pezzo·surface";
  
3 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and·sectional·genus·13 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and·sectional·genus·1
  
4 i2·:·time·discriminant·X4 i2·:·time·discriminant·X
5 ·--·used·0.146937s·(cpu);·0.0730333s·(thread);·0s·(gc)5 ·--·used·0.17626s·(cpu);·0.0850651s·(thread);·0s·(gc)
  
6 o2·=·146 o2·=·14
  
7 i3·:·7 i3·:·
⊟
609 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_discriminant_lp__Special__Gushel__Mukai__Fourfold_rp.out
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
1 --·-*-·M2-comint·-*-·hash:·17302209324187387131 --·-*-·M2-comint·-*-·hash:·1730220932418738713
  
2 i1·:·X·=·specialGushelMukaiFourfold·"tau-quadric";2 i1·:·X·=·specialGushelMukaiFourfold·"tau-quadric";
  
3 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·03 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0
  
4 i2·:·time·discriminant·X4 i2·:·time·discriminant·X
5 ·--·used·0.891636s·(cpu);·0.354162s·(thread);·0s·(gc)5 ·--·used·1.26496s·(cpu);·0.367537s·(thread);·0s·(gc)
  
6 o2·=·106 o2·=·10
  
7 i3·:·7 i3·:·
⊟
803 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 o2·:·ProjectiveVariety,·curve·in·PP^55 o2·:·ProjectiveVariety,·curve·in·PP^5
  
6 i3·:·X·=·random({{2},{2},{2}},S);6 i3·:·X·=·random({{2},{2},{2}},S);
  
7 o3·:·ProjectiveVariety,·surface·in·PP^57 o3·:·ProjectiveVariety,·surface·in·PP^5
  
8 i4·:·time·parameterCount(S,X,Verbose=>true)8 i4·:·time·parameterCount(S,X,Verbose=>true)
9 ·--·used·0.220293s·(cpu);·0.169981s·(thread);·0s·(gc)9 ·--·used·0.220138s·(cpu);·0.159861s·(thread);·0s·(gc)
10 S:·rational·normal·curve·of·degree·5·in·PP^510 S:·rational·normal·curve·of·degree·5·in·PP^5
11 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·3·hypersurfaces·of·degree·211 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·3·hypersurfaces·of·degree·2
12 (assumption:·h^1(N_{S,P^5})·=·0)12 (assumption:·h^1(N_{S,P^5})·=·0)
13 h^0(N_{S,P^5})·=·3213 h^0(N_{S,P^5})·=·32
14 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));14 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));
15 in·particular,·h^0(I_{S,P^5}(2))·is·minimal15 in·particular,·h^0(I_{S,P^5}(2))·is·minimal
16 dim·GG(2,9)·=·2116 dim·GG(2,9)·=·21
⊟
849 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Cubic__Fourfold_rp.out
    
Offset 5, 15 lines modifiedOffset 5, 15 lines modified
5 o2·:·ProjectiveVariety,·surface·in·PP^55 o2·:·ProjectiveVariety,·surface·in·PP^5
  
6 i3·:·X·=·specialCubicFourfold·V;6 i3·:·X·=·specialCubicFourfold·V;
  
7 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·07 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0
  
8 i4·:·time·parameterCount(X,Verbose=>true)8 i4·:·time·parameterCount(X,Verbose=>true)
9 ·--·used·0.366041s·(cpu);·0.251843s·(thread);·0s·(gc)9 ·--·used·0.426191s·(cpu);·0.293512s·(thread);·0s·(gc)
10 S:·Veronese·surface·in·PP^510 S:·Veronese·surface·in·PP^5
11 X:·smooth·cubic·hypersurface·in·PP^511 X:·smooth·cubic·hypersurface·in·PP^5
12 (assumption:·h^1(N_{S,P^5})·=·0)12 (assumption:·h^1(N_{S,P^5})·=·0)
13 h^0(N_{S,P^5})·=·2713 h^0(N_{S,P^5})·=·27
14 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));14 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));
15 in·particular,·h^0(I_{S,P^5}(3))·is·minimal15 in·particular,·h^0(I_{S,P^5}(3))·is·minimal
16 h^0(N_{S,P^5})·+·27·=·5416 h^0(N_{S,P^5})·+·27·=·54
⊟
927 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Special__Gushel__Mukai__Fourfold_rp.out
    
Offset 11, 15 lines modifiedOffset 11, 15 lines modified
11 o2·:·ProjectiveVariety,·surface·in·PP^9·(subvariety·of·codimension·4·in·G)11 o2·:·ProjectiveVariety,·surface·in·PP^9·(subvariety·of·codimension·4·in·G)
  
12 i3·:·X·=·specialGushelMukaiFourfold·S;12 i3·:·X·=·specialGushelMukaiFourfold·S;
  
13 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and·sectional·genus·013 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and·sectional·genus·0
  
14 i4·:·time·parameterCount(X,Verbose=>true)14 i4·:·time·parameterCount(X,Verbose=>true)
15 ·--·used·2.83519s·(cpu);·2.13399s·(thread);·0s·(gc)15 ·--·used·3.05616s·(cpu);·2.17337s·(thread);·0s·(gc)
16 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)16 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)
17 X:·GM·fourfold·containing·S17 X:·GM·fourfold·containing·S
18 Y:·del·Pezzo·fivefold·containing·X18 Y:·del·Pezzo·fivefold·containing·X
19 h^1(N_{S,Y})·=·019 h^1(N_{S,Y})·=·0
20 h^0(N_{S,Y})·=·1120 h^0(N_{S,Y})·=·11
21 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));21 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));
22 in·particular,·h^0(I_{S,Y}(2))·is·minimal22 in·particular,·h^0(I_{S,Y}(2))·is·minimal
⊟
692 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out
    
Offset 6, 15 lines modifiedOffset 6, 15 lines modified
  
6 i3·:·?·X6 i3·:·?·X
  
7 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees7 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees
8 ·····1^2·2^58 ·····1^2·2^5
  
9 i4·:·time·parametrizeFanoFourfold·X9 i4·:·time·parametrizeFanoFourfold·X
10 ·--·used·2.02256s·(cpu);·0.681808s·(thread);·0s·(gc)10 ·--·used·2.01818s·(cpu);·0.711885s·(thread);·0s·(gc)
  
11 o4·=·multi-rational·map·consisting·of·one·single·rational·map11 o4·=·multi-rational·map·consisting·of·one·single·rational·map
12 ·····source·variety:·PP^412 ·····source·variety:·PP^4
13 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees·1^2·2^5·13 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees·1^2·2^5·
14 ·····dominance:·true14 ·····dominance:·true
15 ·····degree:·115 ·····degree:·1
  
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1.45 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Cubic__Fourfold.out
    
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7 o3·:·ProjectiveVariety,·surface·in·PP^57 o3·:·ProjectiveVariety,·surface·in·PP^5
  
8 i4·:·X·=·projectiveVariety·ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);8 i4·:·X·=·projectiveVariety·ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
  
9 o4·:·ProjectiveVariety,·hypersurface·in·PP^59 o4·:·ProjectiveVariety,·hypersurface·in·PP^5
  
10 i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);10 i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);
11 ·--·used·0.00682864s·(cpu);·0.00694529s·(thread);·0s·(gc)11 ·--·used·0.009766s·(cpu);·0.00928852s·(thread);·0s·(gc)
  
12 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and·sectional·genus·012 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and·sectional·genus·0
  
13 i6·:·time·describe·F13 i6·:·time·describe·F
14 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it14 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
15 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·986815 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
16 ·--·used·0.395379s·(cpu);·0.168723s·(thread);·0s·(gc)16 ·--·used·0.728678s·(cpu);·0.24257s·(thread);·0s·(gc)
  
17 o6·=·Special·cubic·fourfold·of·discriminant·2617 o6·=·Special·cubic·fourfold·of·discriminant·26
18 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·018 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
19 ·····cut·out·by·13·hypersurfaces·of·degree·319 ·····cut·out·by·13·hypersurfaces·of·degree·3
  
20 i7·:·assert(F·==·X)20 i7·:·assert(F·==·X)
  
⊟
1.62 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Gushel__Mukai__Fourfold.out
    
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7 o3·:·ProjectiveVariety,·surface·in·PP^87 o3·:·ProjectiveVariety,·surface·in·PP^8
  
8 i4·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);8 i4·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
  
9 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^89 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^8
  
10 i5·:·time·F·=·specialGushelMukaiFourfold(S,X);10 i5·:·time·F·=·specialGushelMukaiFourfold(S,X);
11 ·--·used·1.39467s·(cpu);·1.14945s·(thread);·0s·(gc)11 ·--·used·1.43482s·(cpu);·1.16197s·(thread);·0s·(gc)
  
12 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·012 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0
  
13 i6·:·time·describe·F13 i6·:·time·describe·F
14 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it14 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
15 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·986815 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
16 ·--·used·6.60891s·(cpu);·2.64522s·(thread);·0s·(gc)16 ·--·used·7.67224s·(cpu);·2.94689s·(thread);·0s·(gc)
  
17 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')17 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')
18 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·018 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0
19 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)19 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
20 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)20 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
21 ·····Type:·ordinary21 ·····Type:·ordinary
22 ·····(case·1·of·Table·1·in·arXiv:2002.07026)22 ·····(case·1·of·Table·1·in·arXiv:2002.07026)
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1.34 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
    
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5 i2·:·X·=·specialGushelMukaiFourfold(ideal(x_6-x_7,·x_5,·x_3-x_4,·x_1,·x_0-x_4,·x_2*x_7-x_4*x_8),·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));5 i2·:·X·=·specialGushelMukaiFourfold(ideal(x_6-x_7,·x_5,·x_3-x_4,·x_1,·x_0-x_4,·x_2*x_7-x_4*x_8),·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
  
6 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·06 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0
  
7 i3·:·time·toGrass·X7 i3·:·time·toGrass·X
8 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it8 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
9 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·98689 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
10 ·--·used·6.1112s·(cpu);·2.35774s·(thread);·0s·(gc)10 ·--·used·6.86175s·(cpu);·2.49811s·(thread);·0s·(gc)
  
11 o3·=·multi-rational·map·consisting·of·one·single·rational·map11 o3·=·multi-rational·map·consisting·of·one·single·rational·map
12 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces·of·degree·212 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces·of·degree·2
13 ·····target·variety:·GG(1,4)·⊂·PP^913 ·····target·variety:·GG(1,4)·⊂·PP^9
  
14 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))14 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))
  
⊟
1.06 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass_lp__Embedded__Projective__Variety_rp.out
    
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5 i2·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);5 i2·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8);
  
6 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^86 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^8
  
7 i3·:·time·toGrass·X7 i3·:·time·toGrass·X
8 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it8 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
9 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·98689 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
10 ·--·used·6.40265s·(cpu);·2.47054s·(thread);·0s·(gc)10 ·--·used·6.61212s·(cpu);·2.57624s·(thread);·0s·(gc)
  
11 o3·=·multi-rational·map·consisting·of·one·single·rational·map11 o3·=·multi-rational·map·consisting·of·one·single·rational·map
12 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5·hypersurfaces·of·degree·212 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5·hypersurfaces·of·degree·2
13 ·····target·variety:·GG(1,4)·⊂·PP^913 ·····target·variety:·GG(1,4)·⊂·PP^9
  
14 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))14 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))
  
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633 B
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
    
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5 o2·:·ProjectiveVariety,·surface·in·PP^55 o2·:·ProjectiveVariety,·surface·in·PP^5
  
6 i3·:·X·=·specialCubicFourfold·S;6 i3·:·X·=·specialCubicFourfold·S;
  
7 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·07 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0
  
8 i4·:·time·f·=·unirationalParametrization·X;8 i4·:·time·f·=·unirationalParametrization·X;
9 ·--·used·0.847054s·(cpu);·0.3949s·(thread);·0s·(gc)9 ·--·used·1.12796s·(cpu);·0.429726s·(thread);·0s·(gc)
  
10 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)10 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)
  
11 i5·:·degreeSequence·f11 i5·:·degreeSequence·f
  
12 o5·=·{[10]}12 o5·=·{[10]}
  
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1.6 KB
./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
    
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110 ·····························|·1·4·|110 ·····························|·1·4·|
111 ·····containing·a·surface·of·degree·1·and·sectional·genus·0111 ·····containing·a·surface·of·degree·1·and·sectional·genus·0
112 ·····cut·out·by·5·hypersurfaces·of·degree·1112 ·····cut·out·by·5·hypersurfaces·of·degree·1
113 ·····(This·is·a·classical·example·of·rational·fourfold)</code></pre>113 ·····(This·is·a·classical·example·of·rational·fourfold)</code></pre>
114 </td>··········</tr>114 </td>··········</tr>
115 ··········<tr>115 ··········<tr>
116 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedCastelnuovoSurface·X;116 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedCastelnuovoSurface·X;
117 ·--·used·3.59693s·(cpu);·1.06332s·(thread);·0s·(gc)117 ·--·used·4.40484s·(cpu);·1.04889s·(thread);·0s·(gc)
  
118 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X</code></pre>118 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X</code></pre>
119 </td>··········</tr>119 </td>··········</tr>
120 ··········<tr>120 ··········<tr>
121 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>121 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>
122 </td>··········</tr>122 </td>··········</tr>
123 ··········<tr>123 ··········<tr>
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726 B
html2text {}
    
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42 o2·=·Complete·intersection·of·3·quadrics·in·PP^742 o2·=·Complete·intersection·of·3·quadrics·in·PP^7
43 ·····of·discriminant·31·=·det|·8·1·|43 ·····of·discriminant·31·=·det|·8·1·|
44 ·····························|·1·4·|44 ·····························|·1·4·|
45 ·····containing·a·surface·of·degree·1·and·sectional·genus·045 ·····containing·a·surface·of·degree·1·and·sectional·genus·0
46 ·····cut·out·by·5·hypersurfaces·of·degree·146 ·····cut·out·by·5·hypersurfaces·of·degree·1
47 ·····(This·is·a·classical·example·of·rational·fourfold)47 ·····(This·is·a·classical·example·of·rational·fourfold)
48 i3·:·time·U'·=·associatedCastelnuovoSurface·X;48 i3·:·time·U'·=·associatedCastelnuovoSurface·X;
49 ·--·used·3.59693s·(cpu);·1.06332s·(thread);·0s·(gc)49 ·--·used·4.40484s·(cpu);·1.04889s·(thread);·0s·(gc)
  
50 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X50 o3·:·ProjectiveVariety,·Castelnuovo·surface·associated·to·X
51 i4·:·(mu,U,C,f)·=·building·U';51 i4·:·(mu,U,C,f)·=·building·U';
52 i5·:·?·mu52 i5·:·?·mu
  
53 o5·=·multi-rational·map·consisting·of·one·single·rational·map53 o5·=·multi-rational·map·consisting·of·one·single·rational·map
54 ·····source·variety:·5-dimensional·subvariety·of·PP^7·cut·out·by·254 ·····source·variety:·5-dimensional·subvariety·of·PP^7·cut·out·by·2
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109 o2·=·Special·cubic·fourfold·of·discriminant·14109 o2·=·Special·cubic·fourfold·of·discriminant·14
110 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·0110 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·0
111 ·····cut·out·by·6·hypersurfaces·of·degree·2</code></pre>111 ·····cut·out·by·6·hypersurfaces·of·degree·2</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedK3surface·X;114 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedK3surface·X;
115 ·--·used·3.57308s·(cpu);·1.10736s·(thread);·0s·(gc)115 ·--·used·3.99832s·(cpu);·1.12526s·(thread);·0s·(gc)
  
116 o3·:·ProjectiveVariety,·K3·surface·associated·to·X</code></pre>116 o3·:·ProjectiveVariety,·K3·surface·associated·to·X</code></pre>
117 </td>··········</tr>117 </td>··········</tr>
118 ··········<tr>118 ··········<tr>
119 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>119 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>
120 </td>··········</tr>120 </td>··········</tr>
121 ··········<tr>121 ··········<tr>
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42 sectional·genus·042 sectional·genus·0
43 i2·:·describe·X43 i2·:·describe·X
  
44 o2·=·Special·cubic·fourfold·of·discriminant·1444 o2·=·Special·cubic·fourfold·of·discriminant·14
45 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·045 ·····containing·a·(smooth)·surface·of·degree·4·and·sectional·genus·0
46 ·····cut·out·by·6·hypersurfaces·of·degree·246 ·····cut·out·by·6·hypersurfaces·of·degree·2
47 i3·:·time·U'·=·associatedK3surface·X;47 i3·:·time·U'·=·associatedK3surface·X;
48 ·--·used·3.57308s·(cpu);·1.10736s·(thread);·0s·(gc)48 ·--·used·3.99832s·(cpu);·1.12526s·(thread);·0s·(gc)
  
49 o3·:·ProjectiveVariety,·K3·surface·associated·to·X49 o3·:·ProjectiveVariety,·K3·surface·associated·to·X
50 i4·:·(mu,U,C,f)·=·building·U';50 i4·:·(mu,U,C,f)·=·building·U';
51 i5·:·?·mu51 i5·:·?·mu
  
52 o5·=·multi-rational·map·consisting·of·one·single·rational·map52 o5·=·multi-rational·map·consisting·of·one·single·rational·map
53 ·····source·variety:·PP^553 ·····source·variety:·PP^5
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112 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)112 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
113 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)113 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
114 ·····Type:·ordinary114 ·····Type:·ordinary
115 ·····(case·1·of·Table·1·in·arXiv:2002.07026)</code></pre>115 ·····(case·1·of·Table·1·in·arXiv:2002.07026)</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedK3surface·X;118 <td>··············<pre><code·class="language-macaulay2">i3·:·time·U'·=·associatedK3surface·X;
119 ·--·used·7.9956s·(cpu);·3.7328s·(thread);·0s·(gc)119 ·--·used·10.0322s·(cpu);·4.13645s·(thread);·0s·(gc)
  
120 o3·:·ProjectiveVariety,·K3·surface·associated·to·X</code></pre>120 o3·:·ProjectiveVariety,·K3·surface·associated·to·X</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>123 <td>··············<pre><code·class="language-macaulay2">i4·:·(mu,U,C,f)·=·building·U';</code></pre>
124 </td>··········</tr>124 </td>··········</tr>
125 ··········<tr>125 ··········<tr>
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44 o2·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')44 o2·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')
45 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·045 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0
46 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)46 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
47 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)47 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
48 ·····Type:·ordinary48 ·····Type:·ordinary
49 ·····(case·1·of·Table·1·in·arXiv:2002.07026)49 ·····(case·1·of·Table·1·in·arXiv:2002.07026)
50 i3·:·time·U'·=·associatedK3surface·X;50 i3·:·time·U'·=·associatedK3surface·X;
51 ·--·used·7.9956s·(cpu);·3.7328s·(thread);·0s·(gc)51 ·--·used·10.0322s·(cpu);·4.13645s·(thread);·0s·(gc)
  
52 o3·:·ProjectiveVariety,·K3·surface·associated·to·X52 o3·:·ProjectiveVariety,·K3·surface·associated·to·X
53 i4·:·(mu,U,C,f)·=·building·U';53 i4·:·(mu,U,C,f)·=·building·U';
54 i5·:·?·mu54 i5·:·?·mu
  
55 o5·=·multi-rational·map·consisting·of·one·single·rational·map55 o5·=·multi-rational·map·consisting·of·one·single·rational·map
56 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·556 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5
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90 o2·=·Special·cubic·fourfold·of·discriminant·2690 o2·=·Special·cubic·fourfold·of·discriminant·26
91 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·091 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
92 ·····cut·out·by·13·hypersurfaces·of·degree·3</code></pre>92 ·····cut·out·by·13·hypersurfaces·of·degree·3</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·detectCongruence(X,Verbose=>true);95 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·detectCongruence(X,Verbose=>true);
96 ·--·used·2.96759s·(cpu);·1.73627s·(thread);·0s·(gc)96 ·--·used·3.66454s·(cpu);·1.59606s·(thread);·0s·(gc)
97 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a·general·point:·897 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a·general·point:·8
98 number·2-secant·lines·=·798 number·2-secant·lines·=·7
99 number·5-secant·conics·=·199 number·5-secant·conics·=·1
  
100 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^5</code></pre>100 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^5</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
Offset 106, 15 lines modifiedOffset 106, 15 lines modified
  
106 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]106 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]
  
107 o4·:·ProjectiveVariety,·a·point·in·PP^5</code></pre>107 o4·:·ProjectiveVariety,·a·point·in·PP^5</code></pre>
108 </td>··········</tr>108 </td>··········</tr>
109 ··········<tr>109 ··········<tr>
110 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface110 <td>··············<pre><code·class="language-macaulay2">i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface
111 ·--·used·0.372948s·(cpu);·0.237981s·(thread);·0s·(gc)111 ·--·used·0.386309s·(cpu);·0.226969s·(thread);·0s·(gc)
  
112 o5·:·ProjectiveVariety,·curve·in·PP^5</code></pre>112 o5·:·ProjectiveVariety,·curve·in·PP^5</code></pre>
113 </td>··········</tr>113 </td>··········</tr>
114 ··········<tr>114 ··········<tr>
115 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree(C·*·surface·X)·==·5·and·isSubset(p,·C))</code></pre>115 <td>··············<pre><code·class="language-macaulay2">i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree(C·*·surface·X)·==·5·and·isSubset(p,·C))</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ········</table>117 ········</table>
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31 sectional·genus·031 sectional·genus·0
32 i2·:·describe·X32 i2·:·describe·X
  
33 o2·=·Special·cubic·fourfold·of·discriminant·2633 o2·=·Special·cubic·fourfold·of·discriminant·26
34 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·034 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
35 ·····cut·out·by·13·hypersurfaces·of·degree·335 ·····cut·out·by·13·hypersurfaces·of·degree·3
36 i3·:·time·f·=·detectCongruence(X,Verbose=>true);36 i3·:·time·f·=·detectCongruence(X,Verbose=>true);
37 ·--·used·2.96759s·(cpu);·1.73627s·(thread);·0s·(gc)37 ·--·used·3.66454s·(cpu);·1.59606s·(thread);·0s·(gc)
38 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a38 number·lines·contained·in·the·image·of·the·cubic·map·and·passing·through·a
39 general·point:·839 general·point:·8
40 number·2-secant·lines·=·740 number·2-secant·lines·=·7
41 number·5-secant·conics·=·141 number·5-secant·conics·=·1
  
42 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^542 o3·:·Congruence·of·5-secant·conics·to·surface·in·PP^5
43 i4·:·p·:=·point·ambient·X·--·random·point·on·P^543 i4·:·p·:=·point·ambient·X·--·random·point·on·P^5
  
44 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]44 o4·=·point·of·coordinates·[15092,·-9738,·-3620,·-15181,·12688,·1]
  
45 o4·:·ProjectiveVariety,·a·point·in·PP^545 o4·:·ProjectiveVariety,·a·point·in·PP^5
46 i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface46 i5·:·time·C·=·f·p;·--·5-secant·conic·to·the·surface
47 ·--·used·0.372948s·(cpu);·0.237981s·(thread);·0s·(gc)47 ·--·used·0.386309s·(cpu);·0.226969s·(thread);·0s·(gc)
  
48 o5·:·ProjectiveVariety,·curve·in·PP^548 o5·:·ProjectiveVariety,·curve·in·PP^5
49 i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree49 i6·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C·*·surface·X)·==·0·and·degree
50 (C·*·surface·X)·==·5·and·isSubset(p,·C))50 (C·*·surface·X)·==·5·and·isSubset(p,·C))
51 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*51 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
52 ····*·_\x8d_\x8e_\x8t_\x8e_\x8c_\x8t_\x8C_\x8o_\x8n_\x8g_\x8r_\x8u_\x8e_\x8n_\x8c_\x8e_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8,_\x8Z_\x8Z_\x8)·--·detect·and·return·a52 ····*·_\x8d_\x8e_\x8t_\x8e_\x8c_\x8t_\x8C_\x8o_\x8n_\x8g_\x8r_\x8u_\x8e_\x8n_\x8c_\x8e_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8,_\x8Z_\x8Z_\x8)·--·detect·and·return·a
53 ······congruence·of·(2e-1)-secant·curves·of·degree·e·inside·a·del·Pezzo53 ······congruence·of·(2e-1)-secant·curves·of·degree·e·inside·a·del·Pezzo
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93 ·····cut·out·by·19·hypersurfaces·of·degree·293 ·····cut·out·by·19·hypersurfaces·of·degree·2
94 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)94 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)
95 ·····Type:·ordinary95 ·····Type:·ordinary
96 ·····(case·17·of·Table·1·in·arXiv:2002.07026)</code></pre>96 ·····(case·17·of·Table·1·in·arXiv:2002.07026)</code></pre>
97 </td>··········</tr>97 </td>··········</tr>
98 ··········<tr>98 ··········<tr>
99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·detectCongruence(X,Verbose=>true);99 <td>··············<pre><code·class="language-macaulay2">i3·:·time·f·=·detectCongruence(X,Verbose=>true);
100 ·--·used·11.0735s·(cpu);·5.31108s·(thread);·0s·(gc)100 ·--·used·14.1479s·(cpu);·5.55996s·(thread);·0s·(gc)
101 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a·general·point:·7101 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a·general·point:·7
102 number·1-secant·lines·=·6102 number·1-secant·lines·=·6
103 number·3-secant·conics·=·1103 number·3-secant·conics·=·1
  
104 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^8</code></pre>104 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^8</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ··········<tr>106 ··········<tr>
Offset 114, 15 lines modifiedOffset 114, 15 lines modified
  
114 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,·13402,·1]114 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,·13402,·1]
  
115 o5·:·ProjectiveVariety,·a·point·in·PP^8</code></pre>115 o5·:·ProjectiveVariety,·a·point·in·PP^8</code></pre>
116 </td>··········</tr>116 </td>··········</tr>
117 ··········<tr>117 ··········<tr>
118 <td>··············<pre><code·class="language-macaulay2">i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface118 <td>··············<pre><code·class="language-macaulay2">i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface
119 ·--·used·0.294189s·(cpu);·0.190741s·(thread);·0s·(gc)119 ·--·used·0.562943s·(cpu);·0.214401s·(thread);·0s·(gc)
  
120 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)</code></pre>120 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ··········<tr>122 ··········<tr>
123 <td>··············<pre><code·class="language-macaulay2">i7·:·S·=·surface·X;123 <td>··············<pre><code·class="language-macaulay2">i7·:·S·=·surface·X;
  
124 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)</code></pre>124 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)</code></pre>
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37 o2·=·Special·Gushel-Mukai·fourfold·of·discriminant·2037 o2·=·Special·Gushel-Mukai·fourfold·of·discriminant·20
38 ·····containing·a·surface·in·PP^8·of·degree·9·and·sectional·genus·238 ·····containing·a·surface·in·PP^8·of·degree·9·and·sectional·genus·2
39 ·····cut·out·by·19·hypersurfaces·of·degree·239 ·····cut·out·by·19·hypersurfaces·of·degree·2
40 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)40 ·····and·with·class·in·G(1,4)·given·by·6*s_(3,1)+3*s_(2,2)
41 ·····Type:·ordinary41 ·····Type:·ordinary
42 ·····(case·17·of·Table·1·in·arXiv:2002.07026)42 ·····(case·17·of·Table·1·in·arXiv:2002.07026)
43 i3·:·time·f·=·detectCongruence(X,Verbose=>true);43 i3·:·time·f·=·detectCongruence(X,Verbose=>true);
44 ·--·used·11.0735s·(cpu);·5.31108s·(thread);·0s·(gc)44 ·--·used·14.1479s·(cpu);·5.55996s·(thread);·0s·(gc)
45 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a45 number·lines·contained·in·the·image·of·the·quadratic·map·and·passing·through·a
46 general·point:·746 general·point:·7
47 number·1-secant·lines·=·647 number·1-secant·lines·=·6
48 number·3-secant·conics·=·148 number·3-secant·conics·=·1
  
49 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^849 o3·:·Congruence·of·3-secant·conics·to·surface·in·a·fivefold·in·PP^8
50 i4·:·Y·=·ambientFivefold·X;·--·del·Pezzo·fivefold·containing·X50 i4·:·Y·=·ambientFivefold·X;·--·del·Pezzo·fivefold·containing·X
Offset 54, 15 lines modifiedOffset 54, 15 lines modified
54 i5·:·p·:=·point·Y·--·random·point·on·Y54 i5·:·p·:=·point·Y·--·random·point·on·Y
  
55 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,55 o5·=·point·of·coordinates·[7214,·-1460,·7057,·-2440,·15907,·-14345,·-5937,
56 13402,·1]56 13402,·1]
  
57 o5·:·ProjectiveVariety,·a·point·in·PP^857 o5·:·ProjectiveVariety,·a·point·in·PP^8
58 i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface58 i6·:·time·C·=·f·p;·--·3-secant·conic·to·the·surface
59 ·--·used·0.294189s·(cpu);·0.190741s·(thread);·0s·(gc)59 ·--·used·0.562943s·(cpu);·0.214401s·(thread);·0s·(gc)
  
60 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)60 o6·:·ProjectiveVariety,·curve·in·PP^8·(subvariety·of·codimension·4·in·Y)
61 i7·:·S·=·surface·X;61 i7·:·S·=·surface·X;
  
62 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)62 o7·:·ProjectiveVariety,·surface·in·PP^8·(subvariety·of·codimension·3·in·Y)
63 i8·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C*S)·==·0·and·degree(C*S)·==·363 i8·:·assert(dim·C·==·1·and·degree·C·==·2·and·dim(C*S)·==·0·and·degree(C*S)·==·3
64 and·isSubset(p,C)·and·isSubset(C,Y))64 and·isSubset(p,C)·and·isSubset(C,Y))
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80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i1·:·X·=·specialCubicFourfold·&quot;quintic·del·Pezzo·surface&quot;;81 <td>··············<pre><code·class="language-macaulay2">i1·:·X·=·specialCubicFourfold·&quot;quintic·del·Pezzo·surface&quot;;
  
82 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and·sectional·genus·1</code></pre>82 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and·sectional·genus·1</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·time·discriminant·X85 <td>··············<pre><code·class="language-macaulay2">i2·:·time·discriminant·X
86 ·--·used·0.146937s·(cpu);·0.0730333s·(thread);·0s·(gc)86 ·--·used·0.17626s·(cpu);·0.0850651s·(thread);·0s·(gc)
  
87 o2·=·14</code></pre>87 o2·=·14</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
90 ······</div>90 ······</div>
91 ······<div>91 ······<div>
92 ········<h2>See·also</h2>92 ········<h2>See·also</h2>
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21 thanks·to·the·functions·_\x8E_\x8u_\x8l_\x8e_\x8r_\x8C_\x8h_\x8a_\x8r_\x8a_\x8c_\x8t_\x8e_\x8r_\x8i_\x8s_\x8t_\x8i_\x8c·and·_\x8E_\x8u_\x8l_\x8e_\x8r·(the·option·Algorithm21 thanks·to·the·functions·_\x8E_\x8u_\x8l_\x8e_\x8r_\x8C_\x8h_\x8a_\x8r_\x8a_\x8c_\x8t_\x8e_\x8r_\x8i_\x8s_\x8t_\x8i_\x8c·and·_\x8E_\x8u_\x8l_\x8e_\x8r·(the·option·Algorithm
22 allows·you·to·select·the·method).22 allows·you·to·select·the·method).
23 i1·:·X·=·specialCubicFourfold·"quintic·del·Pezzo·surface";23 i1·:·X·=·specialCubicFourfold·"quintic·del·Pezzo·surface";
  
24 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and24 o1·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·5·and
25 sectional·genus·125 sectional·genus·1
26 i2·:·time·discriminant·X26 i2·:·time·discriminant·X
27 ·--·used·0.146937s·(cpu);·0.0730333s·(thread);·0s·(gc)27 ·--·used·0.17626s·(cpu);·0.0850651s·(thread);·0s·(gc)
  
28 o2·=·1428 o2·=·14
29 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
30 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special30 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special
31 ······Gushel-Mukai·fourfold31 ······Gushel-Mukai·fourfold
32 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*32 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
33 ····*·discriminant(HodgeSpecialFourfold)33 ····*·discriminant(HodgeSpecialFourfold)
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80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i1·:·X·=·specialGushelMukaiFourfold·&quot;tau-quadric&quot;;81 <td>··············<pre><code·class="language-macaulay2">i1·:·X·=·specialGushelMukaiFourfold·&quot;tau-quadric&quot;;
  
82 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>82 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ··········<tr>84 ··········<tr>
85 <td>··············<pre><code·class="language-macaulay2">i2·:·time·discriminant·X85 <td>··············<pre><code·class="language-macaulay2">i2·:·time·discriminant·X
86 ·--·used·0.891636s·(cpu);·0.354162s·(thread);·0s·(gc)86 ·--·used·1.26496s·(cpu);·0.367537s·(thread);·0s·(gc)
  
87 o2·=·10</code></pre>87 o2·=·10</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ········</table>89 ········</table>
90 ······</div>90 ······</div>
91 ······<div>91 ······<div>
92 ········<h2>See·also</h2>92 ········<h2>See·also</h2>
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21 the·functions·_\x8c_\x8y_\x8c_\x8l_\x8e_\x8C_\x8l_\x8a_\x8s_\x8s,·_\x8E_\x8u_\x8l_\x8e_\x8r_\x8C_\x8h_\x8a_\x8r_\x8a_\x8c_\x8t_\x8e_\x8r_\x8i_\x8s_\x8t_\x8i_\x8c·and·_\x8E_\x8u_\x8l_\x8e_\x8r·(the·option·Algorithm21 the·functions·_\x8c_\x8y_\x8c_\x8l_\x8e_\x8C_\x8l_\x8a_\x8s_\x8s,·_\x8E_\x8u_\x8l_\x8e_\x8r_\x8C_\x8h_\x8a_\x8r_\x8a_\x8c_\x8t_\x8e_\x8r_\x8i_\x8s_\x8t_\x8i_\x8c·and·_\x8E_\x8u_\x8l_\x8e_\x8r·(the·option·Algorithm
22 allows·you·to·select·the·method).22 allows·you·to·select·the·method).
23 i1·:·X·=·specialGushelMukaiFourfold·"tau-quadric";23 i1·:·X·=·specialGushelMukaiFourfold·"tau-quadric";
  
24 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and24 o1·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and
25 sectional·genus·025 sectional·genus·0
26 i2·:·time·discriminant·X26 i2·:·time·discriminant·X
27 ·--·used·0.891636s·(cpu);·0.354162s·(thread);·0s·(gc)27 ·--·used·1.26496s·(cpu);·0.367537s·(thread);·0s·(gc)
  
28 o2·=·1028 o2·=·10
29 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*29 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
30 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8C_\x8u_\x8b_\x8i_\x8c_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special·cubic30 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8C_\x8u_\x8b_\x8i_\x8c_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special·cubic
31 ······fourfold31 ······fourfold
32 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*32 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
33 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special33 ····*·_\x8d_\x8i_\x8s_\x8c_\x8r_\x8i_\x8m_\x8i_\x8n_\x8a_\x8n_\x8t_\x8(_\x8S_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8G_\x8u_\x8s_\x8h_\x8e_\x8l_\x8M_\x8u_\x8k_\x8a_\x8i_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8)·--·discriminant·of·a·special
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86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·random({{2},{2},{2}},S);87 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·random({{2},{2},{2}},S);
  
88 o3·:·ProjectiveVariety,·surface·in·PP^5</code></pre>88 o3·:·ProjectiveVariety,·surface·in·PP^5</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(S,X,Verbose=>true)91 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(S,X,Verbose=>true)
92 ·--·used·0.220293s·(cpu);·0.169981s·(thread);·0s·(gc)92 ·--·used·0.220138s·(cpu);·0.159861s·(thread);·0s·(gc)
93 S:·rational·normal·curve·of·degree·5·in·PP^593 S:·rational·normal·curve·of·degree·5·in·PP^5
94 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·3·hypersurfaces·of·degree·294 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·3·hypersurfaces·of·degree·2
95 (assumption:·h^1(N_{S,P^5})·=·0)95 (assumption:·h^1(N_{S,P^5})·=·0)
96 h^0(N_{S,P^5})·=·3296 h^0(N_{S,P^5})·=·32
97 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));97 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));
98 in·particular,·h^0(I_{S,P^5}(2))·is·minimal98 in·particular,·h^0(I_{S,P^5}(2))·is·minimal
99 dim·GG(2,9)·=·2199 dim·GG(2,9)·=·21
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24 i1·:·K·=·ZZ/33331;·S·=·PP_K^(1,5);24 i1·:·K·=·ZZ/33331;·S·=·PP_K^(1,5);
  
25 o2·:·ProjectiveVariety,·curve·in·PP^525 o2·:·ProjectiveVariety,·curve·in·PP^5
26 i3·:·X·=·random({{2},{2},{2}},S);26 i3·:·X·=·random({{2},{2},{2}},S);
  
27 o3·:·ProjectiveVariety,·surface·in·PP^527 o3·:·ProjectiveVariety,·surface·in·PP^5
28 i4·:·time·parameterCount(S,X,Verbose=>true)28 i4·:·time·parameterCount(S,X,Verbose=>true)
29 ·--·used·0.220293s·(cpu);·0.169981s·(thread);·0s·(gc)29 ·--·used·0.220138s·(cpu);·0.159861s·(thread);·0s·(gc)
30 S:·rational·normal·curve·of·degree·5·in·PP^530 S:·rational·normal·curve·of·degree·5·in·PP^5
31 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·331 X:·smooth·surface·of·degree·8·and·sectional·genus·5·in·PP^5·cut·out·by·3
32 hypersurfaces·of·degree·232 hypersurfaces·of·degree·2
33 (assumption:·h^1(N_{S,P^5})·=·0)33 (assumption:·h^1(N_{S,P^5})·=·0)
34 h^0(N_{S,P^5})·=·3234 h^0(N_{S,P^5})·=·32
35 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));35 h^1(O_S(2))·=·0,·and·h^0(I_{S,P^5}(2))·=·10·=·h^0(O_(P^5)(2))·-·\chi(O_S(2));
36 in·particular,·h^0(I_{S,P^5}(2))·is·minimal36 in·particular,·h^0(I_{S,P^5}(2))·is·minimal
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88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialCubicFourfold·V;89 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialCubicFourfold·V;
  
90 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0</code></pre>90 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0</code></pre>
91 </td>··········</tr>91 </td>··········</tr>
92 ··········<tr>92 ··········<tr>
93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(X,Verbose=>true)93 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(X,Verbose=>true)
94 ·--·used·0.366041s·(cpu);·0.251843s·(thread);·0s·(gc)94 ·--·used·0.426191s·(cpu);·0.293512s·(thread);·0s·(gc)
95 S:·Veronese·surface·in·PP^595 S:·Veronese·surface·in·PP^5
96 X:·smooth·cubic·hypersurface·in·PP^596 X:·smooth·cubic·hypersurface·in·PP^5
97 (assumption:·h^1(N_{S,P^5})·=·0)97 (assumption:·h^1(N_{S,P^5})·=·0)
98 h^0(N_{S,P^5})·=·2798 h^0(N_{S,P^5})·=·27
99 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));99 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));
100 in·particular,·h^0(I_{S,P^5}(3))·is·minimal100 in·particular,·h^0(I_{S,P^5}(3))·is·minimal
101 h^0(N_{S,P^5})·+·27·=·54101 h^0(N_{S,P^5})·+·27·=·54
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34 o2·:·ProjectiveVariety,·surface·in·PP^534 o2·:·ProjectiveVariety,·surface·in·PP^5
35 i3·:·X·=·specialCubicFourfold·V;35 i3·:·X·=·specialCubicFourfold·V;
  
36 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and36 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and
37 sectional·genus·037 sectional·genus·0
38 i4·:·time·parameterCount(X,Verbose=>true)38 i4·:·time·parameterCount(X,Verbose=>true)
39 ·--·used·0.366041s·(cpu);·0.251843s·(thread);·0s·(gc)39 ·--·used·0.426191s·(cpu);·0.293512s·(thread);·0s·(gc)
40 S:·Veronese·surface·in·PP^540 S:·Veronese·surface·in·PP^5
41 X:·smooth·cubic·hypersurface·in·PP^541 X:·smooth·cubic·hypersurface·in·PP^5
42 (assumption:·h^1(N_{S,P^5})·=·0)42 (assumption:·h^1(N_{S,P^5})·=·0)
43 h^0(N_{S,P^5})·=·2743 h^0(N_{S,P^5})·=·27
44 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));44 h^1(O_S(3))·=·0,·and·h^0(I_{S,P^5}(3))·=·28·=·h^0(O_(P^5)(3))·-·\chi(O_S(3));
45 in·particular,·h^0(I_{S,P^5}(3))·is·minimal45 in·particular,·h^0(I_{S,P^5}(3))·is·minimal
46 h^0(N_{S,P^5})·+·27·=·5446 h^0(N_{S,P^5})·+·27·=·54
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95 ··········<tr>95 ··········<tr>
96 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialGushelMukaiFourfold·S;96 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialGushelMukaiFourfold·S;
  
97 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and·sectional·genus·0</code></pre>97 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and·sectional·genus·0</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(X,Verbose=>true)100 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parameterCount(X,Verbose=>true)
101 ·--·used·2.83519s·(cpu);·2.13399s·(thread);·0s·(gc)101 ·--·used·3.05616s·(cpu);·2.17337s·(thread);·0s·(gc)
102 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)102 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)
103 X:·GM·fourfold·containing·S103 X:·GM·fourfold·containing·S
104 Y:·del·Pezzo·fivefold·containing·X104 Y:·del·Pezzo·fivefold·containing·X
105 h^1(N_{S,Y})·=·0105 h^1(N_{S,Y})·=·0
106 h^0(N_{S,Y})·=·11106 h^0(N_{S,Y})·=·11
107 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));107 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));
108 in·particular,·h^0(I_{S,Y}(2))·is·minimal108 in·particular,·h^0(I_{S,Y}(2))·is·minimal
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36 o2·:·ProjectiveVariety,·surface·in·PP^9·(subvariety·of·codimension·4·in·G)36 o2·:·ProjectiveVariety,·surface·in·PP^9·(subvariety·of·codimension·4·in·G)
37 i3·:·X·=·specialGushelMukaiFourfold·S;37 i3·:·X·=·specialGushelMukaiFourfold·S;
  
38 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and38 o3·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·3·and
39 sectional·genus·039 sectional·genus·0
40 i4·:·time·parameterCount(X,Verbose=>true)40 i4·:·time·parameterCount(X,Verbose=>true)
41 ·--·used·2.83519s·(cpu);·2.13399s·(thread);·0s·(gc)41 ·--·used·3.05616s·(cpu);·2.17337s·(thread);·0s·(gc)
42 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)42 S:·cubic·surface·in·PP^8·cut·out·by·7·hypersurfaces·of·degrees·(1,1,1,1,2,2,2)
43 X:·GM·fourfold·containing·S43 X:·GM·fourfold·containing·S
44 Y:·del·Pezzo·fivefold·containing·X44 Y:·del·Pezzo·fivefold·containing·X
45 h^1(N_{S,Y})·=·045 h^1(N_{S,Y})·=·0
46 h^0(N_{S,Y})·=·1146 h^0(N_{S,Y})·=·11
47 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));47 h^1(O_S(2))·=·0,·and·h^0(I_{S,Y}(2))·=·28·=·h^0(O_Y(2))·-·\chi(O_S(2));
48 in·particular,·h^0(I_{S,Y}(2))·is·minimal48 in·particular,·h^0(I_{S,Y}(2))·is·minimal
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85 <td>··············<pre><code·class="language-macaulay2">i3·:·?·X85 <td>··············<pre><code·class="language-macaulay2">i3·:·?·X
  
86 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees86 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees
87 ·····1^2·2^5</code></pre>87 ·····1^2·2^5</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parametrizeFanoFourfold·X90 <td>··············<pre><code·class="language-macaulay2">i4·:·time·parametrizeFanoFourfold·X
91 ·--·used·2.02256s·(cpu);·0.681808s·(thread);·0s·(gc)91 ·--·used·2.01818s·(cpu);·0.711885s·(thread);·0s·(gc)
  
92 o4·=·multi-rational·map·consisting·of·one·single·rational·map92 o4·=·multi-rational·map·consisting·of·one·single·rational·map
93 ·····source·variety:·PP^493 ·····source·variety:·PP^4
94 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees·1^2·2^5·94 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees·1^2·2^5·
95 ·····dominance:·true95 ·····dominance:·true
96 ·····degree:·196 ·····degree:·1
  
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30 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^930 o2·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^9
31 i3·:·?·X31 i3·:·?·X
  
32 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees32 o3·=·4-dimensional·subvariety·of·PP^9·cut·out·by·7·hypersurfaces·of·degrees
33 ·····1^2·2^533 ·····1^2·2^5
34 i4·:·time·parametrizeFanoFourfold·X34 i4·:·time·parametrizeFanoFourfold·X
35 ·--·used·2.02256s·(cpu);·0.681808s·(thread);·0s·(gc)35 ·--·used·2.01818s·(cpu);·0.711885s·(thread);·0s·(gc)
  
36 o4·=·multi-rational·map·consisting·of·one·single·rational·map36 o4·=·multi-rational·map·consisting·of·one·single·rational·map
37 ·····source·variety:·PP^437 ·····source·variety:·PP^4
38 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·738 ·····target·variety:·4-dimensional·subvariety·of·PP^9·cut·out·by·7
39 hypersurfaces·of·degrees·1^2·2^539 hypersurfaces·of·degrees·1^2·2^5
40 ·····dominance:·true40 ·····dominance:·true
41 ·····degree:·141 ·····degree:·1
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93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i4·:·X·=·projectiveVariety·ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);94 <td>··············<pre><code·class="language-macaulay2">i4·:·X·=·projectiveVariety·ideal(x_1^2*x_3+x_0*x_2*x_3-6*x_1*x_2*x_3-x_0*x_3^2-4*x_1*x_3^2-3*x_2*x_3^2+2*x_0^2*x_4-10*x_0*x_1*x_4+13*x_1^2*x_4-x_0*x_2*x_4-3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
  
95 o4·:·ProjectiveVariety,·hypersurface·in·PP^5</code></pre>95 o4·:·ProjectiveVariety,·hypersurface·in·PP^5</code></pre>
96 </td>··········</tr>96 </td>··········</tr>
97 ··········<tr>97 ··········<tr>
98 <td>··············<pre><code·class="language-macaulay2">i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);98 <td>··············<pre><code·class="language-macaulay2">i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);
99 ·--·used·0.00682864s·(cpu);·0.00694529s·(thread);·0s·(gc)99 ·--·used·0.009766s·(cpu);·0.00928852s·(thread);·0s·(gc)
  
100 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and·sectional·genus·0</code></pre>100 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and·sectional·genus·0</code></pre>
101 </td>··········</tr>101 </td>··········</tr>
102 ··········<tr>102 ··········<tr>
103 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·F103 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·F
104 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it104 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
105 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868105 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
106 ·--·used·0.395379s·(cpu);·0.168723s·(thread);·0s·(gc)106 ·--·used·0.728678s·(cpu);·0.24257s·(thread);·0s·(gc)
  
107 o6·=·Special·cubic·fourfold·of·discriminant·26107 o6·=·Special·cubic·fourfold·of·discriminant·26
108 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0108 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
109 ·····cut·out·by·13·hypersurfaces·of·degree·3</code></pre>109 ·····cut·out·by·13·hypersurfaces·of·degree·3</code></pre>
110 </td>··········</tr>110 </td>··········</tr>
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i7·:·assert(F·==·X)</code></pre>112 <td>··············<pre><code·class="language-macaulay2">i7·:·assert(F·==·X)</code></pre>
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116 3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-116 3*x_1*x_2*x_4+3*x_2^2*x_4+14*x_0*x_3*x_4-8*x_1*x_3*x_4-4*x_3^2*x_4+x_0*x_4^2-
117 7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-117 7*x_1*x_4^2+4*x_2*x_4^2-2*x_3*x_4^2-2*x_4^3-
118 x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-118 x_0*x_1*x_5+x_1^2*x_5+2*x_1*x_2*x_5+3*x_0*x_3*x_5+3*x_1*x_3*x_5-x_3^2*x_5-
119 x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);119 x_0*x_4*x_5-4*x_1*x_4*x_5+3*x_2*x_4*x_5+2*x_3*x_4*x_5-x_1*x_5^2);
  
120 o4·:·ProjectiveVariety,·hypersurface·in·PP^5120 o4·:·ProjectiveVariety,·hypersurface·in·PP^5
121 i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);121 i5·:·time·F·=·specialCubicFourfold(S,X,NumNodes=>3);
122 ·--·used·0.00682864s·(cpu);·0.00694529s·(thread);·0s·(gc)122 ·--·used·0.009766s·(cpu);·0.00928852s·(thread);·0s·(gc)
  
123 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and123 o5·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·7·and
124 sectional·genus·0124 sectional·genus·0
125 i6·:·time·describe·F125 i6·:·time·describe·F
126 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables126 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables
127 based·on·it127 based·on·it
128 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--128 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--
129 debug·9868129 debug·9868
130 ·--·used·0.395379s·(cpu);·0.168723s·(thread);·0s·(gc)130 ·--·used·0.728678s·(cpu);·0.24257s·(thread);·0s·(gc)
  
131 o6·=·Special·cubic·fourfold·of·discriminant·26131 o6·=·Special·cubic·fourfold·of·discriminant·26
132 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0132 ·····containing·a·3-nodal·surface·of·degree·7·and·sectional·genus·0
133 ·····cut·out·by·13·hypersurfaces·of·degree·3133 ·····cut·out·by·13·hypersurfaces·of·degree·3
134 i7·:·assert(F·==·X)134 i7·:·assert(F·==·X)
135 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*135 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
136 ····*·_\x8s_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8C_\x8u_\x8b_\x8i_\x8c_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8(_\x8E_\x8m_\x8b_\x8e_\x8d_\x8d_\x8e_\x8d_\x8P_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·random·special·cubic136 ····*·_\x8s_\x8p_\x8e_\x8c_\x8i_\x8a_\x8l_\x8C_\x8u_\x8b_\x8i_\x8c_\x8F_\x8o_\x8u_\x8r_\x8f_\x8o_\x8l_\x8d_\x8(_\x8E_\x8m_\x8b_\x8e_\x8d_\x8d_\x8e_\x8d_\x8P_\x8r_\x8o_\x8j_\x8e_\x8c_\x8t_\x8i_\x8v_\x8e_\x8V_\x8a_\x8r_\x8i_\x8e_\x8t_\x8y_\x8)·--·random·special·cubic
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90 ··········<tr>90 ··········<tr>
91 <td>··············<pre><code·class="language-macaulay2">i4·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);91 <td>··············<pre><code·class="language-macaulay2">i4·:·X·=·projectiveVariety·ideal(x_4*x_6-x_3*x_7+x_1*x_8,·x_4*x_5-x_2*x_7+x_0*x_8,·x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8,·x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7,·x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
  
92 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^8</code></pre>92 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^8</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·F·=·specialGushelMukaiFourfold(S,X);95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·F·=·specialGushelMukaiFourfold(S,X);
96 ·--·used·1.39467s·(cpu);·1.14945s·(thread);·0s·(gc)96 ·--·used·1.43482s·(cpu);·1.16197s·(thread);·0s·(gc)
  
97 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>97 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·F100 <td>··············<pre><code·class="language-macaulay2">i6·:·time·describe·F
101 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it101 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
102 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868102 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
103 ·--·used·6.60891s·(cpu);·2.64522s·(thread);·0s·(gc)103 ·--·used·7.67224s·(cpu);·2.94689s·(thread);·0s·(gc)
  
104 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')104 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')
105 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0105 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0
106 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)106 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
107 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)107 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
108 ·····Type:·ordinary108 ·····Type:·ordinary
109 ·····(case·1·of·Table·1·in·arXiv:2002.07026)</code></pre>109 ·····(case·1·of·Table·1·in·arXiv:2002.07026)</code></pre>
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34 x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-34 x_2*x_7+x_0*x_8,·x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-
35 x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-35 x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-
36 2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-36 2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
37 3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);37 3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8);
  
38 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^838 o4·:·ProjectiveVariety,·4-dimensional·subvariety·of·PP^8
39 i5·:·time·F·=·specialGushelMukaiFourfold(S,X);39 i5·:·time·F·=·specialGushelMukaiFourfold(S,X);
40 ·--·used·1.39467s·(cpu);·1.14945s·(thread);·0s·(gc)40 ·--·used·1.43482s·(cpu);·1.16197s·(thread);·0s·(gc)
  
41 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and41 o5·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and
42 sectional·genus·042 sectional·genus·0
43 i6·:·time·describe·F43 i6·:·time·describe·F
44 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables44 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables
45 based·on·it45 based·on·it
46 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--46 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--
47 debug·986847 debug·9868
48 ·--·used·6.60891s·(cpu);·2.64522s·(thread);·0s·(gc)48 ·--·used·7.67224s·(cpu);·2.94689s·(thread);·0s·(gc)
  
49 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')49 o6·=·Special·Gushel-Mukai·fourfold·of·discriminant·10(')
50 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·050 ·····containing·a·surface·in·PP^8·of·degree·2·and·sectional·genus·0
51 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)51 ·····cut·out·by·6·hypersurfaces·of·degrees·(1,1,1,1,1,2)
52 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)52 ·····and·with·class·in·G(1,4)·given·by·s_(3,1)+s_(2,2)
53 ·····Type:·ordinary53 ·····Type:·ordinary
54 ·····(case·1·of·Table·1·in·arXiv:2002.07026)54 ·····(case·1·of·Table·1·in·arXiv:2002.07026)
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76 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>76 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and·sectional·genus·0</code></pre>
77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i3·:·time·toGrass·X79 <td>··············<pre><code·class="language-macaulay2">i3·:·time·toGrass·X
80 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it80 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
81 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·986881 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
82 ·--·used·6.1112s·(cpu);·2.35774s·(thread);·0s·(gc)82 ·--·used·6.86175s·(cpu);·2.49811s·(thread);·0s·(gc)
  
83 o3·=·multi-rational·map·consisting·of·one·single·rational·map83 o3·=·multi-rational·map·consisting·of·one·single·rational·map
84 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces·of·degree·284 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces·of·degree·2
85 ·····target·variety:·GG(1,4)·⊂·PP^985 ·····target·variety:·GG(1,4)·⊂·PP^9
  
86 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))</code></pre>86 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
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27 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and27 o2·:·ProjectiveVariety,·GM·fourfold·containing·a·surface·of·degree·2·and
28 sectional·genus·028 sectional·genus·0
29 i3·:·time·toGrass·X29 i3·:·time·toGrass·X
30 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables30 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables
31 based·on·it31 based·on·it
32 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--32 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--
33 debug·986833 debug·9868
34 ·--·used·6.1112s·(cpu);·2.35774s·(thread);·0s·(gc)34 ·--·used·6.86175s·(cpu);·2.49811s·(thread);·0s·(gc)
  
35 o3·=·multi-rational·map·consisting·of·one·single·rational·map35 o3·=·multi-rational·map·consisting·of·one·single·rational·map
36 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces36 ·····source·variety:·4-dimensional·subvariety·of·PP^8·cut·out·by·6·hypersurfaces
37 of·degree·237 of·degree·2
38 ·····target·variety:·GG(1,4)·âŠ‚·PP^938 ·····target·variety:·GG(1,4)·âŠ‚·PP^9
  
39 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))39 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))
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78 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^8</code></pre>78 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^8</code></pre>
79 </td>··········</tr>79 </td>··········</tr>
80 ··········<tr>80 ··········<tr>
81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·toGrass·X81 <td>··············<pre><code·class="language-macaulay2">i3·:·time·toGrass·X
82 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it82 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables·based·on·it
83 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·986883 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--debug·9868
84 ·--·used·6.40265s·(cpu);·2.47054s·(thread);·0s·(gc)84 ·--·used·6.61212s·(cpu);·2.57624s·(thread);·0s·(gc)
  
85 o3·=·multi-rational·map·consisting·of·one·single·rational·map85 o3·=·multi-rational·map·consisting·of·one·single·rational·map
86 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5·hypersurfaces·of·degree·286 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5·hypersurfaces·of·degree·2
87 ·····target·variety:·GG(1,4)·⊂·PP^987 ·····target·variety:·GG(1,4)·⊂·PP^9
  
88 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))</code></pre>88 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))</code></pre>
89 </td>··········</tr>89 </td>··········</tr>
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26 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^826 o2·:·ProjectiveVariety,·5-dimensional·subvariety·of·PP^8
27 i3·:·time·toGrass·X27 i3·:·time·toGrass·X
28 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables28 warning:·clearing·value·of·symbol·x·to·allow·access·to·subscripted·variables
29 based·on·it29 based·on·it
30 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--30 ·······:·debug·with·expression···debug·9868···or·with·command·line·option···--
31 debug·986831 debug·9868
32 ·--·used·6.40265s·(cpu);·2.47054s·(thread);·0s·(gc)32 ·--·used·6.61212s·(cpu);·2.57624s·(thread);·0s·(gc)
  
33 o3·=·multi-rational·map·consisting·of·one·single·rational·map33 o3·=·multi-rational·map·consisting·of·one·single·rational·map
34 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·534 ·····source·variety:·5-dimensional·subvariety·of·PP^8·cut·out·by·5
35 hypersurfaces·of·degree·235 hypersurfaces·of·degree·2
36 ·····target·variety:·GG(1,4)·âŠ‚·PP^936 ·····target·variety:·GG(1,4)·âŠ‚·PP^9
  
37 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))37 o3·:·MultirationalMap·(rational·map·from·X·to·GG(1,4))
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77 ··········<tr>77 ··········<tr>
78 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialCubicFourfold·S;78 <td>··············<pre><code·class="language-macaulay2">i3·:·X·=·specialCubicFourfold·S;
  
79 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0</code></pre>79 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and·sectional·genus·0</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i4·:·time·f·=·unirationalParametrization·X;82 <td>··············<pre><code·class="language-macaulay2">i4·:·time·f·=·unirationalParametrization·X;
83 ·--·used·0.847054s·(cpu);·0.3949s·(thread);·0s·(gc)83 ·--·used·1.12796s·(cpu);·0.429726s·(thread);·0s·(gc)
  
84 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)</code></pre>84 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)</code></pre>
85 </td>··········</tr>85 </td>··········</tr>
86 ··········<tr>86 ··········<tr>
87 <td>··············<pre><code·class="language-macaulay2">i5·:·degreeSequence·f87 <td>··············<pre><code·class="language-macaulay2">i5·:·degreeSequence·f
  
88 o5·=·{[10]}88 o5·=·{[10]}
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19 o2·:·ProjectiveVariety,·surface·in·PP^519 o2·:·ProjectiveVariety,·surface·in·PP^5
20 i3·:·X·=·specialCubicFourfold·S;20 i3·:·X·=·specialCubicFourfold·S;
  
21 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and21 o3·:·ProjectiveVariety,·cubic·fourfold·containing·a·surface·of·degree·4·and
22 sectional·genus·022 sectional·genus·0
23 i4·:·time·f·=·unirationalParametrization·X;23 i4·:·time·f·=·unirationalParametrization·X;
24 ·--·used·0.847054s·(cpu);·0.3949s·(thread);·0s·(gc)24 ·--·used·1.12796s·(cpu);·0.429726s·(thread);·0s·(gc)
  
25 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)25 o4·:·MultirationalMap·(rational·map·from·PP^4·to·X)
26 i5·:·degreeSequence·f26 i5·:·degreeSequence·f
  
27 o5·=·{[10]}27 o5·=·{[10]}
  
28 o5·:·List28 o5·:·List
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11 ················Graph·=>·Graph{1·=>·{3}}11 ················Graph·=>·Graph{1·=>·{3}}
12 ·······························3·=>·{1}12 ·······························3·=>·{1}
  
13 o1·:·MixedGraph13 o1·:·MixedGraph
  
14 i2·:·toString·G14 i2·:·toString·G
  
15 o2·=·new·HashTable·from·{Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),15 o2·=·new·HashTable·from·{Digraph·=>·digraph·({1,·2,·3},·{{1,·2},·{2,·3}}),
16 ·····Graph·=>·graph·({3,·1},·{{1,·3}}),·Digraph·=>·digraph·({1,·2,·3},·{{1,16 ·····Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),·Graph·=>·graph·({3,
17 ·····2},·{2,·3}})}17 ·····1},·{{1,·3}})}
  
18 i3·:·18 i3·:·
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86 ·······························3·=>·{1}86 ·······························3·=>·{1}
  
87 o1·:·MixedGraph</code></pre>87 o1·:·MixedGraph</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i2·:·toString·G90 <td>··············<pre><code·class="language-macaulay2">i2·:·toString·G
  
91 o2·=·new·HashTable·from·{Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),91 o2·=·new·HashTable·from·{Digraph·=>·digraph·({1,·2,·3},·{{1,·2},·{2,·3}}),
92 ·····Graph·=>·graph·({3,·1},·{{1,·3}}),·Digraph·=>·digraph·({1,·2,·3},·{{1,92 ·····Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),·Graph·=>·graph·({3,
93 ·····2},·{2,·3}})}</code></pre>93 ·····1},·{{1,·3}})}</code></pre>
94 </td>··········</tr>94 </td>··········</tr>
95 ········</table>95 ········</table>
96 ······</div>96 ······</div>
97 ······<div>97 ······<div>
98 ········<h2>See·also</h2>98 ········<h2>See·also</h2>
99 ········<ul>99 ········<ul>
100 ··········<li>100 ··········<li>
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25 ················Graph·=>·Graph{1·=>·{3}}25 ················Graph·=>·Graph{1·=>·{3}}
26 ·······························3·=>·{1}26 ·······························3·=>·{1}
  
27 o1·:·MixedGraph27 o1·:·MixedGraph
28 i2·:·toString·G28 i2·:·toString·G
  
29 o2·=·new·HashTable·from·{Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),29 o2·=·new·HashTable·from·{Digraph·=>·digraph·({1,·2,·3},·{{1,·2},·{2,·3}}),
30 ·····Graph·=>·graph·({3,·1},·{{1,·3}}),·Digraph·=>·digraph·({1,·2,·3},·{{1,30 ·····Bigraph·=>·bigraph·({3,·4,·2},·{{4,·3},·{4,·2}}),·Graph·=>·graph·({3,
31 ·····2},·{2,·3}})}31 ·····1},·{{1,·3}})}
32 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*32 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
33 ····*·_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h·--·a·graph·that·has·undirected,·directed·and·bidirected·edges33 ····*·_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h·--·a·graph·that·has·undirected,·directed·and·bidirected·edges
34 ····*·_\x8n_\x8e_\x8t_\x8(_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8)·--·print·a·mixed·graph·as·a·net34 ····*·_\x8n_\x8e_\x8t_\x8(_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8)·--·print·a·mixed·graph·as·a·net
35 ····*·_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g·--·the·class·of·all·strings35 ····*·_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g·--·the·class·of·all·strings
36 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*36 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
37 ····*·_\x8t_\x8o_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8(_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8)·--·print·a·mixed·graph·as·a·string37 ····*·_\x8t_\x8o_\x8S_\x8t_\x8r_\x8i_\x8n_\x8g_\x8(_\x8M_\x8i_\x8x_\x8e_\x8d_\x8G_\x8r_\x8a_\x8p_\x8h_\x8)·--·print·a·mixed·graph·as·a·string
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731 B
./usr/share/doc/Macaulay2/StatePolytope/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
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773 B
./usr/share/doc/Macaulay2/StronglyStableIdeals/dump/rawdocumentation.dump
    
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702 B
./usr/share/doc/Macaulay2/Style/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=57 #:len=5
8 U3R5bGU=8 U3R5bGU=
9 #:len=3449 #:len=344
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3R5bGUgc2hlZXRzIGFuZCBpbWFnZXMg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3R5bGUgc2hlZXRzIGFuZCBpbWFnZXMg
11 Zm9yIHRoZSBkb2N1bWVudGF0aW9uIiwgRGVzY3JpcHRpb24gPT4gMTooIlRoaXMgcGFja2FnZSBp11 Zm9yIHRoZSBkb2N1bWVudGF0aW9uIiwgRGVzY3JpcHRpb24gPT4gMTooIlRoaXMgcGFja2FnZSBp
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747 B
./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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6 #·End·of·header6 #·End·of·header
7 #:len=247 #:len=24
8 c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p8 c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p
9 #:len=3009 #:len=300
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry
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751 B
./usr/share/doc/Macaulay2/SumsOfSquares/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=307 #:len=30
8 bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p8 bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p
9 #:len=3029 #:len=302
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm9111 PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm91
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729 B
./usr/share/doc/Macaulay2/SuperLinearAlgebra/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=67 #:len=6
8 cGFyaXR58 cGFyaXR5
9 #:len=17179 #:len=1717
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg
11 YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2MTAsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ911 YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2MTAsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ9
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759 B
./usr/share/doc/Macaulay2/SwitchingFields/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=337 #:len=33
8 ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp8 ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp
9 #:len=3009 #:len=300
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs
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734 B
./usr/share/doc/Macaulay2/SymbolicPowers/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=157 #:len=15
8 bm9QYWNrZWRBbGxTdWJz8 bm9QYWNrZWRBbGxTdWJz
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826 B
./usr/share/doc/Macaulay2/SymbolicPowers/example-output/_symbolic__Power.out
    
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31 o5·:·Ideal·of·QQ[x..z]31 o5·:·Ideal·of·QQ[x..z]
  
32 i6·:·isHomogeneous·P32 i6·:·isHomogeneous·P
  
33 o6·=·false33 o6·=·false
  
34 i7·:·time·symbolicPower(P,4);34 i7·:·time·symbolicPower(P,4);
35 ·--·used·0.142572s·(cpu);·0.117068s·(thread);·0s·(gc)35 ·--·used·0.182972s·(cpu);·0.156294s·(thread);·0s·(gc)
  
36 o7·:·Ideal·of·QQ[x..z]36 o7·:·Ideal·of·QQ[x..z]
  
37 i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})37 i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})
  
38 ·············2·········3·········2·····238 ·············2·········3·········2·····2
39 o8·=·ideal·(y··-·x*z,·x··-·y*z,·x·y·-·z·)39 o8·=·ideal·(y··-·x*z,·x··-·y*z,·x·y·-·z·)
Offset 47, 12 lines modifiedOffset 47, 12 lines modified
47 o8·:·Ideal·of·QQ[x..z]47 o8·:·Ideal·of·QQ[x..z]
  
48 i9·:·isHomogeneous·Q48 i9·:·isHomogeneous·Q
  
49 o9·=·true49 o9·=·true
  
50 i10·:·time·symbolicPower(Q,4);50 i10·:·time·symbolicPower(Q,4);
51 ·--·used·0.0670086s·(cpu);·0.0448657s·(thread);·0s·(gc)51 ·--·used·0.073273s·(cpu);·0.0449149s·(thread);·0s·(gc)
  
52 o10·:·Ideal·of·QQ[x..z]52 o10·:·Ideal·of·QQ[x..z]
  
53 i11·:·53 i11·:·
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2.27 KB
./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
    
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132 ··········<tr>132 ··········<tr>
133 <td>··············<pre><code·class="language-macaulay2">i6·:·isHomogeneous·P133 <td>··············<pre><code·class="language-macaulay2">i6·:·isHomogeneous·P
  
134 o6·=·false</code></pre>134 o6·=·false</code></pre>
135 </td>··········</tr>135 </td>··········</tr>
136 ··········<tr>136 ··········<tr>
137 <td>··············<pre><code·class="language-macaulay2">i7·:·time·symbolicPower(P,4);137 <td>··············<pre><code·class="language-macaulay2">i7·:·time·symbolicPower(P,4);
138 ·--·used·0.142572s·(cpu);·0.117068s·(thread);·0s·(gc)138 ·--·used·0.182972s·(cpu);·0.156294s·(thread);·0s·(gc)
  
139 o7·:·Ideal·of·QQ[x..z]</code></pre>139 o7·:·Ideal·of·QQ[x..z]</code></pre>
140 </td>··········</tr>140 </td>··········</tr>
141 ··········<tr>141 ··········<tr>
142 <td>··············<pre><code·class="language-macaulay2">i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})142 <td>··············<pre><code·class="language-macaulay2">i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})
  
143 ·············2·········3·········2·····2143 ·············2·········3·········2·····2
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151 ··········<tr>151 ··········<tr>
152 <td>··············<pre><code·class="language-macaulay2">i9·:·isHomogeneous·Q152 <td>··············<pre><code·class="language-macaulay2">i9·:·isHomogeneous·Q
  
153 o9·=·true</code></pre>153 o9·=·true</code></pre>
154 </td>··········</tr>154 </td>··········</tr>
155 ··········<tr>155 ··········<tr>
156 <td>··············<pre><code·class="language-macaulay2">i10·:·time·symbolicPower(Q,4);156 <td>··············<pre><code·class="language-macaulay2">i10·:·time·symbolicPower(Q,4);
157 ·--·used·0.0670086s·(cpu);·0.0448657s·(thread);·0s·(gc)157 ·--·used·0.073273s·(cpu);·0.0449149s·(thread);·0s·(gc)
  
158 o10·:·Ideal·of·QQ[x..z]</code></pre>158 o10·:·Ideal·of·QQ[x..z]</code></pre>
159 </td>··········</tr>159 </td>··········</tr>
160 ········</table>160 ········</table>
161 ······</div>161 ······</div>
162 ······<div>162 ······<div>
163 ········<h2>See·also</h2>163 ········<h2>See·also</h2>
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1.03 KB
html2text {}
    
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60 o5·=·ideal·(y··-·x*z,·x·y·-·z·,·x··-·y*z)60 o5·=·ideal·(y··-·x*z,·x·y·-·z·,·x··-·y*z)
  
61 o5·:·Ideal·of·QQ[x..z]61 o5·:·Ideal·of·QQ[x..z]
62 i6·:·isHomogeneous·P62 i6·:·isHomogeneous·P
  
63 o6·=·false63 o6·=·false
64 i7·:·time·symbolicPower(P,4);64 i7·:·time·symbolicPower(P,4);
65 ·--·used·0.142572s·(cpu);·0.117068s·(thread);·0s·(gc)65 ·--·used·0.182972s·(cpu);·0.156294s·(thread);·0s·(gc)
  
66 o7·:·Ideal·of·QQ[x..z]66 o7·:·Ideal·of·QQ[x..z]
67 i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})67 i8·:·Q·=·ker·map(QQ[t],QQ[x,y,z,·Degrees·=>·{3,4,5}],{t^3,t^4,t^5})
  
68 ·············2·········3·········2·····268 ·············2·········3·········2·····2
69 o8·=·ideal·(y··-·x*z,·x··-·y*z,·x·y·-·z·)69 o8·=·ideal·(y··-·x*z,·x··-·y*z,·x·y·-·z·)
  
70 o8·:·Ideal·of·QQ[x..z]70 o8·:·Ideal·of·QQ[x..z]
71 i9·:·isHomogeneous·Q71 i9·:·isHomogeneous·Q
  
72 o9·=·true72 o9·=·true
73 i10·:·time·symbolicPower(Q,4);73 i10·:·time·symbolicPower(Q,4);
74 ·--·used·0.0670086s·(cpu);·0.0448657s·(thread);·0s·(gc)74 ·--·used·0.073273s·(cpu);·0.0449149s·(thread);·0s·(gc)
  
75 o10·:·Ideal·of·QQ[x..z]75 o10·:·Ideal·of·QQ[x..z]
76 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*76 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
77 ····*·_\x8s_\x8y_\x8m_\x8b_\x8P_\x8o_\x8w_\x8e_\x8r_\x8P_\x8r_\x8i_\x8m_\x8e_\x8P_\x8o_\x8s_\x8C_\x8h_\x8a_\x8r77 ····*·_\x8s_\x8y_\x8m_\x8b_\x8P_\x8o_\x8w_\x8e_\x8r_\x8P_\x8r_\x8i_\x8m_\x8e_\x8P_\x8o_\x8s_\x8C_\x8h_\x8a_\x8r
78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·s\x8sy\x8ym\x8mb\x8bo\x8ol\x8li\x8ic\x8cP\x8Po\x8ow\x8we\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·s\x8sy\x8ym\x8mb\x8bo\x8ol\x8li\x8ic\x8cP\x8Po\x8ow\x8we\x8er\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
79 ····*·symbolicPower(Ideal,ZZ)79 ····*·symbolicPower(Ideal,ZZ)
80 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*80 *\x8**\x8**\x8**\x8**\x8*·F\x8Fo\x8or\x8r·t\x8th\x8he\x8e·p\x8pr\x8ro\x8og\x8gr\x8ra\x8am\x8mm\x8me\x8er\x8r·*\x8**\x8**\x8**\x8**\x8*
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769 B
./usr/share/doc/Macaulay2/SymmetricPolynomials/dump/rawdocumentation.dump
    
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2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
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7 #:len=327 #:len=32
8 YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk=8 YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk=
9 #:len=9369 #:len=936
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu
11 dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzAsIElu11 dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzAsIElu
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755 B
./usr/share/doc/Macaulay2/TSpreadIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=317 #:len=31
8 Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ==8 Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ==
9 #:len=2689 #:len=268
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4Miwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4Miwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU
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723 B
./usr/share/doc/Macaulay2/TangentCone/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=117 #:len=11
8 VGFuZ2VudENvbmU=8 VGFuZ2VudENvbmU=
9 #:len=3129 #:len=312
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB010 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB0
11 aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj11 aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj
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765 B
./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=397 #:len=39
8 YWN0aW9uT25EaXJlY3RJbWFnZShJZGVhbCxDaGFpbkNvbXBsZXgp8 YWN0aW9uT25EaXJlY3RJbWFnZShJZGVhbCxDaGFpbkNvbXBsZXgp
9 #:len=3189 #:len=318
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA2Mywgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA2Mywgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uT25EaXJlY3RJbWFnZSxJZGVhbCxDaGFp11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uT25EaXJlY3RJbWFnZSxJZGVhbCxDaGFp
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576 B
./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out
    
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10 o3·=·0··<--·E··<--·010 o3·=·0··<--·E··<--·0
11 ····················11 ····················
12 ·····-1·····0······112 ·····-1·····0······1
  
13 o3·:·ChainComplex13 o3·:·ChainComplex
  
14 i4·:·time·T=tateExtension·W;14 i4·:·time·T=tateExtension·W;
15 ·--·used·0.342439s·(cpu);·0.151823s·(thread);·0s·(gc)15 ·--·used·0.337868s·(cpu);·0.168801s·(thread);·0s·(gc)
  
16 i5·:·cohomologyMatrix(T,-{3,3},{3,3})16 i5·:·cohomologyMatrix(T,-{3,3},{3,3})
  
17 o5·=·|·8h··4h··0·4··8··12·16·|17 o5·=·|·8h··4h··0·4··8··12·16·|
18 ·····|·6h··3h··0·3··6··9··12·|18 ·····|·6h··3h··0·3··6··9··12·|
19 ·····|·4h··2h··0·2··4··6··8··|19 ·····|·4h··2h··0·2··4··6··8··|
20 ·····|·2h··h···0·1··2··3··4··|20 ·····|·2h··h···0·1··2··3··4··|
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1.17 KB
./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html
    
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85 ····················85 ····················
86 ·····-1·····0······186 ·····-1·····0······1
  
87 o3·:·ChainComplex</code></pre>87 o3·:·ChainComplex</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i4·:·time·T=tateExtension·W;90 <td>··············<pre><code·class="language-macaulay2">i4·:·time·T=tateExtension·W;
91 ·--·used·0.342439s·(cpu);·0.151823s·(thread);·0s·(gc)</code></pre>91 ·--·used·0.337868s·(cpu);·0.168801s·(thread);·0s·(gc)</code></pre>
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i5·:·cohomologyMatrix(T,-{3,3},{3,3})94 <td>··············<pre><code·class="language-macaulay2">i5·:·cohomologyMatrix(T,-{3,3},{3,3})
  
95 o5·=·|·8h··4h··0·4··8··12·16·|95 o5·=·|·8h··4h··0·4··8··12·16·|
96 ·····|·6h··3h··0·3··6··9··12·|96 ·····|·6h··3h··0·3··6··9··12·|
97 ·····|·4h··2h··0·2··4··6··8··|97 ·····|·4h··2h··0·2··4··6··8··|
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468 B
html2text {}
    
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24 ·············124 ·············1
25 o3·=·0··<--·E··<--·025 o3·=·0··<--·E··<--·0
  
26 ·····-1·····0······126 ·····-1·····0······1
  
27 o3·:·ChainComplex27 o3·:·ChainComplex
28 i4·:·time·T=tateExtension·W;28 i4·:·time·T=tateExtension·W;
29 ·--·used·0.342439s·(cpu);·0.151823s·(thread);·0s·(gc)29 ·--·used·0.337868s·(cpu);·0.168801s·(thread);·0s·(gc)
30 i5·:·cohomologyMatrix(T,-{3,3},{3,3})30 i5·:·cohomologyMatrix(T,-{3,3},{3,3})
  
31 o5·=·|·8h··4h··0·4··8··12·16·|31 o5·=·|·8h··4h··0·4··8··12·16·|
32 ·····|·6h··3h··0·3··6··9··12·|32 ·····|·6h··3h··0·3··6··9··12·|
33 ·····|·4h··2h··0·2··4··6··8··|33 ·····|·4h··2h··0·2··4··6··8··|
34 ·····|·2h··h···0·1··2··3··4··|34 ·····|·2h··h···0·1··2··3··4··|
35 ·····|·0···0···0·0··0··0··0··|35 ·····|·0···0···0·0··0··0··0··|
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759 B
./usr/share/doc/Macaulay2/TensorComplexes/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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6 #·End·of·header6 #·End·of·header
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9 #:len=2859 #:len=285
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747 B
./usr/share/doc/Macaulay2/TerraciniLoci/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=267 #:len=26
8 dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=8 dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=
9 #:len=2789 #:len=278
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy
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729 B
./usr/share/doc/Macaulay2/TestIdeals/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
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6 #·End·of·header6 #·End·of·header
7 #:len=177 #:len=17
8 ZnJvYmVuaXVzUHJlaW1hZ2U=8 ZnJvYmVuaXVzUHJlaW1hZ2U=
9 #:len=9349 #:len=934
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647 B
./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
    
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63 o15·:·Ideal·of·R63 o15·:·Ideal·of·R
  
64 i16·:·I3·=·ideal(x^50*y^50*z^50);64 i16·:·I3·=·ideal(x^50*y^50*z^50);
  
65 o16·:·Ideal·of·R65 o16·:·Ideal·of·R
  
66 i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});66 i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});
67 ·--·used·0.607482s·(cpu);·0.520091s·(thread);·0s·(gc)67 ·--·used·0.755444s·(cpu);·0.585445s·(thread);·0s·(gc)
  
68 o17·:·Ideal·of·R68 o17·:·Ideal·of·R
  
69 i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);69 i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);
70 ·--·used·1.82798s·(cpu);·1.63249s·(thread);·0s·(gc)70 ·--·used·2.1206s·(cpu);·1.84924s·(thread);·0s·(gc)
  
71 o18·:·Ideal·of·R71 o18·:·Ideal·of·R
  
72 i19·:·J1·==·J272 i19·:·J1·==·J2
  
73 o19·=·true73 o19·=·true
  
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682 B
./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out
    
Offset 7, 20 lines modifiedOffset 7, 20 lines modified
7 i3·:·g·=·map(T,·S,·{x^3,·x^2*y,·x*y^2,·y^3});7 i3·:·g·=·map(T,·S,·{x^3,·x^2*y,·x*y^2,·y^3});
  
8 o3·:·RingMap·T·<--·S8 o3·:·RingMap·T·<--·S
  
9 i4·:·R·=·S/(ker·g);9 i4·:·R·=·S/(ker·g);
  
10 i5·:·time·isCohenMacaulay(R)10 i5·:·time·isCohenMacaulay(R)
11 ·--·used·0.000109515s·(cpu);·0.00153143s·(thread);·0s·(gc)11 ·--·used·4.3863e-05s·(cpu);·0.00164567s·(thread);·0s·(gc)
  
12 o5·=·true12 o5·=·true
  
13 i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)13 i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)
14 ·--·used·0.00234999s·(cpu);·0.00403571s·(thread);·0s·(gc)14 ·--·used·0.0023181s·(cpu);·0.0037297s·(thread);·0s·(gc)
  
15 o6·=·true15 o6·=·true
  
16 i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);16 i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);
  
17 i8·:·isCohenMacaulay(R)17 i8·:·isCohenMacaulay(R)
  
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1.47 KB
./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out
    
Offset 60, 49 lines modifiedOffset 60, 49 lines modified
60 i19·:·R·=·ZZ/5[x,y,z]/(y^2*z·+·x*y*z-x^3)60 i19·:·R·=·ZZ/5[x,y,z]/(y^2*z·+·x*y*z-x^3)
  
61 o19·=·R61 o19·=·R
  
62 o19·:·QuotientRing62 o19·:·QuotientRing
  
63 i20·:·time·isFInjective(R)63 i20·:·time·isFInjective(R)
64 ·--·used·0.019982s·(cpu);·0.0201818s·(thread);·0s·(gc)64 ·--·used·0.0228976s·(cpu);·0.0237592s·(thread);·0s·(gc)
  
65 o20·=·true65 o20·=·true
  
66 i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)66 i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)
67 ·--·used·1.46863s·(cpu);·0.974333s·(thread);·0s·(gc)67 ·--·used·1.88888s·(cpu);·1.14442s·(thread);·0s·(gc)
  
68 o21·=·true68 o21·=·true
  
69 i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);69 i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);
  
70 i23·:·time·isFInjective(R)70 i23·:·time·isFInjective(R)
71 ·--·used·0.101434s·(cpu);·0.0729141s·(thread);·0s·(gc)71 ·--·used·0.14885s·(cpu);·0.0849583s·(thread);·0s·(gc)
  
72 o23·=·false72 o23·=·false
  
73 i24·:·time·isFInjective(R,·AtOrigin·=>·true)73 i24·:·time·isFInjective(R,·AtOrigin·=>·true)
74 ·--·used·0.0608068s·(cpu);·0.0619928s·(thread);·0s·(gc)74 ·--·used·0.0755154s·(cpu);·0.0777611s·(thread);·0s·(gc)
  
75 o24·=·true75 o24·=·true
  
76 i25·:·S·=·ZZ/3[xs,·ys,·zs,·xt,·yt,·zt];76 i25·:·S·=·ZZ/3[xs,·ys,·zs,·xt,·yt,·zt];
  
77 i26·:·EP1·=·ZZ/3[x,y,z,s,t]/(x^3·+·y^2*z·-·x*z^2);77 i26·:·EP1·=·ZZ/3[x,y,z,s,t]/(x^3·+·y^2*z·-·x*z^2);
  
78 i27·:·f·=·map(EP1,·S,·{x*s,·y*s,·z*s,·x*t,·y*t,·z*t});78 i27·:·f·=·map(EP1,·S,·{x*s,·y*s,·z*s,·x*t,·y*t,·z*t});
  
79 o27·:·RingMap·EP1·<--·S79 o27·:·RingMap·EP1·<--·S
  
80 i28·:·R·=·S/(ker·f);80 i28·:·R·=·S/(ker·f);
  
81 i29·:·time·isFInjective(R)81 i29·:·time·isFInjective(R)
82 ·--·used·0.689618s·(cpu);·0.601557s·(thread);·0s·(gc)82 ·--·used·0.878975s·(cpu);·0.727581s·(thread);·0s·(gc)
  
83 o29·=·false83 o29·=·false
  
84 i30·:·time·isFInjective(R,·AssumeCM·=>·true)84 i30·:·time·isFInjective(R,·AssumeCM·=>·true)
85 ·--·used·0.261858s·(cpu);·0.21387s·(thread);·0s·(gc)85 ·--·used·0.291161s·(cpu);·0.216869s·(thread);·0s·(gc)
  
86 o30·=·true86 o30·=·true
  
87 i31·:·87 i31·:·
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832 B
./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out
    
Offset 79, 20 lines modifiedOffset 79, 20 lines modified
79 i25·:·I·=·minors(2,·matrix·{{x,·y,·z},·{u,·v,·w}});79 i25·:·I·=·minors(2,·matrix·{{x,·y,·z},·{u,·v,·w}});
  
80 o25·:·Ideal·of·S80 o25·:·Ideal·of·S
  
81 i26·:·debugLevel·=·1;81 i26·:·debugLevel·=·1;
  
82 i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)82 i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)
83 ·--·used·0.0864527s·(cpu);·0.0569409s·(thread);·0s·(gc)83 ·--·used·0.0995687s·(cpu);·0.0597315s·(thread);·0s·(gc)
84 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing·DepthOfSearch·will·let·the·function·search·more·deeply.84 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing·DepthOfSearch·will·let·the·function·search·more·deeply.
  
85 o27·=·false85 o27·=·false
  
86 i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)86 i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)
87 ·--·used·0.112503s·(cpu);·0.113655s·(thread);·0s·(gc)87 ·--·used·0.189973s·(cpu);·0.151838s·(thread);·0s·(gc)
  
88 o28·=·true88 o28·=·true
  
89 i29·:·debugLevel·=·0;89 i29·:·debugLevel·=·0;
  
90 i30·:·90 i30·:·
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644 B
./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out
    
Offset 81, 21 lines modifiedOffset 81, 21 lines modified
81 i22·:·testIdeal({3/4,·2/3,·3/5},·L)81 i22·:·testIdeal({3/4,·2/3,·3/5},·L)
  
82 o22·=·ideal·(y,·x)82 o22·=·ideal·(y,·x)
  
83 o22·:·Ideal·of·R83 o22·:·Ideal·of·R
  
84 i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)84 i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)
85 ·--·used·0.204528s·(cpu);·0.146163s·(thread);·0s·(gc)85 ·--·used·0.239999s·(cpu);·0.159148s·(thread);·0s·(gc)
  
86 o23·=·ideal·(y,·x)86 o23·=·ideal·(y,·x)
  
87 o23·:·Ideal·of·R87 o23·:·Ideal·of·R
  
88 i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)88 i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)
89 ·--·used·0.31089s·(cpu);·0.219376s·(thread);·0s·(gc)89 ·--·used·0.382388s·(cpu);·0.262986s·(thread);·0s·(gc)
  
90 o24·=·ideal·(y,·x)90 o24·=·ideal·(y,·x)
  
91 o24·:·Ideal·of·R91 o24·:·Ideal·of·R
  
92 i25·:·92 i25·:·
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1.69 KB
./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
    
Offset 205, 21 lines modifiedOffset 205, 21 lines modified
205 ··········<tr>205 ··········<tr>
206 <td>··············<pre><code·class="language-macaulay2">i16·:·I3·=·ideal(x^50*y^50*z^50);206 <td>··············<pre><code·class="language-macaulay2">i16·:·I3·=·ideal(x^50*y^50*z^50);
  
207 o16·:·Ideal·of·R</code></pre>207 o16·:·Ideal·of·R</code></pre>
208 </td>··········</tr>208 </td>··········</tr>
209 ··········<tr>209 ··········<tr>
210 <td>··············<pre><code·class="language-macaulay2">i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});210 <td>··············<pre><code·class="language-macaulay2">i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});
211 ·--·used·0.607482s·(cpu);·0.520091s·(thread);·0s·(gc)211 ·--·used·0.755444s·(cpu);·0.585445s·(thread);·0s·(gc)
  
212 o17·:·Ideal·of·R</code></pre>212 o17·:·Ideal·of·R</code></pre>
213 </td>··········</tr>213 </td>··········</tr>
214 ··········<tr>214 ··········<tr>
215 <td>··············<pre><code·class="language-macaulay2">i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);215 <td>··············<pre><code·class="language-macaulay2">i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);
216 ·--·used·1.82798s·(cpu);·1.63249s·(thread);·0s·(gc)216 ·--·used·2.1206s·(cpu);·1.84924s·(thread);·0s·(gc)
  
217 o18·:·Ideal·of·R</code></pre>217 o18·:·Ideal·of·R</code></pre>
218 </td>··········</tr>218 </td>··········</tr>
219 ··········<tr>219 ··········<tr>
220 <td>··············<pre><code·class="language-macaulay2">i19·:·J1·==·J2220 <td>··············<pre><code·class="language-macaulay2">i19·:·J1·==·J2
  
221 o19·=·true</code></pre>221 o19·=·true</code></pre>
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717 B
html2text {}
    
Offset 107, 19 lines modifiedOffset 107, 19 lines modified
107 i15·:·I2·=·ideal(x^20*y^100,·x·+·z^100);107 i15·:·I2·=·ideal(x^20*y^100,·x·+·z^100);
  
108 o15·:·Ideal·of·R108 o15·:·Ideal·of·R
109 i16·:·I3·=·ideal(x^50*y^50*z^50);109 i16·:·I3·=·ideal(x^50*y^50*z^50);
  
110 o16·:·Ideal·of·R110 o16·:·Ideal·of·R
111 i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});111 i17·:·time·J1·=·frobeniusRoot(1,·{8,·10,·12},·{I1,·I2,·I3});
112 ·--·used·0.607482s·(cpu);·0.520091s·(thread);·0s·(gc)112 ·--·used·0.755444s·(cpu);·0.585445s·(thread);·0s·(gc)
  
113 o17·:·Ideal·of·R113 o17·:·Ideal·of·R
114 i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);114 i18·:·time·J2·=·frobeniusRoot(1,·I1^8*I2^10*I3^12);
115 ·--·used·1.82798s·(cpu);·1.63249s·(thread);·0s·(gc)115 ·--·used·2.1206s·(cpu);·1.84924s·(thread);·0s·(gc)
  
116 o18·:·Ideal·of·R116 o18·:·Ideal·of·R
117 i19·:·J1·==·J2117 i19·:·J1·==·J2
  
118 o19·=·true118 o19·=·true
119 For·legacy·reasons,·the·last·ideal·in·the·list·can·be·specified·separately,119 For·legacy·reasons,·the·last·ideal·in·the·list·can·be·specified·separately,
120 using·frobeniusRoot(e,·\{a_1,\ldots,a_n\},·\{I_1,\ldots,I_n\},·I).·The·last120 using·frobeniusRoot(e,·\{a_1,\ldots,a_n\},·\{I_1,\ldots,I_n\},·I).·The·last
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1.72 KB
./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
    
Offset 89, 21 lines modifiedOffset 89, 21 lines modified
89 o3·:·RingMap·T·&lt;--·S</code></pre>89 o3·:·RingMap·T·&lt;--·S</code></pre>
90 </td>··········</tr>90 </td>··········</tr>
91 ··········<tr>91 ··········<tr>
92 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·S/(ker·g);</code></pre>92 <td>··············<pre><code·class="language-macaulay2">i4·:·R·=·S/(ker·g);</code></pre>
93 </td>··········</tr>93 </td>··········</tr>
94 ··········<tr>94 ··········<tr>
95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isCohenMacaulay(R)95 <td>··············<pre><code·class="language-macaulay2">i5·:·time·isCohenMacaulay(R)
96 ·--·used·0.000109515s·(cpu);·0.00153143s·(thread);·0s·(gc)96 ·--·used·4.3863e-05s·(cpu);·0.00164567s·(thread);·0s·(gc)
  
97 o5·=·true</code></pre>97 o5·=·true</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)100 <td>··············<pre><code·class="language-macaulay2">i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)
101 ·--·used·0.00234999s·(cpu);·0.00403571s·(thread);·0s·(gc)101 ·--·used·0.0023181s·(cpu);·0.0037297s·(thread);·0s·(gc)
  
102 o6·=·true</code></pre>102 o6·=·true</code></pre>
103 </td>··········</tr>103 </td>··········</tr>
104 ········</table>104 ········</table>
105 ········<table·class="examples">105 ········<table·class="examples">
106 ··········<tr>106 ··········<tr>
107 <td>··············<pre><code·class="language-macaulay2">i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);</code></pre>107 <td>··············<pre><code·class="language-macaulay2">i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);</code></pre>
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684 B
html2text {}
    
Offset 24, 19 lines modifiedOffset 24, 19 lines modified
24 i1·:·T·=·ZZ/5[x,y];24 i1·:·T·=·ZZ/5[x,y];
25 i2·:·S·=·ZZ/5[a,b,c,d];25 i2·:·S·=·ZZ/5[a,b,c,d];
26 i3·:·g·=·map(T,·S,·{x^3,·x^2*y,·x*y^2,·y^3});26 i3·:·g·=·map(T,·S,·{x^3,·x^2*y,·x*y^2,·y^3});
  
27 o3·:·RingMap·T·<--·S27 o3·:·RingMap·T·<--·S
28 i4·:·R·=·S/(ker·g);28 i4·:·R·=·S/(ker·g);
29 i5·:·time·isCohenMacaulay(R)29 i5·:·time·isCohenMacaulay(R)
30 ·--·used·0.000109515s·(cpu);·0.00153143s·(thread);·0s·(gc)30 ·--·used·4.3863e-05s·(cpu);·0.00164567s·(thread);·0s·(gc)
  
31 o5·=·true31 o5·=·true
32 i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)32 i6·:·time·isCohenMacaulay(R,·AtOrigin·=>·true)
33 ·--·used·0.00234999s·(cpu);·0.00403571s·(thread);·0s·(gc)33 ·--·used·0.0023181s·(cpu);·0.0037297s·(thread);·0s·(gc)
  
34 o6·=·true34 o6·=·true
35 i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);35 i7·:·R·=·QQ[x,y,u,v]/(x*u,·x*v,·y*u,·y*v);
36 i8·:·isCohenMacaulay(R)36 i8·:·isCohenMacaulay(R)
  
37 o8·=·false37 o8·=·false
38 The·function·isCohenMacaulay·considers·$R$·as·a·quotient·of·a·polynomial·ring,38 The·function·isCohenMacaulay·considers·$R$·as·a·quotient·of·a·polynomial·ring,
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5.82 KB
./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
    
Offset 182, 41 lines modifiedOffset 182, 41 lines modified
  
182 o19·=·R182 o19·=·R
  
183 o19·:·QuotientRing</code></pre>183 o19·:·QuotientRing</code></pre>
184 </td>··········</tr>184 </td>··········</tr>
185 ··········<tr>185 ··········<tr>
186 <td>··············<pre><code·class="language-macaulay2">i20·:·time·isFInjective(R)186 <td>··············<pre><code·class="language-macaulay2">i20·:·time·isFInjective(R)
187 ·--·used·0.019982s·(cpu);·0.0201818s·(thread);·0s·(gc)187 ·--·used·0.0228976s·(cpu);·0.0237592s·(thread);·0s·(gc)
  
188 o20·=·true</code></pre>188 o20·=·true</code></pre>
189 </td>··········</tr>189 </td>··········</tr>
190 ··········<tr>190 ··········<tr>
191 <td>··············<pre><code·class="language-macaulay2">i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)191 <td>··············<pre><code·class="language-macaulay2">i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)
192 ·--·used·1.46863s·(cpu);·0.974333s·(thread);·0s·(gc)192 ·--·used·1.88888s·(cpu);·1.14442s·(thread);·0s·(gc)
  
193 o21·=·true</code></pre>193 o21·=·true</code></pre>
194 </td>··········</tr>194 </td>··········</tr>
195 ········</table>195 ········</table>
196 ········<div>196 ········<div>
197 ··········<p>If·the·option·<span·class="tt">AtOrigin</span>·(default·value·<span·class="tt">false</span>)·is·set·to·<span·class="tt">true</span>,·<span·class="tt">isFInjective</span>·will·only·check·$F$-injectivity·at·the·origin.·Otherwise,·it·will·check·$F$-injectivity·globally.·Note·that·checking·$F$-injectivity·at·the·origin·can·be·slower·than·checking·it·globally.·Consider·the·following·example·of·a·non-$F$-injective·ring.</p>197 ··········<p>If·the·option·<span·class="tt">AtOrigin</span>·(default·value·<span·class="tt">false</span>)·is·set·to·<span·class="tt">true</span>,·<span·class="tt">isFInjective</span>·will·only·check·$F$-injectivity·at·the·origin.·Otherwise,·it·will·check·$F$-injectivity·globally.·Note·that·checking·$F$-injectivity·at·the·origin·can·be·slower·than·checking·it·globally.·Consider·the·following·example·of·a·non-$F$-injective·ring.</p>
198 ········</div>198 ········</div>
199 ········<table·class="examples">199 ········<table·class="examples">
200 ··········<tr>200 ··········<tr>
201 <td>··············<pre><code·class="language-macaulay2">i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);</code></pre>201 <td>··············<pre><code·class="language-macaulay2">i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ··········<tr>203 ··········<tr>
204 <td>··············<pre><code·class="language-macaulay2">i23·:·time·isFInjective(R)204 <td>··············<pre><code·class="language-macaulay2">i23·:·time·isFInjective(R)
205 ·--·used·0.101434s·(cpu);·0.0729141s·(thread);·0s·(gc)205 ·--·used·0.14885s·(cpu);·0.0849583s·(thread);·0s·(gc)
  
206 o23·=·false</code></pre>206 o23·=·false</code></pre>
207 </td>··········</tr>207 </td>··········</tr>
208 ··········<tr>208 ··········<tr>
209 <td>··············<pre><code·class="language-macaulay2">i24·:·time·isFInjective(R,·AtOrigin·=>·true)209 <td>··············<pre><code·class="language-macaulay2">i24·:·time·isFInjective(R,·AtOrigin·=>·true)
210 ·--·used·0.0608068s·(cpu);·0.0619928s·(thread);·0s·(gc)210 ·--·used·0.0755154s·(cpu);·0.0777611s·(thread);·0s·(gc)
  
211 o24·=·true</code></pre>211 o24·=·true</code></pre>
212 </td>··········</tr>212 </td>··········</tr>
213 ········</table>213 ········</table>
214 ········<div>214 ········<div>
215 ··········<p>If·the·option·<span·class="tt">AssumeCM</span>·(default·value·<span·class="tt">false</span>)·is·set·to·<span·class="tt">true</span>,·then·<span·class="tt">isFInjective</span>·only·checks·the·Frobenius·action·on·top·cohomology·(which·is·typically·much·faster).·Note·that·it·can·give·an·incorrect·answer·if·the·non-injective·Frobenius·occurs·in·a·lower·degree.··Consider·the·example·of·the·cone·over·a·supersingular·elliptic·curve·times·$\mathbb{P}^1$.</p>215 ··········<p>If·the·option·<span·class="tt">AssumeCM</span>·(default·value·<span·class="tt">false</span>)·is·set·to·<span·class="tt">true</span>,·then·<span·class="tt">isFInjective</span>·only·checks·the·Frobenius·action·on·top·cohomology·(which·is·typically·much·faster).·Note·that·it·can·give·an·incorrect·answer·if·the·non-injective·Frobenius·occurs·in·a·lower·degree.··Consider·the·example·of·the·cone·over·a·supersingular·elliptic·curve·times·$\mathbb{P}^1$.</p>
216 ········</div>216 ········</div>
Offset 233, 21 lines modifiedOffset 233, 21 lines modified
233 o27·:·RingMap·EP1·&lt;--·S</code></pre>233 o27·:·RingMap·EP1·&lt;--·S</code></pre>
234 </td>··········</tr>234 </td>··········</tr>
235 ··········<tr>235 ··········<tr>
236 <td>··············<pre><code·class="language-macaulay2">i28·:·R·=·S/(ker·f);</code></pre>236 <td>··············<pre><code·class="language-macaulay2">i28·:·R·=·S/(ker·f);</code></pre>
237 </td>··········</tr>237 </td>··········</tr>
238 ··········<tr>238 ··········<tr>
239 <td>··············<pre><code·class="language-macaulay2">i29·:·time·isFInjective(R)239 <td>··············<pre><code·class="language-macaulay2">i29·:·time·isFInjective(R)
240 ·--·used·0.689618s·(cpu);·0.601557s·(thread);·0s·(gc)240 ·--·used·0.878975s·(cpu);·0.727581s·(thread);·0s·(gc)
  
241 o29·=·false</code></pre>241 o29·=·false</code></pre>
242 </td>··········</tr>242 </td>··········</tr>
243 ··········<tr>243 ··········<tr>
244 <td>··············<pre><code·class="language-macaulay2">i30·:·time·isFInjective(R,·AssumeCM·=>·true)244 <td>··············<pre><code·class="language-macaulay2">i30·:·time·isFInjective(R,·AssumeCM·=>·true)
245 ·--·used·0.261858s·(cpu);·0.21387s·(thread);·0s·(gc)245 ·--·used·0.291161s·(cpu);·0.216869s·(thread);·0s·(gc)
  
246 o30·=·true</code></pre>246 o30·=·true</code></pre>
247 </td>··········</tr>247 </td>··········</tr>
248 ········</table>248 ········</table>
249 ········<div>249 ········<div>
250 ··········<p>If·the·option·<span·class="tt">AssumedReduced</span>·is·set·to·<span·class="tt">true</span>·(its·default·behavior),·then·the·bottom·local·cohomology·is·avoided·(this·means·the·Frobenius·action·on·the·top·potentially·nonzero·Ext·is·not·computed).</p>250 ··········<p>If·the·option·<span·class="tt">AssumedReduced</span>·is·set·to·<span·class="tt">true</span>·(its·default·behavior),·then·the·bottom·local·cohomology·is·avoided·(this·means·the·Frobenius·action·on·the·top·potentially·nonzero·Ext·is·not·computed).</p>
251 ········</div>251 ········</div>
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Offset 82, 52 lines modifiedOffset 82, 52 lines modified
82 much·faster.82 much·faster.
83 i19·:·R·=·ZZ/5[x,y,z]/(y^2*z·+·x*y*z-x^3)83 i19·:·R·=·ZZ/5[x,y,z]/(y^2*z·+·x*y*z-x^3)
  
84 o19·=·R84 o19·=·R
  
85 o19·:·QuotientRing85 o19·:·QuotientRing
86 i20·:·time·isFInjective(R)86 i20·:·time·isFInjective(R)
87 ·--·used·0.019982s·(cpu);·0.0201818s·(thread);·0s·(gc)87 ·--·used·0.0228976s·(cpu);·0.0237592s·(thread);·0s·(gc)
  
88 o20·=·true88 o20·=·true
89 i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)89 i21·:·time·isFInjective(R,·CanonicalStrategy·=>·null)
90 ·--·used·1.46863s·(cpu);·0.974333s·(thread);·0s·(gc)90 ·--·used·1.88888s·(cpu);·1.14442s·(thread);·0s·(gc)
  
91 o21·=·true91 o21·=·true
92 If·the·option·AtOrigin·(default·value·false)·is·set·to·true,·isFInjective·will92 If·the·option·AtOrigin·(default·value·false)·is·set·to·true,·isFInjective·will
93 only·check·$F$-injectivity·at·the·origin.·Otherwise,·it·will·check·$F$-93 only·check·$F$-injectivity·at·the·origin.·Otherwise,·it·will·check·$F$-
94 injectivity·globally.·Note·that·checking·$F$-injectivity·at·the·origin·can·be94 injectivity·globally.·Note·that·checking·$F$-injectivity·at·the·origin·can·be
95 slower·than·checking·it·globally.·Consider·the·following·example·of·a·non-$F$-95 slower·than·checking·it·globally.·Consider·the·following·example·of·a·non-$F$-
96 injective·ring.96 injective·ring.
97 i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);97 i22·:·R·=·ZZ/7[x,y,z]/((x-1)^5·+·(y+1)^5·+·z^5);
98 i23·:·time·isFInjective(R)98 i23·:·time·isFInjective(R)
99 ·--·used·0.101434s·(cpu);·0.0729141s·(thread);·0s·(gc)99 ·--·used·0.14885s·(cpu);·0.0849583s·(thread);·0s·(gc)
  
100 o23·=·false100 o23·=·false
101 i24·:·time·isFInjective(R,·AtOrigin·=>·true)101 i24·:·time·isFInjective(R,·AtOrigin·=>·true)
102 ·--·used·0.0608068s·(cpu);·0.0619928s·(thread);·0s·(gc)102 ·--·used·0.0755154s·(cpu);·0.0777611s·(thread);·0s·(gc)
  
103 o24·=·true103 o24·=·true
104 If·the·option·AssumeCM·(default·value·false)·is·set·to·true,·then·isFInjective104 If·the·option·AssumeCM·(default·value·false)·is·set·to·true,·then·isFInjective
105 only·checks·the·Frobenius·action·on·top·cohomology·(which·is·typically·much105 only·checks·the·Frobenius·action·on·top·cohomology·(which·is·typically·much
106 faster).·Note·that·it·can·give·an·incorrect·answer·if·the·non-injective106 faster).·Note·that·it·can·give·an·incorrect·answer·if·the·non-injective
107 Frobenius·occurs·in·a·lower·degree.·Consider·the·example·of·the·cone·over·a107 Frobenius·occurs·in·a·lower·degree.·Consider·the·example·of·the·cone·over·a
108 supersingular·elliptic·curve·times·$\mathbb{P}^1$.108 supersingular·elliptic·curve·times·$\mathbb{P}^1$.
109 i25·:·S·=·ZZ/3[xs,·ys,·zs,·xt,·yt,·zt];109 i25·:·S·=·ZZ/3[xs,·ys,·zs,·xt,·yt,·zt];
110 i26·:·EP1·=·ZZ/3[x,y,z,s,t]/(x^3·+·y^2*z·-·x*z^2);110 i26·:·EP1·=·ZZ/3[x,y,z,s,t]/(x^3·+·y^2*z·-·x*z^2);
111 i27·:·f·=·map(EP1,·S,·{x*s,·y*s,·z*s,·x*t,·y*t,·z*t});111 i27·:·f·=·map(EP1,·S,·{x*s,·y*s,·z*s,·x*t,·y*t,·z*t});
  
112 o27·:·RingMap·EP1·<--·S112 o27·:·RingMap·EP1·<--·S
113 i28·:·R·=·S/(ker·f);113 i28·:·R·=·S/(ker·f);
114 i29·:·time·isFInjective(R)114 i29·:·time·isFInjective(R)
115 ·--·used·0.689618s·(cpu);·0.601557s·(thread);·0s·(gc)115 ·--·used·0.878975s·(cpu);·0.727581s·(thread);·0s·(gc)
  
116 o29·=·false116 o29·=·false
117 i30·:·time·isFInjective(R,·AssumeCM·=>·true)117 i30·:·time·isFInjective(R,·AssumeCM·=>·true)
118 ·--·used·0.261858s·(cpu);·0.21387s·(thread);·0s·(gc)118 ·--·used·0.291161s·(cpu);·0.216869s·(thread);·0s·(gc)
  
119 o30·=·true119 o30·=·true
120 If·the·option·AssumedReduced·is·set·to·true·(its·default·behavior),·then·the120 If·the·option·AssumedReduced·is·set·to·true·(its·default·behavior),·then·the
121 bottom·local·cohomology·is·avoided·(this·means·the·Frobenius·action·on·the·top121 bottom·local·cohomology·is·avoided·(this·means·the·Frobenius·action·on·the·top
122 potentially·nonzero·Ext·is·not·computed).122 potentially·nonzero·Ext·is·not·computed).
123 If·the·option·AssumeNormal·(default·value·false)·is·set·to·true,·then·the123 If·the·option·AssumeNormal·(default·value·false)·is·set·to·true,·then·the
124 bottom·two·local·cohomology·modules·(or,·rather,·their·duals)·need·not·be124 bottom·two·local·cohomology·modules·(or,·rather,·their·duals)·need·not·be
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Offset 230, 22 lines modifiedOffset 230, 22 lines modified
230 o25·:·Ideal·of·S</code></pre>230 o25·:·Ideal·of·S</code></pre>
231 </td>··········</tr>231 </td>··········</tr>
232 ··········<tr>232 ··········<tr>
233 <td>··············<pre><code·class="language-macaulay2">i26·:·debugLevel·=·1;</code></pre>233 <td>··············<pre><code·class="language-macaulay2">i26·:·debugLevel·=·1;</code></pre>
234 </td>··········</tr>234 </td>··········</tr>
235 ··········<tr>235 ··········<tr>
236 <td>··············<pre><code·class="language-macaulay2">i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)236 <td>··············<pre><code·class="language-macaulay2">i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)
237 ·--·used·0.0864527s·(cpu);·0.0569409s·(thread);·0s·(gc)237 ·--·used·0.0995687s·(cpu);·0.0597315s·(thread);·0s·(gc)
238 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing·DepthOfSearch·will·let·the·function·search·more·deeply.238 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing·DepthOfSearch·will·let·the·function·search·more·deeply.
  
239 o27·=·false</code></pre>239 o27·=·false</code></pre>
240 </td>··········</tr>240 </td>··········</tr>
241 ··········<tr>241 ··········<tr>
242 <td>··············<pre><code·class="language-macaulay2">i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)242 <td>··············<pre><code·class="language-macaulay2">i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)
243 ·--·used·0.112503s·(cpu);·0.113655s·(thread);·0s·(gc)243 ·--·used·0.189973s·(cpu);·0.151838s·(thread);·0s·(gc)
  
244 o28·=·true</code></pre>244 o28·=·true</code></pre>
245 </td>··········</tr>245 </td>··········</tr>
246 ··········<tr>246 ··········<tr>
247 <td>··············<pre><code·class="language-macaulay2">i29·:·debugLevel·=·0;</code></pre>247 <td>··············<pre><code·class="language-macaulay2">i29·:·debugLevel·=·0;</code></pre>
248 </td>··········</tr>248 </td>··········</tr>
249 ········</table>249 ········</table>
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113 also·use·the·option·DepthOfSearch·to·increase·the·depth·of·search.113 also·use·the·option·DepthOfSearch·to·increase·the·depth·of·search.
114 i24·:·S·=·ZZ/7[x,y,z,u,v,w];114 i24·:·S·=·ZZ/7[x,y,z,u,v,w];
115 i25·:·I·=·minors(2,·matrix·{{x,·y,·z},·{u,·v,·w}});115 i25·:·I·=·minors(2,·matrix·{{x,·y,·z},·{u,·v,·w}});
  
116 o25·:·Ideal·of·S116 o25·:·Ideal·of·S
117 i26·:·debugLevel·=·1;117 i26·:·debugLevel·=·1;
118 i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)118 i27·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·1)
119 ·--·used·0.0864527s·(cpu);·0.0569409s·(thread);·0s·(gc)119 ·--·used·0.0995687s·(cpu);·0.0597315s·(thread);·0s·(gc)
120 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing120 isFRegular:·This·ring·does·not·appear·to·be·F-regular.··Increasing
121 DepthOfSearch·will·let·the·function·search·more·deeply.121 DepthOfSearch·will·let·the·function·search·more·deeply.
  
122 o27·=·false122 o27·=·false
123 i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)123 i28·:·time·isFRegular(S/I,·QGorensteinIndex·=>·infinity,·DepthOfSearch·=>·2)
124 ·--·used·0.112503s·(cpu);·0.113655s·(thread);·0s·(gc)124 ·--·used·0.189973s·(cpu);·0.151838s·(thread);·0s·(gc)
  
125 o28·=·true125 o28·=·true
126 i29·:·debugLevel·=·0;126 i29·:·debugLevel·=·0;
127 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*127 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
128 ····*·_\x8t_\x8e_\x8s_\x8t_\x8I_\x8d_\x8e_\x8a_\x8l·--·compute·a·test·ideal·in·a·Q-Gorenstein·ring128 ····*·_\x8t_\x8e_\x8s_\x8t_\x8I_\x8d_\x8e_\x8a_\x8l·--·compute·a·test·ideal·in·a·Q-Gorenstein·ring
129 ····*·_\x8i_\x8s_\x8F_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l·--·whether·a·ring·is·F-rational129 ····*·_\x8i_\x8s_\x8F_\x8R_\x8a_\x8t_\x8i_\x8o_\x8n_\x8a_\x8l·--·whether·a·ring·is·F-rational
130 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sF\x8FR\x8Re\x8eg\x8gu\x8ul\x8la\x8ar\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*130 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·i\x8is\x8sF\x8FR\x8Re\x8eg\x8gu\x8ul\x8la\x8ar\x8r:\x8:·*\x8**\x8**\x8**\x8**\x8*
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219 ········</table>219 ········</table>
220 ········<div>220 ········<div>
221 ··········<p>It·is·often·more·efficient·to·pass·a·list,·as·opposed·to·finding·a·common·denominator·and·passing·a·single·element,·since·<span·class="tt">testIdeal</span>·can·do·things·in·a·more·intelligent·way·for·such·a·list.</p>221 ··········<p>It·is·often·more·efficient·to·pass·a·list,·as·opposed·to·finding·a·common·denominator·and·passing·a·single·element,·since·<span·class="tt">testIdeal</span>·can·do·things·in·a·more·intelligent·way·for·such·a·list.</p>
222 ········</div>222 ········</div>
223 ········<table·class="examples">223 ········<table·class="examples">
224 ··········<tr>224 ··········<tr>
225 <td>··············<pre><code·class="language-macaulay2">i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)225 <td>··············<pre><code·class="language-macaulay2">i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)
226 ·--·used·0.204528s·(cpu);·0.146163s·(thread);·0s·(gc)226 ·--·used·0.239999s·(cpu);·0.159148s·(thread);·0s·(gc)
  
227 o23·=·ideal·(y,·x)227 o23·=·ideal·(y,·x)
  
228 o23·:·Ideal·of·R</code></pre>228 o23·:·Ideal·of·R</code></pre>
229 </td>··········</tr>229 </td>··········</tr>
230 ··········<tr>230 ··········<tr>
231 <td>··············<pre><code·class="language-macaulay2">i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)231 <td>··············<pre><code·class="language-macaulay2">i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)
232 ·--·used·0.31089s·(cpu);·0.219376s·(thread);·0s·(gc)232 ·--·used·0.382388s·(cpu);·0.262986s·(thread);·0s·(gc)
  
233 o24·=·ideal·(y,·x)233 o24·=·ideal·(y,·x)
  
234 o24·:·Ideal·of·R</code></pre>234 o24·:·Ideal·of·R</code></pre>
235 </td>··········</tr>235 </td>··········</tr>
236 ········</table>236 ········</table>
237 ········<div>237 ········<div>
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101 o22·=·ideal·(y,·x)101 o22·=·ideal·(y,·x)
  
102 o22·:·Ideal·of·R102 o22·:·Ideal·of·R
103 It·is·often·more·efficient·to·pass·a·list,·as·opposed·to·finding·a·common103 It·is·often·more·efficient·to·pass·a·list,·as·opposed·to·finding·a·common
104 denominator·and·passing·a·single·element,·since·testIdeal·can·do·things·in·a104 denominator·and·passing·a·single·element,·since·testIdeal·can·do·things·in·a
105 more·intelligent·way·for·such·a·list.105 more·intelligent·way·for·such·a·list.
106 i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)106 i23·:·time·testIdeal({3/4,·2/3,·3/5},·L)
107 ·--·used·0.204528s·(cpu);·0.146163s·(thread);·0s·(gc)107 ·--·used·0.239999s·(cpu);·0.159148s·(thread);·0s·(gc)
  
108 o23·=·ideal·(y,·x)108 o23·=·ideal·(y,·x)
  
109 o23·:·Ideal·of·R109 o23·:·Ideal·of·R
110 i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)110 i24·:·time·testIdeal(1/60,·x^45*y^40*(x·+·y)^36)
111 ·--·used·0.31089s·(cpu);·0.219376s·(thread);·0s·(gc)111 ·--·used·0.382388s·(cpu);·0.262986s·(thread);·0s·(gc)
  
112 o24·=·ideal·(y,·x)112 o24·=·ideal·(y,·x)
  
113 o24·:·Ideal·of·R113 o24·:·Ideal·of·R
114 The·option·AssumeDomain·(default·value·false)·is·used·when·finding·a·test114 The·option·AssumeDomain·(default·value·false)·is·used·when·finding·a·test
115 element.·The·option·FrobeniusRootStrategy·(default·value·Substitution)·is115 element.·The·option·FrobeniusRootStrategy·(default·value·Substitution)·is
116 passed·to·internal·_\x8f_\x8r_\x8o_\x8b_\x8e_\x8n_\x8i_\x8u_\x8s_\x8R_\x8o_\x8o_\x8t·calls.116 passed·to·internal·_\x8f_\x8r_\x8o_\x8b_\x8e_\x8n_\x8i_\x8u_\x8s_\x8R_\x8o_\x8o_\x8t·calls.
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8 bmV3IFRPSCBmcm9tIFRoaW5n8 bmV3IFRPSCBmcm9tIFRoaW5n
9 #:len=2199 #:len=219
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU
⊟
749 B
./usr/share/doc/Macaulay2/ThinSincereQuivers/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=207 #:len=20
8 aXNUaWdodChUb3JpY1F1aXZlcik=8 aXNUaWdodChUb3JpY1F1aXZlcik=
9 #:len=2689 #:len=268
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA1OSwgc3ltYm9sIERvY3VtZW50VGFn10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA1OSwgc3ltYm9sIERvY3VtZW50VGFn
11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNUaWdodCxUb3JpY1F1aXZlciksImlzVGlnaHQo11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNUaWdodCxUb3JpY1F1aXZlciksImlzVGlnaHQo
⊟
716 B
./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump
    
Offset 1, 11 lines modifiedOffset 1, 11 lines modified
1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=97 #:len=9
8 dGdiKExpc3Qp8 dGdiKExpc3Qp
9 #:len=2139 #:len=213
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY4LCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk
⊟
1.89 KB
./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out
    
Offset 34, 17 lines modifiedOffset 34, 14 lines modified
34 ·······························0·5·2····3·234 ·······························0·5·2····3·2
35 ·································2······235 ·································2······2
36 ··················(1,·7)·=>·-·x·x··+·x·x36 ··················(1,·7)·=>·-·x·x··+·x·x
37 ·······························0·4····4·237 ·······························0·4····4·2
38 ····································238 ····································2
39 ··················(2,·3)·=>·x·x··-·x39 ··················(2,·3)·=>·x·x··-·x
40 ·····························0·4····240 ·····························0·4····2
41 ········································2 
42 ··················(2,·5)·=>·x·x·x··-·x·x 
43 ·····························0·5·2····3·2 
44 ··················(4,·6)·=>·x·x··-·x·x41 ··················(4,·6)·=>·x·x··-·x·x
45 ·····························0·5····3·242 ·····························0·5····3·2
46 ·······························2······343 ·······························2······3
47 ··················(8,·9)·=>·-·x·x··+·x44 ··················(8,·9)·=>·-·x·x··+·x
48 ·······························5·2····445 ·······························5·2····4
49 ··························246 ··························2
50 ··················0·=>·-·x··+·x·x47 ··················0·=>·-·x··+·x·x
Offset 103, 15 lines modifiedOffset 100, 14 lines modified
  
103 o8·=·LineageTable{(1,·2)·=>·null·······}100 o8·=·LineageTable{(1,·2)·=>·null·······}
104 ··················(1,·4)·=>·null101 ··················(1,·4)·=>·null
105 ··················(1,·7)·=>·null102 ··················(1,·7)·=>·null
106 ····································2103 ····································2
107 ··················(2,·3)·=>·x·x··-·x104 ··················(2,·3)·=>·x·x··-·x
108 ·····························0·4····2105 ·····························0·4····2
109 ··················(2,·5)·=>·null 
110 ··················(4,·6)·=>·x·x··-·x·x106 ··················(4,·6)·=>·x·x··-·x·x
111 ·····························0·5····3·2107 ·····························0·5····3·2
112 ·····························2······3108 ·····························2······3
113 ··················(8,·9)·=>·x·x··-·x109 ··················(8,·9)·=>·x·x··-·x
114 ·····························5·2····4110 ·····························5·2····4
115 ························2111 ························2
116 ··················0·=>·x··-·x·x112 ··················0·=>·x··-·x·x
Offset 141, 15 lines modifiedOffset 137, 14 lines modified
  
141 o9·=·LineageTable{(1,·2)·=>·null·······}137 o9·=·LineageTable{(1,·2)·=>·null·······}
142 ··················(1,·4)·=>·null138 ··················(1,·4)·=>·null
143 ··················(1,·7)·=>·null139 ··················(1,·7)·=>·null
144 ····································2140 ····································2
145 ··················(2,·3)·=>·x·x··-·x141 ··················(2,·3)·=>·x·x··-·x
146 ·····························0·4····2142 ·····························0·4····2
147 ··················(2,·5)·=>·null 
148 ··················(4,·6)·=>·x·x··-·x·x143 ··················(4,·6)·=>·x·x··-·x·x
149 ·····························0·5····3·2144 ·····························0·5····3·2
150 ·····························2······3145 ·····························2······3
151 ··················(8,·9)·=>·x·x··-·x146 ··················(8,·9)·=>·x·x··-·x
152 ·····························5·2····4147 ·····························5·2····4
153 ························2148 ························2
154 ··················0·=>·x··-·x·x149 ··················0·=>·x··-·x·x
⊟
656 B
./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
    
Offset 2, 16 lines modifiedOffset 2, 15 lines modified
  
2 i1·:·R·=·ZZ/101[a,b,c];2 i1·:·R·=·ZZ/101[a,b,c];
  
3 i2·:·allowableThreads=·2;3 i2·:·allowableThreads=·2;
  
4 i3·:·T·=·reduce·tgb(·ideal·"abc+c2,ab2-b3c+ac,b2")4 i3·:·T·=·reduce·tgb(·ideal·"abc+c2,ab2-b3c+ac,b2")
  
5 o3·=·LineageTable{(((0,·1),·0),·0)·=>·null}5 o3·=·LineageTable{((0,·2),·0)·=>·null}
6 ··················((0,·1),·0)·=>·null 
7 ··································26 ··································2
8 ··················((1,·2),·0)·=>·c7 ··················((1,·2),·0)·=>·c
9 ··················(0,·1)·=>·null8 ··················(0,·1)·=>·null
10 ··················(0,·2)·=>·null9 ··················(0,·2)·=>·null
11 ··················(1,·2)·=>·a*c10 ··················(1,·2)·=>·a*c
12 ··················0·=>·null11 ··················0·=>·null
13 ··················1·=>·null12 ··················1·=>·null
⊟
1.25 KB
./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
    
Offset 2, 16 lines modifiedOffset 2, 20 lines modified
  
2 i1·:·R·=·ZZ/101[a,b,c];2 i1·:·R·=·ZZ/101[a,b,c];
  
3 i2·:·allowableThreads=·2;3 i2·:·allowableThreads=·2;
  
4 i3·:·T·=·tgb·ideal·"abc+c2,ab2-b3c+ac,b2"4 i3·:·T·=·tgb·ideal·"abc+c2,ab2-b3c+ac,b2"
  
 5 ···································3
 6 o3·=·LineageTable{((0,·2),·0)·=>·-c······}
 7 ····································2
 8 ··················((0,·2),·1)·=>·a*c
5 ···································29 ···································2
6 o3·=·LineageTable{((1,·2),·0)·=>·-c······}10 ··················((1,·2),·0)·=>·-c
7 ·····························211 ·····························2
8 ··················(0,·1)·=>·a·c12 ··················(0,·1)·=>·a·c
9 ·······························213 ·······························2
10 ··················(0,·2)·=>·b*c14 ··················(0,·2)·=>·b*c
11 ··················(1,·2)·=>·-a*c15 ··················(1,·2)·=>·-a*c
12 ································216 ································2
13 ··················0·=>·a*b*c·+·c17 ··················0·=>·a*b*c·+·c
Offset 20, 16 lines modifiedOffset 24, 18 lines modified
20 ························224 ························2
21 ··················2·=>·b25 ··················2·=>·b
  
22 o3·:·LineageTable26 o3·:·LineageTable
  
23 i4·:·reduce·T27 i4·:·reduce·T
  
 28 o4·=·LineageTable{((0,·2),·0)·=>·null}
 29 ··················((0,·2),·1)·=>·null
24 ··································230 ··································2
25 o4·=·LineageTable{((1,·2),·0)·=>·c·}31 ··················((1,·2),·0)·=>·c
26 ··················(0,·1)·=>·null32 ··················(0,·1)·=>·null
27 ··················(0,·2)·=>·null33 ··················(0,·2)·=>·null
28 ··················(1,·2)·=>·a*c34 ··················(1,·2)·=>·a*c
29 ··················0·=>·null35 ··················0·=>·null
30 ··················1·=>·null36 ··················1·=>·null
31 ························237 ························2
32 ··················2·=>·b38 ··················2·=>·b
⊟
1.88 KB
./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out
    
Offset 6, 28 lines modifiedOffset 6, 34 lines modified
  
6 o2·:·Ideal·of·R6 o2·:·Ideal·of·R
  
7 i3·:·allowableThreads··=·4;7 i3·:·allowableThreads··=·4;
  
8 i4·:·H·=·tgb·I8 i4·:·H·=·tgb·I
  
 9 ······································4·9······3·9
 10 o4·=·LineageTable{((0,·3),·1)·=>·-·23y·z··+·44y·z·}
 11 ···································4·4······2·8
 12 ··················((0,·3),·2)·=>·9y·z··-·45y·z
 13 ··········································2·4
 14 ··················((1,·2),·(0,·2))·=>·-36y·z
9 ····································2·415 ·····································3·4
10 o4·=·LineageTable{((0,·2),·1)·=>·33y·z·······}16 ··················((1,·2),·1)·=>·-40y·z
11 ·································5·2······3·417 ·································5·2······3·4
12 ··················(0,·1)·=>·-·25y·z··-·19y·z18 ··················(0,·1)·=>·-·25y·z··-·19y·z
13 ·································3·5·····2·419 ·································5······2·4
14 ··················(0,·2)·=>·-·24y·z··+·9y·z20 ··················(0,·2)·=>·-·25y·z·+·9y·z
 21 ······························5·2······5
 22 ··················(0,·3)·=>·5y·z··+·28y·z
15 ·······························5·······3·423 ·································5·······4·5
16 ··················(0,·3)·=>·28y·z·-·24y·z 
17 ·······························3·4······2·7 
18 ··················(1,·2)·=>·21y·z··-·14y·z24 ··················(1,·2)·=>·-·34y·z·-·45y·z
19 ·································2·6······2·525 ·································4·5······2·8
20 ··················(1,·3)·=>·-·14y·z··-·38y·z26 ··················(1,·3)·=>·-·10y·z··+·23y·z
21 ·······························2·5·····2·427 ······························2·5·····2·4
22 ··················(2,·3)·=>·44y·z··-·9y·z28 ··················(2,·3)·=>·2y·z··-·9y·z
23 ·······························229 ·······························2
24 ··················0·=>·2x·+·10y·z30 ··················0·=>·2x·+·10y·z
25 ·························2···········331 ·························2···········3
26 ··················1·=>·8x·y·+·10x*y*z32 ··················1·=>·8x·y·+·10x*y*z
27 ···························3·2·······333 ···························3·2·······3
28 ··················2·=>·5x*y·z··+·9x*z34 ··················2·=>·5x*y·z··+·9x*z
29 ···························3·········335 ···························3·········3
⊟
1.5 KB
./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
    
Offset 84, 16 lines modifiedOffset 84, 15 lines modified
84 </td>··········</tr>84 </td>··········</tr>
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i2·:·allowableThreads=·2;</code></pre>86 <td>··············<pre><code·class="language-macaulay2">i2·:·allowableThreads=·2;</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·T·=·reduce·tgb(·ideal·&quot;abc+c2,ab2-b3c+ac,b2&quot;)89 <td>··············<pre><code·class="language-macaulay2">i3·:·T·=·reduce·tgb(·ideal·&quot;abc+c2,ab2-b3c+ac,b2&quot;)
  
90 o3·=·LineageTable{(((0,·1),·0),·0)·=>·null}90 o3·=·LineageTable{((0,·2),·0)·=>·null}
91 ··················((0,·1),·0)·=>·null 
92 ··································291 ··································2
93 ··················((1,·2),·0)·=>·c92 ··················((1,·2),·0)·=>·c
94 ··················(0,·1)·=>·null93 ··················(0,·1)·=>·null
95 ··················(0,·2)·=>·null94 ··················(0,·2)·=>·null
96 ··················(1,·2)·=>·a*c95 ··················(1,·2)·=>·a*c
97 ··················0·=>·null96 ··················0·=>·null
98 ··················1·=>·null97 ··················1·=>·null
⊟
714 B
html2text {}
    
Offset 20, 16 lines modifiedOffset 20, 15 lines modified
20 This·simple·function·just·returns·the·Gr\"obner·basis·computed·with·threaded20 This·simple·function·just·returns·the·Gr\"obner·basis·computed·with·threaded
21 Gr\"obner·basis·function·_\x8t_\x8g_\x8b·in·the·expected·Macaulay2·format,·so·that·further21 Gr\"obner·basis·function·_\x8t_\x8g_\x8b·in·the·expected·Macaulay2·format,·so·that·further
22 computation·are·one·step·easier·to·set·up.22 computation·are·one·step·easier·to·set·up.
23 i1·:·R·=·ZZ/101[a,b,c];23 i1·:·R·=·ZZ/101[a,b,c];
24 i2·:·allowableThreads=·2;24 i2·:·allowableThreads=·2;
25 i3·:·T·=·reduce·tgb(·ideal·"abc+c2,ab2-b3c+ac,b2")25 i3·:·T·=·reduce·tgb(·ideal·"abc+c2,ab2-b3c+ac,b2")
  
26 o3·=·LineageTable{(((0,·1),·0),·0)·=>·null}26 o3·=·LineageTable{((0,·2),·0)·=>·null}
27 ··················((0,·1),·0)·=>·null 
28 ··································227 ··································2
29 ··················((1,·2),·0)·=>·c28 ··················((1,·2),·0)·=>·c
30 ··················(0,·1)·=>·null29 ··················(0,·1)·=>·null
31 ··················(0,·2)·=>·null30 ··················(0,·2)·=>·null
32 ··················(1,·2)·=>·a*c31 ··················(1,·2)·=>·a*c
33 ··················0·=>·null32 ··················0·=>·null
34 ··················1·=>·null33 ··················1·=>·null
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77 </td>··········</tr>77 </td>··········</tr>
78 ··········<tr>78 ··········<tr>
79 <td>··············<pre><code·class="language-macaulay2">i2·:·allowableThreads=·2;</code></pre>79 <td>··············<pre><code·class="language-macaulay2">i2·:·allowableThreads=·2;</code></pre>
80 </td>··········</tr>80 </td>··········</tr>
81 ··········<tr>81 ··········<tr>
82 <td>··············<pre><code·class="language-macaulay2">i3·:·T·=·tgb·ideal·&quot;abc+c2,ab2-b3c+ac,b2&quot;82 <td>··············<pre><code·class="language-macaulay2">i3·:·T·=·tgb·ideal·&quot;abc+c2,ab2-b3c+ac,b2&quot;
  
 83 ···································3
 84 o3·=·LineageTable{((0,·2),·0)·=>·-c······}
 85 ····································2
 86 ··················((0,·2),·1)·=>·a*c
83 ···································287 ···································2
84 o3·=·LineageTable{((1,·2),·0)·=>·-c······}88 ··················((1,·2),·0)·=>·-c
85 ·····························289 ·····························2
86 ··················(0,·1)·=>·a·c90 ··················(0,·1)·=>·a·c
87 ·······························291 ·······························2
88 ··················(0,·2)·=>·b*c92 ··················(0,·2)·=>·b*c
89 ··················(1,·2)·=>·-a*c93 ··················(1,·2)·=>·-a*c
90 ································294 ································2
91 ··················0·=>·a*b*c·+·c95 ··················0·=>·a*b*c·+·c
Offset 96, 16 lines modifiedOffset 100, 18 lines modified
96 ··················2·=>·b100 ··················2·=>·b
  
97 o3·:·LineageTable</code></pre>101 o3·:·LineageTable</code></pre>
98 </td>··········</tr>102 </td>··········</tr>
99 ··········<tr>103 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·reduce·T104 <td>··············<pre><code·class="language-macaulay2">i4·:·reduce·T
  
 105 o4·=·LineageTable{((0,·2),·0)·=>·null}
 106 ··················((0,·2),·1)·=>·null
101 ··································2107 ··································2
102 o4·=·LineageTable{((1,·2),·0)·=>·c·}108 ··················((1,·2),·0)·=>·c
103 ··················(0,·1)·=>·null109 ··················(0,·1)·=>·null
104 ··················(0,·2)·=>·null110 ··················(0,·2)·=>·null
105 ··················(1,·2)·=>·a*c111 ··················(1,·2)·=>·a*c
106 ··················0·=>·null112 ··················0·=>·null
107 ··················1·=>·null113 ··················1·=>·null
108 ························2114 ························2
109 ··················2·=>·b115 ··················2·=>·b
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21 remainder·on·the·division·by·the·remaining·values·H.21 remainder·on·the·division·by·the·remaining·values·H.
22 If·values·H·constitute·a·Gr\"obner·basis·of·the·ideal·they·generate,·this22 If·values·H·constitute·a·Gr\"obner·basis·of·the·ideal·they·generate,·this
23 method·returns·a·reduced·Gr\"obner·basis.23 method·returns·a·reduced·Gr\"obner·basis.
24 i1·:·R·=·ZZ/101[a,b,c];24 i1·:·R·=·ZZ/101[a,b,c];
25 i2·:·allowableThreads=·2;25 i2·:·allowableThreads=·2;
26 i3·:·T·=·tgb·ideal·"abc+c2,ab2-b3c+ac,b2"26 i3·:·T·=·tgb·ideal·"abc+c2,ab2-b3c+ac,b2"
  
 27 ···································3
 28 o3·=·LineageTable{((0,·2),·0)·=>·-c······}
 29 ····································2
 30 ··················((0,·2),·1)·=>·a*c
27 ···································231 ···································2
28 o3·=·LineageTable{((1,·2),·0)·=>·-c······}32 ··················((1,·2),·0)·=>·-c
29 ·····························233 ·····························2
30 ··················(0,·1)·=>·a·c34 ··················(0,·1)·=>·a·c
31 ·······························235 ·······························2
32 ··················(0,·2)·=>·b*c36 ··················(0,·2)·=>·b*c
33 ··················(1,·2)·=>·-a*c37 ··················(1,·2)·=>·-a*c
34 ································238 ································2
35 ··················0·=>·a*b*c·+·c39 ··················0·=>·a*b*c·+·c
Offset 38, 16 lines modifiedOffset 42, 18 lines modified
38 ··················1·=>·-·b·c·+·a*b··+·a*c42 ··················1·=>·-·b·c·+·a*b··+·a*c
39 ························243 ························2
40 ··················2·=>·b44 ··················2·=>·b
  
41 o3·:·LineageTable45 o3·:·LineageTable
42 i4·:·reduce·T46 i4·:·reduce·T
  
 47 o4·=·LineageTable{((0,·2),·0)·=>·null}
 48 ··················((0,·2),·1)·=>·null
43 ··································249 ··································2
44 o4·=·LineageTable{((1,·2),·0)·=>·c·}50 ··················((1,·2),·0)·=>·c
45 ··················(0,·1)·=>·null51 ··················(0,·1)·=>·null
46 ··················(0,·2)·=>·null52 ··················(0,·2)·=>·null
47 ··················(1,·2)·=>·a*c53 ··················(1,·2)·=>·a*c
48 ··················0·=>·null54 ··················0·=>·null
49 ··················1·=>·null55 ··················1·=>·null
50 ························256 ························2
51 ··················2·=>·b57 ··················2·=>·b
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Offset 92, 28 lines modifiedOffset 92, 34 lines modified
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i3·:·allowableThreads··=·4;</code></pre>94 <td>··············<pre><code·class="language-macaulay2">i3·:·allowableThreads··=·4;</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i4·:·H·=·tgb·I97 <td>··············<pre><code·class="language-macaulay2">i4·:·H·=·tgb·I
  
 98 ······································4·9······3·9
 99 o4·=·LineageTable{((0,·3),·1)·=>·-·23y·z··+·44y·z·}
 100 ···································4·4······2·8
 101 ··················((0,·3),·2)·=>·9y·z··-·45y·z
 102 ··········································2·4
 103 ··················((1,·2),·(0,·2))·=>·-36y·z
98 ····································2·4104 ·····································3·4
99 o4·=·LineageTable{((0,·2),·1)·=>·33y·z·······}105 ··················((1,·2),·1)·=>·-40y·z
100 ·································5·2······3·4106 ·································5·2······3·4
101 ··················(0,·1)·=>·-·25y·z··-·19y·z107 ··················(0,·1)·=>·-·25y·z··-·19y·z
102 ·································3·5·····2·4108 ·································5······2·4
103 ··················(0,·2)·=>·-·24y·z··+·9y·z109 ··················(0,·2)·=>·-·25y·z·+·9y·z
 110 ······························5·2······5
 111 ··················(0,·3)·=>·5y·z··+·28y·z
104 ·······························5·······3·4112 ·································5·······4·5
105 ··················(0,·3)·=>·28y·z·-·24y·z 
106 ·······························3·4······2·7 
107 ··················(1,·2)·=>·21y·z··-·14y·z113 ··················(1,·2)·=>·-·34y·z·-·45y·z
108 ·································2·6······2·5114 ·································4·5······2·8
109 ··················(1,·3)·=>·-·14y·z··-·38y·z115 ··················(1,·3)·=>·-·10y·z··+·23y·z
110 ·······························2·5·····2·4116 ······························2·5·····2·4
111 ··················(2,·3)·=>·44y·z··-·9y·z117 ··················(2,·3)·=>·2y·z··-·9y·z
112 ·······························2118 ·······························2
113 ··················0·=>·2x·+·10y·z119 ··················0·=>·2x·+·10y·z
114 ·························2···········3120 ·························2···········3
115 ··················1·=>·8x·y·+·10x*y*z121 ··················1·=>·8x·y·+·10x*y*z
116 ···························3·2·······3122 ···························3·2·······3
117 ··················2·=>·5x*y·z··+·9x*z123 ··················2·=>·5x*y·z··+·9x*z
118 ···························3·········3124 ···························3·········3
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27 i2·:·I·=·ideal·{2*x·+·10*y^2*z,·8*x^2*y·+·10*x*y*z^3,·5*x*y^3*z^2·+·9*x*z^3,27 i2·:·I·=·ideal·{2*x·+·10*y^2*z,·8*x^2*y·+·10*x*y*z^3,·5*x*y^3*z^2·+·9*x*z^3,
28 9*x*y^3*z·+·10*x*y^3};28 9*x*y^3*z·+·10*x*y^3};
  
29 o2·:·Ideal·of·R29 o2·:·Ideal·of·R
30 i3·:·allowableThreads··=·4;30 i3·:·allowableThreads··=·4;
31 i4·:·H·=·tgb·I31 i4·:·H·=·tgb·I
  
 32 ······································4·9······3·9
 33 o4·=·LineageTable{((0,·3),·1)·=>·-·23y·z··+·44y·z·}
 34 ···································4·4······2·8
 35 ··················((0,·3),·2)·=>·9y·z··-·45y·z
 36 ··········································2·4
 37 ··················((1,·2),·(0,·2))·=>·-36y·z
32 ····································2·438 ·····································3·4
33 o4·=·LineageTable{((0,·2),·1)·=>·33y·z·······}39 ··················((1,·2),·1)·=>·-40y·z
34 ·································5·2······3·440 ·································5·2······3·4
35 ··················(0,·1)·=>·-·25y·z··-·19y·z41 ··················(0,·1)·=>·-·25y·z··-·19y·z
36 ·································3·5·····2·442 ·································5······2·4
37 ··················(0,·2)·=>·-·24y·z··+·9y·z43 ··················(0,·2)·=>·-·25y·z·+·9y·z
 44 ······························5·2······5
 45 ··················(0,·3)·=>·5y·z··+·28y·z
38 ·······························5·······3·446 ·································5·······4·5
39 ··················(0,·3)·=>·28y·z·-·24y·z 
40 ·······························3·4······2·7 
41 ··················(1,·2)·=>·21y·z··-·14y·z47 ··················(1,·2)·=>·-·34y·z·-·45y·z
42 ·································2·6······2·548 ·································4·5······2·8
43 ··················(1,·3)·=>·-·14y·z··-·38y·z49 ··················(1,·3)·=>·-·10y·z··+·23y·z
44 ·······························2·5·····2·450 ······························2·5·····2·4
45 ··················(2,·3)·=>·44y·z··-·9y·z51 ··················(2,·3)·=>·2y·z··-·9y·z
46 ·······························252 ·······························2
47 ··················0·=>·2x·+·10y·z53 ··················0·=>·2x·+·10y·z
48 ·························2···········354 ·························2···········3
49 ··················1·=>·8x·y·+·10x*y*z55 ··················1·=>·8x·y·+·10x*y*z
50 ···························3·2·······356 ···························3·2·······3
51 ··················2·=>·5x*y·z··+·9x*z57 ··················2·=>·5x*y·z··+·9x*z
52 ···························3·········358 ···························3·········3
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87 ·······························0·5·2····3·287 ·······························0·5·2····3·2
88 ·································2······288 ·································2······2
89 ··················(1,·7)·=>·-·x·x··+·x·x89 ··················(1,·7)·=>·-·x·x··+·x·x
90 ·······························0·4····4·290 ·······························0·4····4·2
91 ····································291 ····································2
92 ··················(2,·3)·=>·x·x··-·x92 ··················(2,·3)·=>·x·x··-·x
93 ·····························0·4····293 ·····························0·4····2
94 ········································2 
95 ··················(2,·5)·=>·x·x·x··-·x·x 
96 ·····························0·5·2····3·2 
97 ··················(4,·6)·=>·x·x··-·x·x94 ··················(4,·6)·=>·x·x··-·x·x
98 ·····························0·5····3·295 ·····························0·5····3·2
99 ·······························2······396 ·······························2······3
100 ··················(8,·9)·=>·-·x·x··+·x97 ··················(8,·9)·=>·-·x·x··+·x
101 ·······························5·2····498 ·······························5·2····4
102 ··························299 ··························2
103 ··················0·=>·-·x··+·x·x100 ··················0·=>·-·x··+·x·x
Offset 178, 15 lines modifiedOffset 175, 14 lines modified
  
178 o8·=·LineageTable{(1,·2)·=>·null·······}175 o8·=·LineageTable{(1,·2)·=>·null·······}
179 ··················(1,·4)·=>·null176 ··················(1,·4)·=>·null
180 ··················(1,·7)·=>·null177 ··················(1,·7)·=>·null
181 ····································2178 ····································2
182 ··················(2,·3)·=>·x·x··-·x179 ··················(2,·3)·=>·x·x··-·x
183 ·····························0·4····2180 ·····························0·4····2
184 ··················(2,·5)·=>·null 
185 ··················(4,·6)·=>·x·x··-·x·x181 ··················(4,·6)·=>·x·x··-·x·x
186 ·····························0·5····3·2182 ·····························0·5····3·2
187 ·····························2······3183 ·····························2······3
188 ··················(8,·9)·=>·x·x··-·x184 ··················(8,·9)·=>·x·x··-·x
189 ·····························5·2····4185 ·····························5·2····4
190 ························2186 ························2
191 ··················0·=>·x··-·x·x187 ··················0·=>·x··-·x·x
Offset 217, 15 lines modifiedOffset 213, 14 lines modified
  
217 o9·=·LineageTable{(1,·2)·=>·null·······}213 o9·=·LineageTable{(1,·2)·=>·null·······}
218 ··················(1,·4)·=>·null214 ··················(1,·4)·=>·null
219 ··················(1,·7)·=>·null215 ··················(1,·7)·=>·null
220 ····································2216 ····································2
221 ··················(2,·3)·=>·x·x··-·x217 ··················(2,·3)·=>·x·x··-·x
222 ·····························0·4····2218 ·····························0·4····2
223 ··················(2,·5)·=>·null 
224 ··················(4,·6)·=>·x·x··-·x·x219 ··················(4,·6)·=>·x·x··-·x·x
225 ·····························0·5····3·2220 ·····························0·5····3·2
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1020 B
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48 o4·=·25248 o4·=·252
  
49 o4·:·QQ49 o4·:·QQ
  
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129 Polar·Degrees:·{45,·98,·81,·28}129 Polar·Degrees:·{45,·98,·81,·28}
130 ED·Degree·=·252130 ED·Degree·=·252
  
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136 o4·=·252136 o4·=·252
  
137 o4·:·QQ</code></pre>137 o4·:·QQ</code></pre>
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145 Polar·Degrees:·{45,·98,·81,·28}145 Polar·Degrees:·{45,·98,·81,·28}
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68 Polar·Degrees:·{45,·98,·81,·28}68 Polar·Degrees:·{45,·98,·81,·28}
69 ED·Degree·=·25269 ED·Degree·=·252
  
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72 o4·=·25272 o4·=·252
  
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76 The·toric·variety·has·degree·=·2876 The·toric·variety·has·degree·=·28
77 The·dual·variety·has·degree·=·45,·and·codimension·=·177 The·dual·variety·has·degree·=·45,·and·codimension·=·1
78 Chern-Mather·Volumes:·(V_0,..,V_(d-1))·=·{20,·23,·31,·28}78 Chern-Mather·Volumes:·(V_0,..,V_(d-1))·=·{20,·23,·31,·28}
79 Polar·Degrees:·{45,·98,·81,·28}79 Polar·Degrees:·{45,·98,·81,·28}
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735 B
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11 cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgyMywgSW5wdXRzID0+IHtTUEFOe1RU11 cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgyMywgSW5wdXRzID0+IHtTUEFOe1RU
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2 #:version=1.12 #:version=1.1
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11 YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV011 YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV0
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./usr/share/doc/Macaulay2/TriangularSets/example-output/_triangularize.out
    
Offset 2, 19 lines modifiedOffset 2, 19 lines modified
  
2 i1·:·R·=·QQ[a..h,·MonomialOrder=>Lex];2 i1·:·R·=·QQ[a..h,·MonomialOrder=>Lex];
  
3 i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};3 i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};
  
4 i3·:·TT·=·triangularize(R,F,{})4 i3·:·TT·=·triangularize(R,F,{})
  
5 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{c,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-5 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-·b*c,·c*f·-
6 ·····------------------------------------------------------------------------6 ·····------------------------------------------------------------------------
7 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g}·/·{d,7 ·····d*e,·e*h·-·f*g}·/·{d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·e,·f},·{a*d·-
8 ·····------------------------------------------------------------------------8 ·····------------------------------------------------------------------------
9 ·····f,·h},·{b,·d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·g,·h}}9 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{b,·d,·f,·h},·{c,·d,·g,·h}}
  
10 o3·:·List10 o3·:·List
  
11 i4·:·first·TT11 i4·:·first·TT
  
12 o4·=·{c,·d,·f,·h}12 o4·=·{c,·d,·f,·h}
  
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./usr/share/doc/Macaulay2/TriangularSets/html/_triangularize.html
    
Offset 92, 19 lines modifiedOffset 92, 19 lines modified
92 </td>··········</tr>92 </td>··········</tr>
93 ··········<tr>93 ··········<tr>
94 <td>··············<pre><code·class="language-macaulay2">i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};</code></pre>94 <td>··············<pre><code·class="language-macaulay2">i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i3·:·TT·=·triangularize(R,F,{})97 <td>··············<pre><code·class="language-macaulay2">i3·:·TT·=·triangularize(R,F,{})
  
98 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{c,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-98 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-·b*c,·c*f·-
99 ·····------------------------------------------------------------------------99 ·····------------------------------------------------------------------------
100 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g}·/·{d,100 ·····d*e,·e*h·-·f*g}·/·{d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·e,·f},·{a*d·-
101 ·····------------------------------------------------------------------------101 ·····------------------------------------------------------------------------
102 ·····f,·h},·{b,·d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·g,·h}}102 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{b,·d,·f,·h},·{c,·d,·g,·h}}
  
103 o3·:·List</code></pre>103 o3·:·List</code></pre>
104 </td>··········</tr>104 </td>··········</tr>
105 ··········<tr>105 ··········<tr>
106 <td>··············<pre><code·class="language-macaulay2">i4·:·first·TT106 <td>··············<pre><code·class="language-macaulay2">i4·:·first·TT
  
107 o4·=·{c,·d,·f,·h}107 o4·=·{c,·d,·f,·h}
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html2text {}
    
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31 U_1)\cup\cdots\cup·Z(T_r/U_r).$$·These·simpler·sets,·called·_\x8t_\x8r_\x8i_\x8a_\x8n_\x8g_\x8u_\x8l_\x8a_\x8r_\x8·_\x8s_\x8y_\x8s_\x8t_\x8e_\x8m_\x8s,31 U_1)\cup\cdots\cup·Z(T_r/U_r).$$·These·simpler·sets,·called·_\x8t_\x8r_\x8i_\x8a_\x8n_\x8g_\x8u_\x8l_\x8a_\x8r_\x8·_\x8s_\x8y_\x8s_\x8t_\x8e_\x8m_\x8s,
32 have·very·nice·algorithmic·properties.32 have·very·nice·algorithmic·properties.
33 As·a·first·example·we·consider·a·case·without·inequations·($H=\emptyset$).33 As·a·first·example·we·consider·a·case·without·inequations·($H=\emptyset$).
34 i1·:·R·=·QQ[a..h,·MonomialOrder=>Lex];34 i1·:·R·=·QQ[a..h,·MonomialOrder=>Lex];
35 i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};35 i2·:·F·=·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g};
36 i3·:·TT·=·triangularize(R,F,{})36 i3·:·TT·=·triangularize(R,F,{})
  
37 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{c,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-37 o3·=·{{c,·d,·f,·h},·{b,·d,·e,·f},·{a*d·-·b*c,·e,·f}·/·d,·{a*d·-·b*c,·c*f·-
38 ·····------------------------------------------------------------------------38 ·····------------------------------------------------------------------------
39 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{a*d·-·b*c,·c*f·-·d*e,·e*h·-·f*g}·/·{d,39 ·····d*e,·e*h·-·f*g}·/·{d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·e,·f},·{a*d·-
40 ·····------------------------------------------------------------------------40 ·····------------------------------------------------------------------------
41 ·····f,·h},·{b,·d,·f,·h},·{c,·d,·e*h·-·f*g}·/·h,·{c,·d,·g,·h}}41 ·····b*c,·c*f·-·d*e,·g,·h}·/·{d,·f},·{b,·d,·f,·h},·{c,·d,·g,·h}}
  
42 o3·:·List42 o3·:·List
43 i4·:·first·TT43 i4·:·first·TT
  
44 o4·=·{c,·d,·f,·h}44 o4·=·{c,·d,·f,·h}
  
45 o4·:·TriaSystem45 o4·:·TriaSystem
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./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
3 #:file=rawdocumentation-dcba-8.db3 #:file=rawdocumentation-dcba-8.db
4 #:uid=1111,user=pbuilder1,gid=1111,group=pbuilder1,mode=6444 #:uid=2222,user=pbuilder2,gid=2222,group=pbuilder2,mode=644
5 #:format=standard5 #:format=standard
6 #·End·of·header6 #·End·of·header
7 #:len=177 #:len=17
8 YWxsVHJpYW5ndWxhdGlvbnM=8 YWxsVHJpYW5ndWxhdGlvbnM=
9 #:len=2859 #:len=285
10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcyLCBzeW1ib2wgRG9jdW1lbnRUYWcg10 bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcyLCBzeW1ib2wgRG9jdW1lbnRUYWcg
11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJhbGxUcmlhbmd1bGF0aW9ucyIsImFsbFRyaWFuZ3Vs11 PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJhbGxUcmlhbmd1bGF0aW9ucyIsImFsbFRyaWFuZ3Vs
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./usr/share/doc/Macaulay2/Triangulations/example-output/___Triangulations.out
    
Offset 17, 15 lines modifiedOffset 17, 15 lines modified
17 ·····|·-1·1··2··-1·-1·1·-1·1·0·0·|17 ·····|·-1·1··2··-1·-1·1·-1·1·0·0·|
18 ·····|·1··0··-1·0··0··0·0··0·0·0·|18 ·····|·1··0··-1·0··0··0·0··0·0·0·|
  
19 ··············4·······1019 ··············4·······10
20 o2·:·Matrix·ZZ··<--·ZZ20 o2·:·Matrix·ZZ··<--·ZZ
  
21 i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);21 i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);
22 ·--·.123063s·elapsed22 ·--·.270068s·elapsed
  
23 i4·:·select(Ts,·T·->·isStar·T)23 i4·:·select(Ts,·T·->·isStar·T)
  
24 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,24 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,
25 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
26 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,26 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,
27 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
Offset 50, 14 lines modifiedOffset 50, 14 lines modified
50 i7·:·T·=·regularFineTriangulation·A50 i7·:·T·=·regularFineTriangulation·A
  
51 o7·=·triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,·1,·6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},·{0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,·7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,·3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}51 o7·=·triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,·1,·6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},·{0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,·7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,·3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}
  
52 o7·:·Triangulation52 o7·:·Triangulation
  
53 i8·:·elapsedTime·Ts2·=·generateTriangulations·T;53 i8·:·elapsedTime·Ts2·=·generateTriangulations·T;
54 ·--·.997632s·elapsed54 ·--·.993287s·elapsed
  
55 i9·:·#Ts2·==·#Ts55 i9·:·#Ts2·==·#Ts
  
56 o9·=·true56 o9·=·true
  
57 i10·:·57 i10·:·
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./usr/share/doc/Macaulay2/Triangulations/example-output/_generate__Triangulations.out
    
Offset 21, 57 lines modifiedOffset 21, 15 lines modified
  
21 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·6},·{3,·5,·6,·7}}21 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·6},·{3,·5,·6,·7}}
  
22 o3·:·Triangulation22 o3·:·Triangulation
  
23 i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.23 i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.
  
24 o4·=·{triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7}, 
25 ·····------------------------------------------------------------------------ 
26 ·····{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7}, 
27 ·····------------------------------------------------------------------------ 
28 ·····{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},·triangulation 
29 ·····------------------------------------------------------------------------ 
30 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·6}, 
31 ·····------------------------------------------------------------------------ 
32 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·6}, 
33 ·····------------------------------------------------------------------------ 
34 ·····{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation·{{0,·1,·3,·5}, 
35 ·····------------------------------------------------------------------------ 
36 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}}, 
37 ·····------------------------------------------------------------------------ 
38 ·····triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5}, 
39 ·····------------------------------------------------------------------------ 
40 ·····{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·4}, 
41 ·····------------------------------------------------------------------------ 
42 ·····{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},·triangulation 
43 ·····------------------------------------------------------------------------ 
44 ·····{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·7}, 
45 ·····------------------------------------------------------------------------ 
46 ·····{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},·{0,·3,·4,·6}, 
47 ·····------------------------------------------------------------------------ 
48 ·····{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
49 ·····------------------------------------------------------------------------ 
50 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}}, 
51 ·····------------------------------------------------------------------------ 
52 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7}, 
53 ·····------------------------------------------------------------------------ 
54 ·····{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},·{0,·1,·4,·6}, 
55 ·····------------------------------------------------------------------------ 
56 ·····{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation 
57 ·····------------------------------------------------------------------------ 
58 ·····{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},·{2,·3,·4,·7}, 
59 ·····------------------------------------------------------------------------ 
60 ·····{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},·{0,·2,·3,·6}, 
61 ·····------------------------------------------------------------------------ 
62 ·····{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
63 ·····------------------------------------------------------------------------ 
64 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}}, 
65 ·····------------------------------------------------------------------------ 
66 ·····triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},24 o4·=·{triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},
67 ·····------------------------------------------------------------------------25 ·····------------------------------------------------------------------------
68 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},26 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
69 ·····------------------------------------------------------------------------27 ·····------------------------------------------------------------------------
70 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation28 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
71 ·····------------------------------------------------------------------------29 ·····------------------------------------------------------------------------
72 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},30 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},
73 ·····------------------------------------------------------------------------31 ·····------------------------------------------------------------------------
Offset 227, 57 lines modifiedOffset 185, 63 lines modified
227 ·····------------------------------------------------------------------------185 ·····------------------------------------------------------------------------
228 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},186 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},
229 ·····------------------------------------------------------------------------187 ·····------------------------------------------------------------------------
230 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation188 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation
231 ·····------------------------------------------------------------------------189 ·····------------------------------------------------------------------------
232 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},190 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},
233 ·····------------------------------------------------------------------------191 ·····------------------------------------------------------------------------
 192 ·····{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},
234 ·····{1,·4,·6,·7}}} 
  
235 o4·:·List 
  
236 i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets 
  
237 o5·=·{{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·5,·7}, 
238 ·····------------------------------------------------------------------------193 ·····------------------------------------------------------------------------
239 ·····{0,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},194 ·····{0,·2,·6,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},
240 ·····------------------------------------------------------------------------195 ·····------------------------------------------------------------------------
241 ·····{0,·4,·5,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},196 ·····{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},
242 ·····------------------------------------------------------------------------197 ·····------------------------------------------------------------------------
243 ·····{0,·3,·5,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},198 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},
244 ·····------------------------------------------------------------------------199 ·····------------------------------------------------------------------------
245 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},200 ·····{0,·4,·5,·6},·{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
246 ·····------------------------------------------------------------------------201 ·····------------------------------------------------------------------------
247 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},202 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
248 ·····------------------------------------------------------------------------203 ·····------------------------------------------------------------------------
249 ·····{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5},·{2,·3,·5,·6},204 ·····{{0,·1,·3,·5},·{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},
250 ·····------------------------------------------------------------------------205 ·····------------------------------------------------------------------------
251 ·····{3,·5,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},206 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},
252 ·····------------------------------------------------------------------------207 ·····------------------------------------------------------------------------
253 ·····{2,·4,·6,·7},·{3,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},208 ·····{1,·2,·3,·5},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},
254 ·····------------------------------------------------------------------------209 ·····------------------------------------------------------------------------
255 ·····{2,·3,·4,·6},·{3,·4,·5,·7},·{3,·4,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·6},210 ·····{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},
256 ·····------------------------------------------------------------------------211 ·····------------------------------------------------------------------------
257 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·{{0,·1,·2,·5},212 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},
258 ·····------------------------------------------------------------------------213 ·····------------------------------------------------------------------------
259 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}},214 ·····{3,·4,·5,·7},·{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},
260 ·····------------------------------------------------------------------------215 ·····------------------------------------------------------------------------
261 ·····{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7},·{2,·4,·5,·6},216 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation
262 ·····------------------------------------------------------------------------217 ·····------------------------------------------------------------------------
263 ·····{2,·5,·6,·7}},·{{0,·1,·2,·6},·{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},218 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},
264 ·····------------------------------------------------------------------------219 ·····------------------------------------------------------------------------
265 ·····{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},220 ·····{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},
266 ·····------------------------------------------------------------------------221 ·····------------------------------------------------------------------------
267 ·····{1,·4,·5,·7},·{2,·3,·4,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·6},·{0,·1,·4,·6},222 ·····{2,·3,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},
268 ·····------------------------------------------------------------------------223 ·····------------------------------------------------------------------------
269 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·5},224 ·····{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},
270 ·····------------------------------------------------------------------------225 ·····------------------------------------------------------------------------
271 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},226 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},
272 ·····------------------------------------------------------------------------227 ·····------------------------------------------------------------------------
 228 ·····{2,·3,·4,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},
 229 ·····------------------------------------------------------------------------
 230 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation
 231 ·····------------------------------------------------------------------------
 232 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},
 233 ·····------------------------------------------------------------------------
 234 ·····{2,·5,·6,·7}}}
  
 235 o4·:·List
  
 236 i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets
  
273 ·····{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},237 o5·=·{{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},
274 ·····------------------------------------------------------------------------238 ·····------------------------------------------------------------------------
275 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},239 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},
276 ·····------------------------------------------------------------------------240 ·····------------------------------------------------------------------------
277 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},241 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},
278 ·····------------------------------------------------------------------------242 ·····------------------------------------------------------------------------
279 ·····{0,·3,·5,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·{{0,·1,·2,·7},·{0,·1,·4,·7},243 ·····{0,·3,·5,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·{{0,·1,·2,·7},·{0,·1,·4,·7},
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./usr/share/doc/Macaulay2/Triangulations/html/_generate__Triangulations.html
    
Offset 116, 57 lines modifiedOffset 116, 15 lines modified
116 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·6},·{3,·5,·6,·7}}116 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·6},·{3,·5,·6,·7}}
  
117 o3·:·Triangulation</code></pre>117 o3·:·Triangulation</code></pre>
118 </td>··········</tr>118 </td>··········</tr>
119 ··········<tr>119 ··········<tr>
120 <td>··············<pre><code·class="language-macaulay2">i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.120 <td>··············<pre><code·class="language-macaulay2">i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.
  
121 o4·=·{triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7}, 
122 ·····------------------------------------------------------------------------ 
123 ·····{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7}, 
124 ·····------------------------------------------------------------------------ 
125 ·····{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},·triangulation 
126 ·····------------------------------------------------------------------------ 
127 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·6}, 
128 ·····------------------------------------------------------------------------ 
129 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·6}, 
130 ·····------------------------------------------------------------------------ 
131 ·····{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation·{{0,·1,·3,·5}, 
132 ·····------------------------------------------------------------------------ 
133 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}}, 
134 ·····------------------------------------------------------------------------ 
135 ·····triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5}, 
136 ·····------------------------------------------------------------------------ 
137 ·····{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·4}, 
138 ·····------------------------------------------------------------------------ 
139 ·····{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},·triangulation 
140 ·····------------------------------------------------------------------------ 
141 ·····{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·7}, 
142 ·····------------------------------------------------------------------------ 
143 ·····{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},·{0,·3,·4,·6}, 
144 ·····------------------------------------------------------------------------ 
145 ·····{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
146 ·····------------------------------------------------------------------------ 
147 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}}, 
148 ·····------------------------------------------------------------------------ 
149 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7}, 
150 ·····------------------------------------------------------------------------ 
151 ·····{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},·{0,·1,·4,·6}, 
152 ·····------------------------------------------------------------------------ 
153 ·····{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation 
154 ·····------------------------------------------------------------------------ 
155 ·····{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},·{2,·3,·4,·7}, 
156 ·····------------------------------------------------------------------------ 
157 ·····{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},·{0,·2,·3,·6}, 
158 ·····------------------------------------------------------------------------ 
159 ·····{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
160 ·····------------------------------------------------------------------------ 
161 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}}, 
162 ·····------------------------------------------------------------------------ 
163 ·····triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},121 o4·=·{triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},
164 ·····------------------------------------------------------------------------122 ·····------------------------------------------------------------------------
165 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},123 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
166 ·····------------------------------------------------------------------------124 ·····------------------------------------------------------------------------
167 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation125 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
168 ·····------------------------------------------------------------------------126 ·····------------------------------------------------------------------------
169 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},127 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},
170 ·····------------------------------------------------------------------------128 ·····------------------------------------------------------------------------
Offset 322, 58 lines modifiedOffset 280, 64 lines modified
322 ·····------------------------------------------------------------------------280 ·····------------------------------------------------------------------------
323 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},281 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},
324 ·····------------------------------------------------------------------------282 ·····------------------------------------------------------------------------
325 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation283 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation
326 ·····------------------------------------------------------------------------284 ·····------------------------------------------------------------------------
327 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},285 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},
328 ·····------------------------------------------------------------------------286 ·····------------------------------------------------------------------------
 287 ·····{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},
 288 ·····------------------------------------------------------------------------
 289 ·····{0,·2,·6,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},
 290 ·····------------------------------------------------------------------------
329 ·····{1,·4,·6,·7}}} 
  
330 o4·:·List</code></pre> 
331 </td>··········</tr> 
332 ··········<tr> 
333 <td>··············<pre><code·class="language-macaulay2">i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets 
  
334 o5·=·{{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·5,·7},291 ·····{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},
335 ·····------------------------------------------------------------------------292 ·····------------------------------------------------------------------------
336 ·····{0,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},293 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},
337 ·····------------------------------------------------------------------------294 ·····------------------------------------------------------------------------
338 ·····{0,·4,·5,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},295 ·····{0,·4,·5,·6},·{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
339 ·····------------------------------------------------------------------------296 ·····------------------------------------------------------------------------
340 ·····{0,·3,·5,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},297 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
341 ·····------------------------------------------------------------------------298 ·····------------------------------------------------------------------------
342 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},299 ·····{{0,·1,·3,·5},·{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},
343 ·····------------------------------------------------------------------------300 ·····------------------------------------------------------------------------
344 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},301 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},
345 ·····------------------------------------------------------------------------302 ·····------------------------------------------------------------------------
346 ·····{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5},·{2,·3,·5,·6},303 ·····{1,·2,·3,·5},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},
347 ·····------------------------------------------------------------------------304 ·····------------------------------------------------------------------------
348 ·····{3,·5,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},305 ·····{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},
349 ·····------------------------------------------------------------------------306 ·····------------------------------------------------------------------------
350 ·····{2,·4,·6,·7},·{3,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},307 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},
351 ·····------------------------------------------------------------------------308 ·····------------------------------------------------------------------------
352 ·····{2,·3,·4,·6},·{3,·4,·5,·7},·{3,·4,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·6},309 ·····{3,·4,·5,·7},·{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},
353 ·····------------------------------------------------------------------------310 ·····------------------------------------------------------------------------
354 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·{{0,·1,·2,·5},311 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation
355 ·····------------------------------------------------------------------------312 ·····------------------------------------------------------------------------
356 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}},313 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},
357 ·····------------------------------------------------------------------------314 ·····------------------------------------------------------------------------
358 ·····{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7},·{2,·4,·5,·6},315 ·····{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},
359 ·····------------------------------------------------------------------------316 ·····------------------------------------------------------------------------
360 ·····{2,·5,·6,·7}},·{{0,·1,·2,·6},·{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},317 ·····{2,·3,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},
361 ·····------------------------------------------------------------------------318 ·····------------------------------------------------------------------------
362 ·····{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},319 ·····{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},
363 ·····------------------------------------------------------------------------320 ·····------------------------------------------------------------------------
364 ·····{1,·4,·5,·7},·{2,·3,·4,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·6},·{0,·1,·4,·6},321 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},
365 ·····------------------------------------------------------------------------322 ·····------------------------------------------------------------------------
366 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·5},323 ·····{2,·3,·4,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},
367 ·····------------------------------------------------------------------------324 ·····------------------------------------------------------------------------
368 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},325 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation
369 ·····------------------------------------------------------------------------326 ·····------------------------------------------------------------------------
 327 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},
 328 ·····------------------------------------------------------------------------
 329 ·····{2,·5,·6,·7}}}
  
 330 o4·:·List</code></pre>
 331 </td>··········</tr>
 332 ··········<tr>
 333 <td>··············<pre><code·class="language-macaulay2">i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets
  
370 ·····{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},334 o5·=·{{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},
371 ·····------------------------------------------------------------------------335 ·····------------------------------------------------------------------------
372 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},336 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},
373 ·····------------------------------------------------------------------------337 ·····------------------------------------------------------------------------
374 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},338 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},
375 ·····------------------------------------------------------------------------339 ·····------------------------------------------------------------------------
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Offset 58, 57 lines modifiedOffset 58, 15 lines modified
  
58 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,58 o3·=·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·6},·{3,
59 4,·5,·6},·{3,·5,·6,·7}}59 4,·5,·6},·{3,·5,·6,·7}}
  
60 o3·:·Triangulation60 o3·:·Triangulation
61 i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.61 i4·:·Ts1·=·generateTriangulations·A·--·list·of·Triangulation's.
  
62 o4·=·{triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7}, 
63 ·····------------------------------------------------------------------------ 
64 ·····{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7}, 
65 ·····------------------------------------------------------------------------ 
66 ·····{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},·triangulation 
67 ·····------------------------------------------------------------------------ 
68 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·6}, 
69 ·····------------------------------------------------------------------------ 
70 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·6}, 
71 ·····------------------------------------------------------------------------ 
72 ·····{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation·{{0,·1,·3,·5}, 
73 ·····------------------------------------------------------------------------ 
74 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}}, 
75 ·····------------------------------------------------------------------------ 
76 ·····triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5}, 
77 ·····------------------------------------------------------------------------ 
78 ·····{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·4}, 
79 ·····------------------------------------------------------------------------ 
80 ·····{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},·triangulation 
81 ·····------------------------------------------------------------------------ 
82 ·····{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},·{3,·4,·5,·7}, 
83 ·····------------------------------------------------------------------------ 
84 ·····{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},·{0,·3,·4,·6}, 
85 ·····------------------------------------------------------------------------ 
86 ·····{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
87 ·····------------------------------------------------------------------------ 
88 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}}, 
89 ·····------------------------------------------------------------------------ 
90 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7}, 
91 ·····------------------------------------------------------------------------ 
92 ·····{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},·{0,·1,·4,·6}, 
93 ·····------------------------------------------------------------------------ 
94 ·····{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation 
95 ·····------------------------------------------------------------------------ 
96 ·····{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},·{2,·3,·4,·7}, 
97 ·····------------------------------------------------------------------------ 
98 ·····{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},·{0,·2,·3,·6}, 
99 ·····------------------------------------------------------------------------ 
100 ·····{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation·{{0,·1,·2,·5}, 
101 ·····------------------------------------------------------------------------ 
102 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}}, 
103 ·····------------------------------------------------------------------------ 
104 ·····triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},62 o4·=·{triangulation·{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},
105 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
106 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},64 ·····{1,·2,·3,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
107 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
108 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation66 ·····{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
109 ·····------------------------------------------------------------------------67 ·····------------------------------------------------------------------------
110 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},68 ·····{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},·{0,·4,·5,·7},
111 ·····------------------------------------------------------------------------69 ·····------------------------------------------------------------------------
Offset 264, 56 lines modifiedOffset 222, 62 lines modified
264 ·····------------------------------------------------------------------------222 ·····------------------------------------------------------------------------
265 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},223 ·····{1,·4,·5,·7},·{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·5},·{0,·2,·3,·5},
266 ·····------------------------------------------------------------------------224 ·····------------------------------------------------------------------------
267 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation225 ·····{0,·2,·5,·6},·{0,·4,·5,·6},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation
268 ·····------------------------------------------------------------------------226 ·····------------------------------------------------------------------------
269 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},227 ·····{{0,·1,·2,·4},·{1,·2,·3,·7},·{1,·2,·4,·6},·{1,·2,·6,·7},·{1,·4,·5,·7},
270 ·····------------------------------------------------------------------------228 ·····------------------------------------------------------------------------
 229 ·····{1,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},
271 ·····{1,·4,·6,·7}}} 
  
272 o4·:·List 
273 i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets 
  
274 o5·=·{{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·4,·5,·7}, 
275 ·····------------------------------------------------------------------------230 ·····------------------------------------------------------------------------
276 ·····{0,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},231 ·····{0,·2,·6,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·triangulation·{{0,·1,·3,·7},
277 ·····------------------------------------------------------------------------232 ·····------------------------------------------------------------------------
278 ·····{0,·4,·5,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},233 ·····{0,·1,·5,·7},·{0,·2,·3,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{2,·4,·6,·7}},
279 ·····------------------------------------------------------------------------234 ·····------------------------------------------------------------------------
280 ·····{0,·3,·5,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},235 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},·{0,·3,·5,·7},
281 ·····------------------------------------------------------------------------236 ·····------------------------------------------------------------------------
282 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},237 ·····{0,·4,·5,·6},·{0,·5,·6,·7}},·triangulation·{{0,·1,·3,·7},·{0,·1,·4,·7},
283 ·····------------------------------------------------------------------------238 ·····------------------------------------------------------------------------
284 ·····{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},·{0,·5,·6,·7}},239 ·····{0,·2,·3,·6},·{0,·3,·6,·7},·{0,·4,·6,·7},·{1,·4,·5,·7}},·triangulation
285 ·····------------------------------------------------------------------------240 ·····------------------------------------------------------------------------
286 ·····{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},·{1,·2,·3,·5},·{2,·3,·5,·6},241 ·····{{0,·1,·3,·5},·{0,·2,·3,·6},·{0,·3,·5,·7},·{0,·3,·6,·7},·{0,·4,·5,·6},
287 ·····------------------------------------------------------------------------242 ·····------------------------------------------------------------------------
288 ·····{3,·5,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},243 ·····{0,·5,·6,·7}},·triangulation·{{0,·1,·2,·5},·{0,·2,·5,·6},·{0,·4,·5,·6},
289 ·····------------------------------------------------------------------------244 ·····------------------------------------------------------------------------
290 ·····{2,·4,·6,·7},·{3,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},245 ·····{1,·2,·3,·5},·{2,·3,·5,·6},·{3,·5,·6,·7}},·triangulation·{{0,·1,·3,·4},
291 ·····------------------------------------------------------------------------246 ·····------------------------------------------------------------------------
292 ·····{2,·3,·4,·6},·{3,·4,·5,·7},·{3,·4,·6,·7}},·{{0,·1,·3,·4},·{0,·2,·3,·6},247 ·····{0,·2,·3,·4},·{1,·3,·4,·5},·{2,·3,·4,·7},·{2,·4,·6,·7},·{3,·4,·5,·7}},
293 ·····------------------------------------------------------------------------248 ·····------------------------------------------------------------------------
 249 ·····triangulation·{{0,·1,·3,·5},·{0,·2,·3,·4},·{0,·3,·4,·5},·{2,·3,·4,·6},
 250 ·····------------------------------------------------------------------------
 251 ·····{3,·4,·5,·7},·{3,·4,·6,·7}},·triangulation·{{0,·1,·3,·4},·{0,·2,·3,·6},
 252 ·····------------------------------------------------------------------------
294 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·{{0,·1,·2,·5},253 ·····{0,·3,·4,·6},·{1,·3,·4,·5},·{3,·4,·5,·6},·{3,·5,·6,·7}},·triangulation
295 ·····------------------------------------------------------------------------254 ·····------------------------------------------------------------------------
296 ·····{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},·{3,·5,·6,·7}},255 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·5},·{2,·3,·5,·6},·{2,·4,·5,·6},
 256 ·····------------------------------------------------------------------------
 257 ·····{3,·5,·6,·7}},·triangulation·{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},
297 ·····------------------------------------------------------------------------258 ·····------------------------------------------------------------------------
298 ·····{{0,·1,·2,·4},·{1,·2,·3,·5},·{1,·2,·4,·5},·{2,·3,·5,·7},·{2,·4,·5,·6},259 ·····{2,·3,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},·triangulation·{{0,·1,·2,·6},
299 ·····------------------------------------------------------------------------260 ·····------------------------------------------------------------------------
300 ·····{2,·5,·6,·7}},·{{0,·1,·2,·6},·{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},261 ·····{0,·1,·4,·6},·{1,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},
301 ·····------------------------------------------------------------------------262 ·····------------------------------------------------------------------------
302 ·····{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},263 ·····triangulation·{{0,·1,·2,·4},·{1,·2,·3,·4},·{1,·3,·4,·7},·{1,·4,·5,·7},
303 ·····------------------------------------------------------------------------264 ·····------------------------------------------------------------------------
304 ·····{1,·4,·5,·7},·{2,·3,·4,·7},·{2,·4,·6,·7}},·{{0,·1,·3,·6},·{0,·1,·4,·6},265 ·····{2,·3,·4,·7},·{2,·4,·6,·7}},·triangulation·{{0,·1,·3,·6},·{0,·1,·4,·6},
305 ·····------------------------------------------------------------------------266 ·····------------------------------------------------------------------------
306 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·{{0,·1,·2,·5},267 ·····{0,·2,·3,·6},·{1,·3,·6,·7},·{1,·4,·5,·6},·{1,·5,·6,·7}},·triangulation
307 ·····------------------------------------------------------------------------268 ·····------------------------------------------------------------------------
308 ·····{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},·{2,·5,·6,·7}},269 ·····{{0,·1,·2,·5},·{0,·2,·4,·5},·{1,·2,·3,·7},·{1,·2,·5,·7},·{2,·4,·5,·6},
309 ·····------------------------------------------------------------------------270 ·····------------------------------------------------------------------------
 271 ·····{2,·5,·6,·7}}}
  
 272 o4·:·List
 273 i5·:·Ts2·=·generateTriangulations(A,·T)·--·list·of·list·of·subsets
  
310 ·····{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},274 o5·=·{{{0,·1,·2,·7},·{0,·1,·5,·7},·{0,·2,·4,·7},·{0,·4,·5,·7},·{1,·2,·3,·7},
311 ·····------------------------------------------------------------------------275 ·····------------------------------------------------------------------------
312 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},276 ·····{2,·4,·6,·7}},·{{0,·1,·3,·7},·{0,·1,·4,·7},·{0,·2,·3,·7},·{0,·2,·6,·7},
313 ·····------------------------------------------------------------------------277 ·····------------------------------------------------------------------------
314 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},278 ·····{0,·4,·6,·7},·{1,·4,·5,·7}},·{{0,·1,·3,·5},·{0,·2,·3,·7},·{0,·2,·6,·7},
315 ·····------------------------------------------------------------------------279 ·····------------------------------------------------------------------------
316 ·····{0,·3,·5,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·{{0,·1,·2,·7},·{0,·1,·4,·7},280 ·····{0,·3,·5,·7},·{0,·4,·5,·7},·{0,·4,·6,·7}},·{{0,·1,·2,·7},·{0,·1,·4,·7},
317 ·····------------------------------------------------------------------------281 ·····------------------------------------------------------------------------
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./usr/share/doc/Macaulay2/Triangulations/html/index.html
    
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158 ·····|·1··0··-1·0··0··0·0··0·0·0·|158 ·····|·1··0··-1·0··0··0·0··0·0·0·|
  
159 ··············4·······10159 ··············4·······10
160 o2·:·Matrix·ZZ··&lt;--·ZZ</code></pre>160 o2·:·Matrix·ZZ··&lt;--·ZZ</code></pre>
161 </td>··········</tr>161 </td>··········</tr>
162 ··········<tr>162 ··········<tr>
163 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);163 <td>··············<pre><code·class="language-macaulay2">i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);
164 ·--·.123063s·elapsed</code></pre>164 ·--·.270068s·elapsed</code></pre>
165 </td>··········</tr>165 </td>··········</tr>
166 ··········<tr>166 ··········<tr>
167 <td>··············<pre><code·class="language-macaulay2">i4·:·select(Ts,·T·->·isStar·T)167 <td>··············<pre><code·class="language-macaulay2">i4·:·select(Ts,·T·->·isStar·T)
  
168 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,168 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,
169 ·····------------------------------------------------------------------------169 ·····------------------------------------------------------------------------
170 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,170 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,
Offset 196, 15 lines modifiedOffset 196, 15 lines modified
  
196 o7·=·triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,·1,·6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},·{0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,·7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,·3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}196 o7·=·triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,·1,·6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},·{0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,·7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,·3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}
  
197 o7·:·Triangulation</code></pre>197 o7·:·Triangulation</code></pre>
198 </td>··········</tr>198 </td>··········</tr>
199 ··········<tr>199 ··········<tr>
200 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·Ts2·=·generateTriangulations·T;200 <td>··············<pre><code·class="language-macaulay2">i8·:·elapsedTime·Ts2·=·generateTriangulations·T;
201 ·--·.997632s·elapsed</code></pre>201 ·--·.993287s·elapsed</code></pre>
202 </td>··········</tr>202 </td>··········</tr>
203 ··········<tr>203 ··········<tr>
204 <td>··············<pre><code·class="language-macaulay2">i9·:·#Ts2·==·#Ts204 <td>··············<pre><code·class="language-macaulay2">i9·:·#Ts2·==·#Ts
  
205 o9·=·true</code></pre>205 o9·=·true</code></pre>
206 </td>··········</tr>206 </td>··········</tr>
207 ········</table>207 ········</table>
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54 ·····|·0··0··0··1··0··0·-1·0·0·0·|54 ·····|·0··0··0··1··0··0·-1·0·0·0·|
55 ·····|·-1·1··2··-1·-1·1·-1·1·0·0·|55 ·····|·-1·1··2··-1·-1·1·-1·1·0·0·|
56 ·····|·1··0··-1·0··0··0·0··0·0·0·|56 ·····|·1··0··-1·0··0··0·0··0·0·0·|
  
57 ··············4·······1057 ··············4·······10
58 o2·:·Matrix·ZZ··<--·ZZ58 o2·:·Matrix·ZZ··<--·ZZ
59 i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);59 i3·:·elapsedTime·Ts·=·allTriangulations(A,·Fine·=>·true);
60 ·--·.123063s·elapsed60 ·--·.270068s·elapsed
61 i4·:·select(Ts,·T·->·isStar·T)61 i4·:·select(Ts,·T·->·isStar·T)
  
62 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,62 o4·=·{triangulation·{{0,·1,·2,·3,·9},·{0,·1,·2,·6,·9},·{0,·1,·3,·7,·9},·{0,
63 ·····------------------------------------------------------------------------63 ·····------------------------------------------------------------------------
64 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,64 ·····1,·6,·7,·9},·{0,·2,·3,·6,·9},·{0,·3,·4,·6,·9},·{0,·3,·4,·8,·9},·{0,·3,
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ·····5,·7,·9},·{0,·3,·5,·8,·9},·{0,·4,·6,·8,·9},·{0,·5,·6,·7,·9},·{0,·5,·6,66 ·····5,·7,·9},·{0,·3,·5,·8,·9},·{0,·4,·6,·8,·9},·{0,·5,·6,·7,·9},·{0,·5,·6,
Offset 86, 15 lines modifiedOffset 86, 15 lines modified
86 6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},86 6,·7,·9},·{0,·2,·3,·4,·6},·{0,·2,·3,·4,·9},·{0,·2,·4,·6,·9},·{0,·3,·4,·7,·8},
87 {0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,87 {0,·3,·4,·7,·9},·{0,·3,·5,·7,·8},·{0,·4,·6,·7,·8},·{0,·4,·6,·7,·9},·{0,·5,·6,
88 7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,88 7,·8},·{1,·2,·3,·7,·9},·{1,·2,·6,·7,·9},·{2,·3,·4,·7,·8},·{2,·3,·4,·7,·9},·{2,
89 3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}89 3,·5,·7,·8},·{2,·4,·6,·7,·8},·{2,·4,·6,·7,·9},·{2,·5,·6,·7,·8}}
  
90 o7·:·Triangulation90 o7·:·Triangulation
91 i8·:·elapsedTime·Ts2·=·generateTriangulations·T;91 i8·:·elapsedTime·Ts2·=·generateTriangulations·T;
92 ·--·.997632s·elapsed92 ·--·.993287s·elapsed
93 i9·:·#Ts2·==·#Ts93 i9·:·#Ts2·==·#Ts
  
94 o9·=·true94 o9·=·true
95 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*95 *\x8**\x8**\x8**\x8**\x8*·S\x8Se\x8ee\x8e·a\x8al\x8ls\x8so\x8o·*\x8**\x8**\x8**\x8**\x8*
96 ····*·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·--·for·computations·with·convex·polyhedra,·cones,·and·fans96 ····*·_\x8P_\x8o_\x8l_\x8y_\x8h_\x8e_\x8d_\x8r_\x8a·--·for·computations·with·convex·polyhedra,·cones,·and·fans
97 ····*·_\x8T_\x8o_\x8p_\x8c_\x8o_\x8m·--·interface·to·selected·functions·from·topcom·package97 ····*·_\x8T_\x8o_\x8p_\x8c_\x8o_\x8m·--·interface·to·selected·functions·from·topcom·package
98 ····*·_\x8R_\x8e_\x8f_\x8l_\x8e_\x8x_\x8i_\x8v_\x8e_\x8P_\x8o_\x8l_\x8y_\x8t_\x8o_\x8p_\x8e_\x8s_\x8D_\x8B·--·simple·access·to·Kreuzer-Skarke·database·of98 ····*·_\x8R_\x8e_\x8f_\x8l_\x8e_\x8x_\x8i_\x8v_\x8e_\x8P_\x8o_\x8l_\x8y_\x8t_\x8o_\x8p_\x8e_\x8s_\x8D_\x8B·--·simple·access·to·Kreuzer-Skarke·database·of
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727 B
./usr/share/doc/Macaulay2/Varieties/dump/rawdocumentation.dump
    
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726 B
./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump
    
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724 B
./usr/share/doc/Macaulay2/VectorGraphics/dump/rawdocumentation.dump
    
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765 B
./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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1.08 KB
./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out
    
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6 o2·=·|·xz·yz·z2·x3·|6 o2·=·|·xz·yz·z2·x3·|
  
7 ·············1······47 ·············1······4
8 o2·:·Matrix·S··<--·S8 o2·:·Matrix·S··<--·S
  
9 i3·:·time·(F,R,G,C)=localHilbertScheme(F0);9 i3·:·time·(F,R,G,C)=localHilbertScheme(F0);
10 ·--·used·0.794787s·(cpu);·0.597638s·(thread);·0s·(gc)10 ·--·used·0.943994s·(cpu);·0.64898s·(thread);·0s·(gc)
  
11 i4·:·T=ring·first·G;11 i4·:·T=ring·first·G;
  
12 i5·:·sum·G12 i5·:·sum·G
  
13 o5·=·|·t_1t_16·············|13 o5·=·|·t_1t_16·············|
14 ·····|·t_9t_16·············|14 ·····|·t_9t_16·············|
15 ·····|·-t_4t_16············|15 ·····|·-t_4t_16············|
16 ·····|·-2t_14t_16+t_15t_16·|16 ·····|·-2t_14t_16+t_15t_16·|
  
17 ·············4······117 ·············4······1
18 o5·:·Matrix·T··<--·T18 o5·:·Matrix·T··<--·T
  
19 i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);19 i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
20 ·--·used·0.521673s·(cpu);·0.378113s·(thread);·0s·(gc)20 ·--·used·0.608517s·(cpu);·0.422352s·(thread);·0s·(gc)
  
21 i7·:·sum·G21 i7·:·sum·G
  
22 o7·=·|·t_1t_16·····························································22 o7·=·|·t_1t_16·····························································
23 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+23 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
24 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t24 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
25 ·····|·2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-25 ·····|·2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
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./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html
    
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59 o2·:·Matrix·S··&lt;--·S</code></pre>59 o2·:·Matrix·S··&lt;--·S</code></pre>
60 </td>··········</tr>60 </td>··········</tr>
61 ········</table>61 ········</table>
62 ········<p>With·the·default·setting·<span·class="tt">SmartLift=>true</span>·we·get·very·nice·equations·for·the·base·space:</p>62 ········<p>With·the·default·setting·<span·class="tt">SmartLift=>true</span>·we·get·very·nice·equations·for·the·base·space:</p>
63 ········<table·class="examples">63 ········<table·class="examples">
64 ··········<tr>64 ··········<tr>
65 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(F,R,G,C)=localHilbertScheme(F0);65 <td>··············<pre><code·class="language-macaulay2">i3·:·time·(F,R,G,C)=localHilbertScheme(F0);
66 ·--·used·0.794787s·(cpu);·0.597638s·(thread);·0s·(gc)</code></pre>66 ·--·used·0.943994s·(cpu);·0.64898s·(thread);·0s·(gc)</code></pre>
67 </td>··········</tr>67 </td>··········</tr>
68 ··········<tr>68 ··········<tr>
69 <td>··············<pre><code·class="language-macaulay2">i4·:·T=ring·first·G;</code></pre>69 <td>··············<pre><code·class="language-macaulay2">i4·:·T=ring·first·G;</code></pre>
70 </td>··········</tr>70 </td>··········</tr>
71 ··········<tr>71 ··········<tr>
72 <td>··············<pre><code·class="language-macaulay2">i5·:·sum·G72 <td>··············<pre><code·class="language-macaulay2">i5·:·sum·G
  
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80 o5·:·Matrix·T··&lt;--·T</code></pre>80 o5·:·Matrix·T··&lt;--·T</code></pre>
81 </td>··········</tr>81 </td>··········</tr>
82 ········</table>82 ········</table>
83 ········<p>With·the·setting·<span·class="tt">SmartLift=>false</span>·the·calculation·is·faster,·but·the·equations·are·no·longer·homogeneous:</p>83 ········<p>With·the·setting·<span·class="tt">SmartLift=>false</span>·the·calculation·is·faster,·but·the·equations·are·no·longer·homogeneous:</p>
84 ········<table·class="examples">84 ········<table·class="examples">
85 ··········<tr>85 ··········<tr>
86 <td>··············<pre><code·class="language-macaulay2">i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);86 <td>··············<pre><code·class="language-macaulay2">i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
87 ·--·used·0.521673s·(cpu);·0.378113s·(thread);·0s·(gc)</code></pre>87 ·--·used·0.608517s·(cpu);·0.422352s·(thread);·0s·(gc)</code></pre>
88 </td>··········</tr>88 </td>··········</tr>
89 ··········<tr>89 ··········<tr>
90 <td>··············<pre><code·class="language-macaulay2">i7·:·sum·G90 <td>··············<pre><code·class="language-macaulay2">i7·:·sum·G
  
91 o7·=·|·t_1t_16·····························································91 o7·=·|·t_1t_16·····························································
92 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+92 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
93 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t93 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
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html2text {}
    
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18 o2·=·|·xz·yz·z2·x3·|18 o2·=·|·xz·yz·z2·x3·|
  
19 ·············1······419 ·············1······4
20 o2·:·Matrix·S··<--·S20 o2·:·Matrix·S··<--·S
21 With·the·default·setting·SmartLift=>true·we·get·very·nice·equations·for·the21 With·the·default·setting·SmartLift=>true·we·get·very·nice·equations·for·the
22 base·space:22 base·space:
23 i3·:·time·(F,R,G,C)=localHilbertScheme(F0);23 i3·:·time·(F,R,G,C)=localHilbertScheme(F0);
24 ·--·used·0.794787s·(cpu);·0.597638s·(thread);·0s·(gc)24 ·--·used·0.943994s·(cpu);·0.64898s·(thread);·0s·(gc)
25 i4·:·T=ring·first·G;25 i4·:·T=ring·first·G;
26 i5·:·sum·G26 i5·:·sum·G
  
27 o5·=·|·t_1t_16·············|27 o5·=·|·t_1t_16·············|
28 ·····|·t_9t_16·············|28 ·····|·t_9t_16·············|
29 ·····|·-t_4t_16············|29 ·····|·-t_4t_16············|
30 ·····|·-2t_14t_16+t_15t_16·|30 ·····|·-2t_14t_16+t_15t_16·|
  
31 ·············4······131 ·············4······1
32 o5·:·Matrix·T··<--·T32 o5·:·Matrix·T··<--·T
33 With·the·setting·SmartLift=>false·the·calculation·is·faster,·but·the·equations33 With·the·setting·SmartLift=>false·the·calculation·is·faster,·but·the·equations
34 are·no·longer·homogeneous:34 are·no·longer·homogeneous:
35 i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);35 i6·:·time·(F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
36 ·--·used·0.521673s·(cpu);·0.378113s·(thread);·0s·(gc)36 ·--·used·0.608517s·(cpu);·0.422352s·(thread);·0s·(gc)
37 i7·:·sum·G37 i7·:·sum·G
  
38 o7·=·|·t_1t_1638 o7·=·|·t_1t_16
39 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+39 ·····|·2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+
40 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t40 ·····|·-t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t
41 ·····|·2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-41 ·····|·2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2-
42 ·····------------------------------------------------------------------------42 ·····------------------------------------------------------------------------
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./usr/share/doc/Macaulay2/VirtualResolutions/dump/rawdocumentation.dump
    
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744 B
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734 B
./usr/share/doc/Macaulay2/WeylAlgebras/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/WeylGroups/dump/rawdocumentation.dump
    
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./usr/share/doc/Macaulay2/WeylGroups/example-output/_above__Bruhat_lp__Basic__List_rp.out
    
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36 ···········································|··2·|36 ···········································|··2·|
37 ···········································|·-1·|37 ···········································|·-1·|
  
38 o3·:·List38 o3·:·List
  
39 i4·:·aboveBruhat(L1)39 i4·:·aboveBruhat(L1)
  
40 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},·{1,·|··240 o4·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},·{1,·|··1
 41 ·············································|··1·|········|··2·|·······|··1
41 ·············································|·-2·|········|··1·|·······|·-142 ·············································|··2·|········|·-1·|·······|·-1
42 ·············································|··3·|········|·-1·|·······|··0 
43 ·····------------------------------------------------------------------------43 ·····------------------------------------------------------------------------
44 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,44 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··0·|},·{2,
45 ·····|············································|··1·|········|··1·|······45 ·····|············································|··3·|········|·-1·|······
46 ·····|············································|·-2·|········|··1·|······46 ·····|············································|·-1·|········|··2·|······
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
48 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,·|··0·|},48 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},
49 ·····|··2·|············································|·-2·|········|·-1·|··49 ·····|·-1·|············································|·-2·|········|··1·|··
50 ·····|·-1·|············································|··1·|········|··2·|··50 ·····|··0·|············································|··3·|········|·-1·|··
51 ·····------------------------------------------------------------------------51 ·····------------------------------------------------------------------------
52 ·····{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-152 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1
53 ·········|··1·|············································|··1·|········|··253 ·········|·-1·|············································|··1·|········|··1
54 ·········|··1·|············································|··2·|········|·-154 ·········|··0·|············································|·-2·|········|··1
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,56 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,
57 ·····|·······|··1·|············································|··3·|·······57 ·····|·······|··2·|············································|·-2·|·······
58 ·····|·······|·-1·|············································|·-1·|·······58 ·····|·······|·-1·|············································|··1·|·······
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····|··0·|},·{2,·|··2·|}}}}60 ·····|··0·|},·{2,·|·-1·|}}}}
61 ·····|·-1·|·······|·-1·|61 ·····|·-1·|·······|··1·|
62 ·····|··2·|·······|··0·|62 ·····|··2·|·······|··1·|
  
63 o4·:·List63 o4·:·List
  
64 i5·:·64 i5·:·
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./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Element_cm__Weyl__Group__Element_rp.out
    
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20 ···········································|·-2·|20 ···········································|·-2·|
21 ···········································|··1·|21 ···········································|··1·|
  
22 o3·:·WeylGroupElement22 o3·:·WeylGroupElement
  
23 i4·:·myInterval=intervalBruhat(w1,w2)23 i4·:·myInterval=intervalBruhat(w1,w2)
  
24 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··0·|},·{1,·|·-1·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{}}}} 
25 ··························································|·-2·|········|·-1·|·······|··2·|··············································|·-3·|········|··1·|············································|·-1·|········|·-1·|··············································|·-1·| 
26 ··························································|··1·|········|··2·|·······|·-1·|··············································|··1·|········|··1·|············································|··2·|········|··2·|···································[·...·truncated·by·diffoscope;·len:·17,·SHA:·5ebd410378763a0377949c88c0bf38c02207bbfbe2135b8d5c0d6fe2c1deafe7·...·]24 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|·-1·|},·{1,·|··0·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}},·{{WeylGroupElement{RootSystem[·...·truncated·by·diffoscope;·len:·25,·SHA:·681e071fee4a8cf2af531899a8824f2395fb7ebd9aba6210a84a4873acf6536f·...·]
 25 ··························································|·-2·|········|··2·|·······|·-1·|··············································|·-1·|········|·-1·|············································|·-3·|········|··1·|··············································|·-1·|
 26 ··························································|··1·|········|·-1·|·······|··2·|··············································|··2·|········|··2·|············································|··1·|········|··1·|··············································|··3·|
  
27 o4·:·HasseDiagram27 o4·:·HasseDiagram
  
28 i5·:·myInterval#128 i5·:·myInterval#1
  
29 o5·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}},29 o5·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}},
30 ·············································|·-3·|········|··1·|····30 ·············································|·-1·|········|·-1·|····
31 ·············································|··1·|········|··1·|····31 ·············································|··2·|········|··2·|····
32 ·····------------------------------------------------------------------------32 ·····------------------------------------------------------------------------
33 ·····{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}}33 ·····{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}}
34 ············································|·-1·|········|·-1·|34 ············································|·-3·|········|··1·|
35 ············································|··2·|········|··2·|35 ············································|··1·|········|··1·|
  
36 o5·:·List36 o5·:·List
  
37 i6·:·37 i6·:·
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./usr/share/doc/Macaulay2/WeylGroups/example-output/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.out
    
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1 --·-*-·M2-comint·-*-·hash:·13307763636601 --·-*-·M2-comint·-*-·hash:·1330776363660
  
2 i1·:·listWeylGroupElements(rootSystemG2,4)2 i1·:·listWeylGroupElements(rootSystemG2,4)
  
3 o1·=·{WeylGroupElement{RootSystem{...8...},·|··4·|},3 o1·=·{WeylGroupElement{RootSystem{...8...},·|·-5·|},
4 ············································|·-3·|··4 ············································|··2·|··
5 ·····------------------------------------------------------------------------5 ·····------------------------------------------------------------------------
6 ·····WeylGroupElement{RootSystem{...8...},·|·-5·|}}6 ·····WeylGroupElement{RootSystem{...8...},·|··4·|}}
7 ···········································|··2·|7 ···········································|·-3·|
  
8 o1·:·List8 o1·:·List
  
9 i2·:·9 i2·:·
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./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_cm__Parabolic_rp.out
    
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10 o2·=·set·{1,·2}10 o2·=·set·{1,·2}
  
11 o2·:·Parabolic11 o2·:·Parabolic
  
12 i3·:·positiveRoots(R,P)12 i3·:·positiveRoots(R,P)
  
13 o3·=·set·{|··1·|,·|··2·|,·|·-1·|}13 o3·=·set·{|·-1·|,·|··1·|,·|··2·|}
 14 ··········|··2·|··|··1·|··|·-1·|
14 ··········|··1·|··|·-1·|··|··2·|15 ··········|·-1·|··|·-1·|··|··0·|
15 ··········|·-1·|··|··0·|··|·-1·| 
16 ··········|··0·|··|··0·|··|··0·|16 ··········|··0·|··|··0·|··|··0·|
  
17 o3·:·Set17 o3·:·Set
  
18 i4·:·18 i4·:·
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./usr/share/doc/Macaulay2/WeylGroups/example-output/_under__Bruhat_lp__Basic__List_rp.out
    
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36 ···········································|·-2·|36 ···········································|·-2·|
37 ···········································|·-1·|37 ···········································|·-1·|
  
38 o3·:·List38 o3·:·List
  
39 i4·:·underBruhat(L1)39 i4·:·underBruhat(L1)
  
40 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··0·|},·{2,·|··240 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,·|·-1
 41 ·············································|·-1·|········|··1·|·······|··2
41 ·············································|·-3·|········|·-1·|·······|·-142 ·············································|·-2·|········|··1·|·······|·-1
42 ·············································|··1·|········|··2·|·······|··0 
43 ·····------------------------------------------------------------------------43 ·····------------------------------------------------------------------------
44 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,44 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},·{1,
45 ·····|············································|·-1·|········|··1·|······ 
46 ·····|············································|·-2·|········|··1·|······45 ·····|············································|··2·|········|··1·|······
 46 ·····|············································|·-1·|········|·-1·|······
47 ·····------------------------------------------------------------------------47 ·····------------------------------------------------------------------------
48 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},48 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},
49 ·····|··2·|············································|··2·|········|··1·|·· 
50 ·····|·-1·|············································|·-1·|········|·-1·|··49 ·····|·-1·|············································|·-1·|········|··2·|··
 50 ·····|··0·|············································|··2·|········|·-1·|··
51 ·····------------------------------------------------------------------------51 ·····------------------------------------------------------------------------
52 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-152 ·····{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,·|··0
53 ·········|·-1·|············································|·-1·|········|··2 
54 ·········|··0·|············································|··2·|········|·-153 ·········|··1·|············································|··2·|········|·-1
 54 ·········|·-1·|············································|·-3·|········|··2
55 ·····------------------------------------------------------------------------55 ·····------------------------------------------------------------------------
56 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,56 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,
57 ·····|·······|··1·|············································|··2·|······· 
58 ·····|·······|·-1·|············································|·-3·|·······57 ·····|·······|··1·|············································|·-3·|·······
 58 ·····|·······|··1·|············································|··1·|·······
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····|··0·|},·{2,·|·-1·|}}}}60 ·····|··0·|},·{2,·|··2·|}}}}
61 ·····|·-1·|·······|··1·|61 ·····|·-1·|·······|·-1·|
62 ·····|··2·|·······|··1·|62 ·····|··2·|·······|··0·|
  
63 o4·:·List63 o4·:·List
  
64 i5·:·64 i5·:·
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./usr/share/doc/Macaulay2/WeylGroups/html/_above__Bruhat_lp__Basic__List_rp.html
    
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110 ···········································|·-1·|110 ···········································|·-1·|
  
111 o3·:·List</code></pre>111 o3·:·List</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i4·:·aboveBruhat(L1)114 <td>··············<pre><code·class="language-macaulay2">i4·:·aboveBruhat(L1)
  
115 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},·{1,·|··2115 o4·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},·{1,·|··1
 116 ·············································|··1·|········|··2·|·······|··1
116 ·············································|·-2·|········|··1·|·······|·-1117 ·············································|··2·|········|·-1·|·······|·-1
117 ·············································|··3·|········|·-1·|·······|··0 
118 ·····------------------------------------------------------------------------118 ·····------------------------------------------------------------------------
119 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,119 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··0·|},·{2,
120 ·····|············································|··1·|········|··1·|······120 ·····|············································|··3·|········|·-1·|······
121 ·····|············································|·-2·|········|··1·|······121 ·····|············································|·-1·|········|··2·|······
122 ·····------------------------------------------------------------------------122 ·····------------------------------------------------------------------------
123 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,·|··0·|},123 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},
124 ·····|··2·|············································|·-2·|········|·-1·|··124 ·····|·-1·|············································|·-2·|········|··1·|··
125 ·····|·-1·|············································|··1·|········|··2·|··125 ·····|··0·|············································|··3·|········|·-1·|··
126 ·····------------------------------------------------------------------------126 ·····------------------------------------------------------------------------
127 ·····{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1127 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1
128 ·········|··1·|············································|··1·|········|··2128 ·········|·-1·|············································|··1·|········|··1
129 ·········|··1·|············································|··2·|········|·-1129 ·········|··0·|············································|·-2·|········|··1
130 ·····------------------------------------------------------------------------130 ·····------------------------------------------------------------------------
131 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,131 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,
132 ·····|·······|··1·|············································|··3·|·······132 ·····|·······|··2·|············································|·-2·|·······
133 ·····|·······|·-1·|············································|·-1·|·······133 ·····|·······|·-1·|············································|··1·|·······
134 ·····------------------------------------------------------------------------134 ·····------------------------------------------------------------------------
135 ·····|··0·|},·{2,·|··2·|}}}}135 ·····|··0·|},·{2,·|·-1·|}}}}
136 ·····|·-1·|·······|·-1·|136 ·····|·-1·|·······|··1·|
137 ·····|··2·|·······|··0·|137 ·····|··2·|·······|··1·|
  
138 o4·:·List</code></pre>138 o4·:·List</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ········</table>140 ········</table>
141 ······</div>141 ······</div>
142 ······<div·class="waystouse">142 ······<div·class="waystouse">
143 ········<h2>Ways·to·use·this·method:</h2>143 ········<h2>Ways·to·use·this·method:</h2>
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49 ·····WeylGroupElement{RootSystem{...8...},·|··1·|}}49 ·····WeylGroupElement{RootSystem{...8...},·|··1·|}}
50 ···········································|··2·|50 ···········································|··2·|
51 ···········································|·-1·|51 ···········································|·-1·|
  
52 o3·:·List52 o3·:·List
53 i4·:·aboveBruhat(L1)53 i4·:·aboveBruhat(L1)
  
54 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},·{1,·|··254 o4·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},·{1,·|··1
 55 ·············································|··1·|········|··2·|·······|··1
55 ·············································|·-2·|········|··1·|·······|·-156 ·············································|··2·|········|·-1·|·······|·-1
56 ·············································|··3·|········|·-1·|·······|··0 
57 ·····------------------------------------------------------------------------57 ·····------------------------------------------------------------------------
58 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,58 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··0·|},·{2,
59 ·····|············································|··1·|········|··1·|59 ·····|············································|··3·|········|·-1·|
60 ·····|············································|·-2·|········|··1·|60 ·····|············································|·-1·|········|··2·|
61 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
62 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,·|··0·|},62 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··1·|},
63 ·····|··2·|············································|·-2·|········|·-1·|63 ·····|·-1·|············································|·-2·|········|··1·|
64 ·····|·-1·|············································|··1·|········|··2·|64 ·····|··0·|············································|··3·|········|·-1·|
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ·····{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-166 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1
67 ·········|··1·|············································|··1·|········|··267 ·········|·-1·|············································|··1·|········|··1
68 ·········|··1·|············································|··2·|········|·-168 ·········|··0·|············································|·-2·|········|··1
69 ·····------------------------------------------------------------------------69 ·····------------------------------------------------------------------------
70 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,70 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··3·|},·{{1,
71 ·····|·······|··1·|············································|··3·|71 ·····|·······|··2·|············································|·-2·|
72 ·····|·······|·-1·|············································|·-1·|72 ·····|·······|·-1·|············································|··1·|
73 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
74 ·····|··0·|},·{2,·|··2·|}}}}74 ·····|··0·|},·{2,·|·-1·|}}}}
75 ·····|·-1·|·······|·-1·|75 ·····|·-1·|·······|··1·|
76 ·····|··2·|·······|··0·|76 ·····|··2·|·······|··1·|
  
77 o4·:·List77 o4·:·List
78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
79 ····*·_\x8a_\x8b_\x8o_\x8v_\x8e_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t_\x8)·--·The·Weyl·group·elements·just·under·the·ones·in79 ····*·_\x8a_\x8b_\x8o_\x8v_\x8e_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t_\x8)·--·The·Weyl·group·elements·just·under·the·ones·in
80 ······the·list·for·the·Bruhat·order80 ······the·list·for·the·Bruhat·order
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96 ···········································|··1·|96 ···········································|··1·|
  
97 o3·:·WeylGroupElement</code></pre>97 o3·:·WeylGroupElement</code></pre>
98 </td>··········</tr>98 </td>··········</tr>
99 ··········<tr>99 ··········<tr>
100 <td>··············<pre><code·class="language-macaulay2">i4·:·myInterval=intervalBruhat(w1,w2)100 <td>··············<pre><code·class="language-macaulay2">i4·:·myInterval=intervalBruhat(w1,w2)
  
101 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··0·|},·{1,·|·-1·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{}}}} 
102 ··························································|·-2·|········|·-1·|·······|··2·|··············································|·-3·|········|··1·|············································|·-1·|········|·-1·|··············································|·-1·| 
103 ··························································|··1·|········|··2·|·······|·-1·|··············································|··1·|········|··1·|············································|··2·|········|··2·|···································[·...·truncated·by·diffoscope;·len:·17,·SHA:·5ebd410378763a0377949c88c0bf38c02207bbfbe2135b8d5c0d6fe2c1deafe7·...·]101 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|·-1·|},·{1,·|··0·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}},·{{WeylGroupElement{RootSystem[·...·truncated·by·diffoscope;·len:·25,·SHA:·681e071fee4a8cf2af531899a8824f2395fb7ebd9aba6210a84a4873acf6536f·...·]
 102 ··························································|·-2·|········|··2·|·······|·-1·|··············································|·-1·|········|·-1·|············································|·-3·|········|··1·|··············································|·-1·|
 103 ··························································|··1·|········|·-1·|·······|··2·|··············································|··2·|········|··2·|············································|··1·|········|··1·|··············································|··3·|
  
104 o4·:·HasseDiagram</code></pre>104 o4·:·HasseDiagram</code></pre>
105 </td>··········</tr>105 </td>··········</tr>
106 ········</table>106 ········</table>
107 ········<div>107 ········<div>
108 ··········<p>Each·row·of·the·Hasse·diagram·contains·the·elements·of·a·certain·length·together·with·their·links·to·the·next·row.</p>108 ··········<p>Each·row·of·the·Hasse·diagram·contains·the·elements·of·a·certain·length·together·with·their·links·to·the·next·row.</p>
109 ········</div>109 ········</div>
110 ········<table·class="examples">110 ········<table·class="examples">
111 ··········<tr>111 ··········<tr>
112 <td>··············<pre><code·class="language-macaulay2">i5·:·myInterval#1112 <td>··············<pre><code·class="language-macaulay2">i5·:·myInterval#1
  
113 o5·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}},113 o5·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}},
114 ·············································|·-3·|········|··1·|····114 ·············································|·-1·|········|·-1·|····
115 ·············································|··1·|········|··1·|····115 ·············································|··2·|········|··2·|····
116 ·····------------------------------------------------------------------------116 ·····------------------------------------------------------------------------
117 ·····{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}}117 ·····{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}}
118 ············································|·-1·|········|·-1·|118 ············································|·-3·|········|··1·|
119 ············································|··2·|········|··2·|119 ············································|··1·|········|··1·|
  
120 o5·:·List</code></pre>120 o5·:·List</code></pre>
121 </td>··········</tr>121 </td>··········</tr>
122 ········</table>122 ········</table>
123 ······</div>123 ······</div>
124 ······<div·class="waystouse">124 ······<div·class="waystouse">
125 ········<h2>Ways·to·use·this·method:</h2>125 ········<h2>Ways·to·use·this·method:</h2>
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37 o3·=·WeylGroupElement{RootSystem{...8...},·|·-1·|}37 o3·=·WeylGroupElement{RootSystem{...8...},·|·-1·|}
38 ···········································|·-2·|38 ···········································|·-2·|
39 ···········································|··1·|39 ···········································|··1·|
  
40 o3·:·WeylGroupElement40 o3·:·WeylGroupElement
41 i4·:·myInterval=intervalBruhat(w1,w2)41 i4·:·myInterval=intervalBruhat(w1,w2)
  
42 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|··042 o4·=·HasseDiagram{{{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{0,·|·-
43 |},·{1,·|·-1·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-43 1·|},·{1,·|··0·|}}}},·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|
44 1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}},·{44 0·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}},·{
45 {WeylGroupElement{RootSystem{...8...},·|·-1·|},·{}}}}45 {WeylGroupElement{RootSystem{...8...},·|·-1·|},·{}}}}
46 ··························································|·-2·|········|·-1·|46 ··························································|·-2·|········|··2·|
47 |··2·|··············································|·-3·|········|··1·|47 |·-1·|··············································|·-1·|········|·-1·|
48 |·-1·|········|·-1·|··············································|·-1·|48 |·-3·|········|··1·|··············································|·-1·|
49 ··························································|··1·|········|··2·|49 ··························································|··1·|········|·-1·|
50 |·-1·|··············································|··1·|········|··1·|50 |··2·|··············································|··2·|········|··2·|
51 |··2·|········|··2·|··············································|··3·|51 |··1·|········|··1·|··············································|··3·|
  
52 o4·:·HasseDiagram52 o4·:·HasseDiagram
53 Each·row·of·the·Hasse·diagram·contains·the·elements·of·a·certain·length53 Each·row·of·the·Hasse·diagram·contains·the·elements·of·a·certain·length
54 together·with·their·links·to·the·next·row.54 together·with·their·links·to·the·next·row.
55 i5·:·myInterval#155 i5·:·myInterval#1
  
56 o5·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}},56 o5·=·{{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}},
57 ·············································|·-3·|········|··1·|57 ·············································|·-1·|········|·-1·|
58 ·············································|··1·|········|··1·|58 ·············································|··2·|········|··2·|
59 ·····------------------------------------------------------------------------59 ·····------------------------------------------------------------------------
60 ·····{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|··0·|}}}}60 ·····{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|·-1·|}}}}
61 ············································|·-1·|········|·-1·|61 ············································|·-3·|········|··1·|
62 ············································|··2·|········|··2·|62 ············································|··1·|········|··1·|
  
63 o5·:·List63 o5·:·List
64 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*64 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
65 ····*·_\x8i_\x8n_\x8t_\x8e_\x8r_\x8v_\x8a_\x8l_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8,_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8)·--·elements·between·two65 ····*·_\x8i_\x8n_\x8t_\x8e_\x8r_\x8v_\x8a_\x8l_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8,_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8)·--·elements·between·two
66 ······given·ones·for·the·Bruhat·order·on·a·Weyl·group66 ······given·ones·for·the·Bruhat·order·on·a·Weyl·group
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71 ······</div>71 ······</div>
72 ······<div>72 ······<div>
73 ········<h2>Description</h2>73 ········<h2>Description</h2>
74 ········<table·class="examples">74 ········<table·class="examples">
75 ··········<tr>75 ··········<tr>
76 <td>··············<pre><code·class="language-macaulay2">i1·:·listWeylGroupElements(rootSystemG2,4)76 <td>··············<pre><code·class="language-macaulay2">i1·:·listWeylGroupElements(rootSystemG2,4)
  
77 o1·=·{WeylGroupElement{RootSystem{...8...},·|··4·|},77 o1·=·{WeylGroupElement{RootSystem{...8...},·|·-5·|},
78 ············································|·-3·|··78 ············································|··2·|··
79 ·····------------------------------------------------------------------------79 ·····------------------------------------------------------------------------
80 ·····WeylGroupElement{RootSystem{...8...},·|·-5·|}}80 ·····WeylGroupElement{RootSystem{...8...},·|··4·|}}
81 ···········································|··2·|81 ···········································|·-3·|
  
82 o1·:·List</code></pre>82 o1·:·List</code></pre>
83 </td>··········</tr>83 </td>··········</tr>
84 ········</table>84 ········</table>
85 ······</div>85 ······</div>
86 ······<div·class="waystouse">86 ······<div·class="waystouse">
87 ········<h2>Ways·to·use·this·method:</h2>87 ········<h2>Ways·to·use·this·method:</h2>
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13 ··········o·R,·an·instance·of·the·type·_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m,13 ··········o·R,·an·instance·of·the·type·_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m,
14 ··········o·k,·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,14 ··········o·k,·an·_\x8i_\x8n_\x8t_\x8e_\x8g_\x8e_\x8r,
15 ····*·Outputs:15 ····*·Outputs:
16 ··········o·a·_\x8l_\x8i_\x8s_\x8t,·of·all·elements·of·length·k·in·the·Weyl·group·of·R16 ··········o·a·_\x8l_\x8i_\x8s_\x8t,·of·all·elements·of·length·k·in·the·Weyl·group·of·R
17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*17 *\x8**\x8**\x8**\x8**\x8*·D\x8De\x8es\x8sc\x8cr\x8ri\x8ip\x8pt\x8ti\x8io\x8on\x8n·*\x8**\x8**\x8**\x8**\x8*
18 i1·:·listWeylGroupElements(rootSystemG2,4)18 i1·:·listWeylGroupElements(rootSystemG2,4)
  
19 o1·=·{WeylGroupElement{RootSystem{...8...},·|··4·|},19 o1·=·{WeylGroupElement{RootSystem{...8...},·|·-5·|},
20 ············································|·-3·|20 ············································|··2·|
21 ·····------------------------------------------------------------------------21 ·····------------------------------------------------------------------------
22 ·····WeylGroupElement{RootSystem{...8...},·|·-5·|}}22 ·····WeylGroupElement{RootSystem{...8...},·|··4·|}}
23 ···········································|··2·|23 ···········································|·-3·|
  
24 o1·:·List24 o1·:·List
25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*25 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
26 ····*·_\x8l_\x8i_\x8s_\x8t_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8s_\x8(_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m_\x8,_\x8Z_\x8Z_\x8)·--·list·all·elements·of·a·given26 ····*·_\x8l_\x8i_\x8s_\x8t_\x8W_\x8e_\x8y_\x8l_\x8G_\x8r_\x8o_\x8u_\x8p_\x8E_\x8l_\x8e_\x8m_\x8e_\x8n_\x8t_\x8s_\x8(_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m_\x8,_\x8Z_\x8Z_\x8)·--·list·all·elements·of·a·given
27 ······length·in·a·Weyl·group27 ······length·in·a·Weyl·group
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85 o2·=·set·{1,·2}85 o2·=·set·{1,·2}
  
86 o2·:·Parabolic</code></pre>86 o2·:·Parabolic</code></pre>
87 </td>··········</tr>87 </td>··········</tr>
88 ··········<tr>88 ··········<tr>
89 <td>··············<pre><code·class="language-macaulay2">i3·:·positiveRoots(R,P)89 <td>··············<pre><code·class="language-macaulay2">i3·:·positiveRoots(R,P)
  
90 o3·=·set·{|··1·|,·|··2·|,·|·-1·|}90 o3·=·set·{|·-1·|,·|··1·|,·|··2·|}
 91 ··········|··2·|··|··1·|··|·-1·|
91 ··········|··1·|··|·-1·|··|··2·|92 ··········|·-1·|··|·-1·|··|··0·|
92 ··········|·-1·|··|··0·|··|·-1·| 
93 ··········|··0·|··|··0·|··|··0·|93 ··········|··0·|··|··0·|··|··0·|
  
94 o3·:·Set</code></pre>94 o3·:·Set</code></pre>
95 </td>··········</tr>95 </td>··········</tr>
96 ········</table>96 ········</table>
97 ······</div>97 ······</div>
98 ······<div·class="waystouse">98 ······<div·class="waystouse">
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24 i2·:·P=parabolic(R,set{1,2})24 i2·:·P=parabolic(R,set{1,2})
  
25 o2·=·set·{1,·2}25 o2·=·set·{1,·2}
  
26 o2·:·Parabolic26 o2·:·Parabolic
27 i3·:·positiveRoots(R,P)27 i3·:·positiveRoots(R,P)
  
28 o3·=·set·{|··1·|,·|··2·|,·|·-1·|}28 o3·=·set·{|·-1·|,·|··1·|,·|··2·|}
 29 ··········|··2·|··|··1·|··|·-1·|
29 ··········|··1·|··|·-1·|··|··2·|30 ··········|·-1·|··|·-1·|··|··0·|
30 ··········|·-1·|··|··0·|··|·-1·| 
31 ··········|··0·|··|··0·|··|··0·|31 ··········|··0·|··|··0·|··|··0·|
  
32 o3·:·Set32 o3·:·Set
33 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*33 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
34 ····*·_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8v_\x8e_\x8R_\x8o_\x8o_\x8t_\x8s_\x8(_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m_\x8,_\x8P_\x8a_\x8r_\x8a_\x8b_\x8o_\x8l_\x8i_\x8c_\x8)·--·the·set·of·all·positive·roots·in·a34 ····*·_\x8p_\x8o_\x8s_\x8i_\x8t_\x8i_\x8v_\x8e_\x8R_\x8o_\x8o_\x8t_\x8s_\x8(_\x8R_\x8o_\x8o_\x8t_\x8S_\x8y_\x8s_\x8t_\x8e_\x8m_\x8,_\x8P_\x8a_\x8r_\x8a_\x8b_\x8o_\x8l_\x8i_\x8c_\x8)·--·the·set·of·all·positive·roots·in·a
35 ······parabolic·sugroups35 ······parabolic·sugroups
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110 ···········································|·-1·|110 ···········································|·-1·|
  
111 o3·:·List</code></pre>111 o3·:·List</code></pre>
112 </td>··········</tr>112 </td>··········</tr>
113 ··········<tr>113 ··········<tr>
114 <td>··············<pre><code·class="language-macaulay2">i4·:·underBruhat(L1)114 <td>··············<pre><code·class="language-macaulay2">i4·:·underBruhat(L1)
  
115 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··0·|},·{2,·|··2115 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,·|·-1
 116 ·············································|·-1·|········|··1·|·······|··2
116 ·············································|·-3·|········|·-1·|·······|·-1117 ·············································|·-2·|········|··1·|·······|·-1
117 ·············································|··1·|········|··2·|·······|··0 
118 ·····------------------------------------------------------------------------118 ·····------------------------------------------------------------------------
119 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,119 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},·{1,
120 ·····|············································|·-1·|········|··1·|······ 
121 ·····|············································|·-2·|········|··1·|······120 ·····|············································|··2·|········|··1·|······
 121 ·····|············································|·-1·|········|·-1·|······
122 ·····------------------------------------------------------------------------122 ·····------------------------------------------------------------------------
123 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},123 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},
124 ·····|··2·|············································|··2·|········|··1·|·· 
125 ·····|·-1·|············································|·-1·|········|·-1·|··124 ·····|·-1·|············································|·-1·|········|··2·|··
 125 ·····|··0·|············································|··2·|········|·-1·|··
126 ·····------------------------------------------------------------------------126 ·····------------------------------------------------------------------------
127 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1127 ·····{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,·|··0
128 ·········|·-1·|············································|·-1·|········|··2 
129 ·········|··0·|············································|··2·|········|·-1128 ·········|··1·|············································|··2·|········|·-1
 129 ·········|·-1·|············································|·-3·|········|··2
130 ·····------------------------------------------------------------------------130 ·····------------------------------------------------------------------------
131 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,131 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,
132 ·····|·······|··1·|············································|··2·|······· 
133 ·····|·······|·-1·|············································|·-3·|·······132 ·····|·······|··1·|············································|·-3·|·······
 133 ·····|·······|··1·|············································|··1·|·······
134 ·····------------------------------------------------------------------------134 ·····------------------------------------------------------------------------
135 ·····|··0·|},·{2,·|·-1·|}}}}135 ·····|··0·|},·{2,·|··2·|}}}}
136 ·····|·-1·|·······|··1·|136 ·····|·-1·|·······|·-1·|
137 ·····|··2·|·······|··1·|137 ·····|··2·|·······|··0·|
  
138 o4·:·List</code></pre>138 o4·:·List</code></pre>
139 </td>··········</tr>139 </td>··········</tr>
140 ········</table>140 ········</table>
141 ······</div>141 ······</div>
142 ······<div·class="waystouse">142 ······<div·class="waystouse">
143 ········<h2>Ways·to·use·this·method:</h2>143 ········<h2>Ways·to·use·this·method:</h2>
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49 ·····WeylGroupElement{RootSystem{...8...},·|··1·|}}49 ·····WeylGroupElement{RootSystem{...8...},·|··1·|}}
50 ···········································|·-2·|50 ···········································|·-2·|
51 ···········································|·-1·|51 ···········································|·-1·|
  
52 o3·:·List52 o3·:·List
53 i4·:·underBruhat(L1)53 i4·:·underBruhat(L1)
  
54 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,·|··0·|},·{2,·|··254 o4·=·{{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,·|·-1
 55 ·············································|·-1·|········|··1·|·······|··2
55 ·············································|·-3·|········|·-1·|·······|·-156 ·············································|·-2·|········|··1·|·······|·-1
56 ·············································|··1·|········|··2·|·······|··0 
57 ·····------------------------------------------------------------------------57 ·····------------------------------------------------------------------------
58 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|··2·|},·{{1,·|·-1·|},·{2,58 ·····|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},·{1,
59 ·····|············································|·-1·|········|··1·| 
60 ·····|············································|·-2·|········|··1·|59 ·····|············································|··2·|········|··1·|
 60 ·····|············································|·-1·|········|·-1·|
61 ·····------------------------------------------------------------------------61 ·····------------------------------------------------------------------------
62 ·····|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-3·|},·{{0,·|··1·|},62 ·····|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-1·|},
63 ·····|··2·|············································|··2·|········|··1·| 
64 ·····|·-1·|············································|·-1·|········|·-1·|63 ·····|·-1·|············································|·-1·|········|··2·|
 64 ·····|··0·|············································|··2·|········|·-1·|
65 ·····------------------------------------------------------------------------65 ·····------------------------------------------------------------------------
66 ·····{1,·|··2·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-2·|},·{{0,·|·-166 ·····{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,·|··0
67 ·········|·-1·|············································|·-1·|········|··2 
68 ·········|··0·|············································|··2·|········|·-167 ·········|··1·|············································|··2·|········|·-1
 68 ·········|·-1·|············································|·-3·|········|··2
69 ·····------------------------------------------------------------------------69 ·····------------------------------------------------------------------------
70 ·····|},·{1,·|··1·|}}},·{WeylGroupElement{RootSystem{...8...},·|·-1·|},·{{1,70 ·····|},·{2,·|·-1·|}}},·{WeylGroupElement{RootSystem{...8...},·|··1·|},·{{0,
71 ·····|·······|··1·|············································|··2·| 
72 ·····|·······|·-1·|············································|·-3·|71 ·····|·······|··1·|············································|·-3·|
 72 ·····|·······|··1·|············································|··1·|
73 ·····------------------------------------------------------------------------73 ·····------------------------------------------------------------------------
74 ·····|··0·|},·{2,·|·-1·|}}}}74 ·····|··0·|},·{2,·|··2·|}}}}
75 ·····|·-1·|·······|··1·|75 ·····|·-1·|·······|·-1·|
76 ·····|··2·|·······|··1·|76 ·····|··2·|·······|··0·|
  
77 o4·:·List77 o4·:·List
78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*78 *\x8**\x8**\x8**\x8**\x8*·W\x8Wa\x8ay\x8ys\x8s·t\x8to\x8o·u\x8us\x8se\x8e·t\x8th\x8hi\x8is\x8s·m\x8me\x8et\x8th\x8ho\x8od\x8d:\x8:·*\x8**\x8**\x8**\x8**\x8*
79 ····*·_\x8u_\x8n_\x8d_\x8e_\x8r_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t_\x8)·--·Weyl·group·elements·just·under·the·ones·in·the79 ····*·_\x8u_\x8n_\x8d_\x8e_\x8r_\x8B_\x8r_\x8u_\x8h_\x8a_\x8t_\x8(_\x8B_\x8a_\x8s_\x8i_\x8c_\x8L_\x8i_\x8s_\x8t_\x8)·--·Weyl·group·elements·just·under·the·ones·in·the
80 ······list·for·the·Bruhat·order80 ······list·for·the·Bruhat·order
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./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
2 #:version=1.12 #:version=1.1
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9 #:len=10379 #:len=1037
10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh10 bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh
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./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out
    
Offset 90, 41 lines modifiedOffset 90, 41 lines modified
90 i22·:·peek·last·ms90 i22·:·peek·last·ms
  
91 o22·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}91 o22·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
92 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}92 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
93 ·······················2·=>·{ideal·0}93 ·······················2·=>·{ideal·0}
  
94 i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")94 i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")
95 ·--·used·1.7596s·(cpu);·1.00549s·(thread);·0s·(gc)95 ·--·used·1.96326s·(cpu);·1.04666s·(thread);·0s·(gc)
  
96 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}96 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
97 o23·:·List97 o23·:·List
  
98 i24·:·peek·last·ms98 i24·:·peek·last·ms
  
99 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}99 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
100 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}100 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
101 ·······················2·=>·{ideal·0}101 ·······················2·=>·{ideal·0}
  
102 i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")102 i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")
103 ·--·used·17.9126s·(cpu);·4.00306s·(thread);·0s·(gc)103 ·--·used·16.555s·(cpu);·3.93355s·(thread);·0s·(gc)
  
104 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}104 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
105 o25·:·List105 o25·:·List
  
106 i26·:·peek·last·ms106 i26·:·peek·last·ms
  
107 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}107 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
108 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}108 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
109 ·······················2·=>·{ideal·0}109 ·······················2·=>·{ideal·0}
  
110 i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")110 i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")
111 ·--·used·20.423s·(cpu);·4.67834s·(thread);·0s·(gc)111 ·--·used·19.2878s·(cpu);·4.41291s·(thread);·0s·(gc)
  
112 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}112 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
113 o27·:·List113 o27·:·List
  
114 i28·:·peek·last·ms114 i28·:·peek·last·ms
  
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./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html
    
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222 ········</table>222 ········</table>
223 ········<div>223 ········<div>
224 ··········<p>Finally·we·remark·that·the·option:·StratsToFind,·may·be·used·with·this·function,·but·should·only·be·used·with·care.·The·default·setting·is·StratsToFind=>&quot;all&quot;,·and·this·is·the·only·value·of·the·option·which·is·guaranteed·to·compute·the·complete·stratification,·the·other·options·may·fail·to·find·all·strata·but·are·provided·to·allow·the·user·to·obtain·partial·information·on·larger·examples·which·may·take·too·long·to·run·on·the·default·&quot;all&quot;·setting.·The·other·possible·values·are·StratsToFind=>&quot;singularOnly&quot;,·and·StratsToFind=>&quot;most&quot;.·The·option··StratsToFind=>&quot;singularOnly&quot;·is·the·fastest,·but·also·the·most·likely·to·return·incomplete·answers,·and·hence·the·output·of·this·command·should·be·treated·as·a·partial·answer·only.·The·option·StratsToFind=>&quot;most&quot;·will·most·often·get·the·full·answer,·but·can·miss·strata,·so·again·the·output·should·be·treated·as·a·partial·answer.·In·the·example·below·all·options·return·the·complete·answer,·but·only·the·output·with·StratsToFind=>&quot;all&quot;·should·be·considered·complete;·StratsToFind=>&quot;all&quot;·is·run·when·no·option·is·given.</p>224 ··········<p>Finally·we·remark·that·the·option:·StratsToFind,·may·be·used·with·this·function,·but·should·only·be·used·with·care.·The·default·setting·is·StratsToFind=>&quot;all&quot;,·and·this·is·the·only·value·of·the·option·which·is·guaranteed·to·compute·the·complete·stratification,·the·other·options·may·fail·to·find·all·strata·but·are·provided·to·allow·the·user·to·obtain·partial·information·on·larger·examples·which·may·take·too·long·to·run·on·the·default·&quot;all&quot;·setting.·The·other·possible·values·are·StratsToFind=>&quot;singularOnly&quot;,·and·StratsToFind=>&quot;most&quot;.·The·option··StratsToFind=>&quot;singularOnly&quot;·is·the·fastest,·but·also·the·most·likely·to·return·incomplete·answers,·and·hence·the·output·of·this·command·should·be·treated·as·a·partial·answer·only.·The·option·StratsToFind=>&quot;most&quot;·will·most·often·get·the·full·answer,·but·can·miss·strata,·so·again·the·output·should·be·treated·as·a·partial·answer.·In·the·example·below·all·options·return·the·complete·answer,·but·only·the·output·with·StratsToFind=>&quot;all&quot;·should·be·considered·complete;·StratsToFind=>&quot;all&quot;·is·run·when·no·option·is·given.</p>
225 ········</div>225 ········</div>
226 ········<table·class="examples">226 ········<table·class="examples">
227 ··········<tr>227 ··········<tr>
228 <td>··············<pre><code·class="language-macaulay2">i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;singularOnly&quot;)228 <td>··············<pre><code·class="language-macaulay2">i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;singularOnly&quot;)
229 ·--·used·1.7596s·(cpu);·1.00549s·(thread);·0s·(gc)229 ·--·used·1.96326s·(cpu);·1.04666s·(thread);·0s·(gc)
  
230 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}230 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
231 o23·:·List</code></pre>231 o23·:·List</code></pre>
232 </td>··········</tr>232 </td>··········</tr>
233 ··········<tr>233 ··········<tr>
234 <td>··············<pre><code·class="language-macaulay2">i24·:·peek·last·ms234 <td>··············<pre><code·class="language-macaulay2">i24·:·peek·last·ms
  
235 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}235 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
236 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}236 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
237 ·······················2·=>·{ideal·0}</code></pre>237 ·······················2·=>·{ideal·0}</code></pre>
238 </td>··········</tr>238 </td>··········</tr>
239 ··········<tr>239 ··········<tr>
240 <td>··············<pre><code·class="language-macaulay2">i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;most&quot;)240 <td>··············<pre><code·class="language-macaulay2">i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;most&quot;)
241 ·--·used·17.9126s·(cpu);·4.00306s·(thread);·0s·(gc)241 ·--·used·16.555s·(cpu);·3.93355s·(thread);·0s·(gc)
  
242 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}242 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
243 o25·:·List</code></pre>243 o25·:·List</code></pre>
244 </td>··········</tr>244 </td>··········</tr>
245 ··········<tr>245 ··········<tr>
246 <td>··············<pre><code·class="language-macaulay2">i26·:·peek·last·ms246 <td>··············<pre><code·class="language-macaulay2">i26·:·peek·last·ms
  
247 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}247 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
248 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}248 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
249 ·······················2·=>·{ideal·0}</code></pre>249 ·······················2·=>·{ideal·0}</code></pre>
250 </td>··········</tr>250 </td>··········</tr>
251 ··········<tr>251 ··········<tr>
252 <td>··············<pre><code·class="language-macaulay2">i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;all&quot;)252 <td>··············<pre><code·class="language-macaulay2">i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>&quot;all&quot;)
253 ·--·used·20.423s·(cpu);·4.67834s·(thread);·0s·(gc)253 ·--·used·19.2878s·(cpu);·4.41291s·(thread);·0s·(gc)
  
254 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}254 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
255 o27·:·List</code></pre>255 o27·:·List</code></pre>
256 </td>··········</tr>256 </td>··········</tr>
257 ··········<tr>257 ··········<tr>
258 <td>··············<pre><code·class="language-macaulay2">i28·:·peek·last·ms258 <td>··············<pre><code·class="language-macaulay2">i28·:·peek·last·ms
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html2text {}
    
Offset 149, 37 lines modifiedOffset 149, 37 lines modified
149 this·command·should·be·treated·as·a·partial·answer·only.·The·option149 this·command·should·be·treated·as·a·partial·answer·only.·The·option
150 StratsToFind=>"most"·will·most·often·get·the·full·answer,·but·can·miss·strata,150 StratsToFind=>"most"·will·most·often·get·the·full·answer,·but·can·miss·strata,
151 so·again·the·output·should·be·treated·as·a·partial·answer.·In·the·example·below151 so·again·the·output·should·be·treated·as·a·partial·answer.·In·the·example·below
152 all·options·return·the·complete·answer,·but·only·the·output·with152 all·options·return·the·complete·answer,·but·only·the·output·with
153 StratsToFind=>"all"·should·be·considered·complete;·StratsToFind=>"all"·is·run153 StratsToFind=>"all"·should·be·considered·complete;·StratsToFind=>"all"·is·run
154 when·no·option·is·given.154 when·no·option·is·given.
155 i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")155 i23·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"singularOnly")
156 ·--·used·1.7596s·(cpu);·1.00549s·(thread);·0s·(gc)156 ·--·used·1.96326s·(cpu);·1.04666s·(thread);·0s·(gc)
  
157 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}157 o23·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
158 o23·:·List158 o23·:·List
159 i24·:·peek·last·ms159 i24·:·peek·last·ms
  
160 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}160 o24·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
161 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}161 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
162 ·······················2·=>·{ideal·0}162 ·······················2·=>·{ideal·0}
163 i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")163 i25·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"most")
164 ·--·used·17.9126s·(cpu);·4.00306s·(thread);·0s·(gc)164 ·--·used·16.555s·(cpu);·3.93355s·(thread);·0s·(gc)
  
165 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}165 o25·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
166 o25·:·List166 o25·:·List
167 i26·:·peek·last·ms167 i26·:·peek·last·ms
  
168 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}168 o26·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
169 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}169 ·······················1·=>·{ideal·P,·ideal·M1,·ideal(4M1·-·P)}
170 ·······················2·=>·{ideal·0}170 ·······················2·=>·{ideal·0}
171 i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")171 i27·:·time·ms=mapStratify(F,Xh,ideal(0_S),StratsToFind=>"all")
172 ·--·used·20.423s·(cpu);·4.67834s·(thread);·0s·(gc)172 ·--·used·19.2878s·(cpu);·4.41291s·(thread);·0s·(gc)
  
173 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}173 o27·=·{MutableHashTable{...5...},·MutableHashTable{...3...}}
  
174 o27·:·List174 o27·:·List
175 i28·:·peek·last·ms175 i28·:·peek·last·ms
  
176 o28·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}176 o28·=·MutableHashTable{0·=>·{ideal·(P,·M1)}····················}
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./usr/share/doc/Macaulay2/gfanInterface/dump/rawdocumentation.dump
    
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1 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Sun·Feb··9·10:54:37·20251 #·GDBM·dump·file·created·by·GDBM·version·1.24.·02/07/2024·on·Mon·Feb·10·12:54:37·2025
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11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp11 ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp
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./usr/share/doc/Macaulay2/gfanInterface/example-output/___Installation_spand_sp__Configuration_spof_spgfan__Interface.out
    
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17 i4·:·prefixDirectory·|·currentLayout#"programs"17 i4·:·prefixDirectory·|·currentLayout#"programs"
  
18 o4·=·/usr/x86_64-Linux-18 o4·=·/usr/x86_64-Linux-
19 ·····Debian-13/libexec/Macaulay2/bin/19 ·····Debian-13/libexec/Macaulay2/bin/
  
20 i5·:·loadPackage("gfanInterface",·Configuration·=>·{·"keepfiles"·=>·true,·"verbose"·=>·true},·Reload·=>·true);20 i5·:·loadPackage("gfanInterface",·Configuration·=>·{·"keepfiles"·=>·true,·"verbose"·=>·true},·Reload·=>·true);
21 ·--·running:·/usr/bin/gfan·gfan·--help·<·/tmp/M2-3757455-0/17221 ·--·running:·/usr/bin/gfan·gfan·--help·<·/tmp/M2-2737956-0/172
22 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial·ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By·default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the·ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done·up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting·Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it·will·compute·one·using·Buchberger's·algorithm.22 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial·ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By·default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the·ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done·up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting·Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it·will·compute·one·using·Buchberger's·algorithm.
23 Options:23 Options:
24 -g:24 -g:
25 ·Tells·the·program·that·the·input·is·already·a·Groebner·basis·(with·the·initial·term·of·each·polynomial·being·the·first·ones·listed).·Use·this·option·if·it·takes·too·much·time·to·compute·the·starting·(standard·degree·lexicographic)·Groebner·basis·and·the·input·is·already·a·Groebner·basis.25 ·Tells·the·program·that·the·input·is·already·a·Groebner·basis·(with·the·initial·term·of·each·polynomial·being·the·first·ones·listed).·Use·this·option·if·it·takes·too·much·time·to·compute·the·starting·(standard·degree·lexicographic)·Groebner·basis·and·the·input·is·already·a·Groebner·basis.
  
26 --symmetry:26 --symmetry:
27 ·Tells·the·program·to·read·in·generators·for·a·group·of·symmetries·(subgroup·of·$S_n$)·after·having·read·in·the·ideal.·The·program·checks·that·the·ideal·stays·fixed·when·permuting·the·variables·with·respect·to·elements·in·the·group.·The·program·uses·breadth·first·search·to·compute·the·set·of·reduced·Groebner·bases·up·to·symmetry·with·respect·to·the·specified·subgroup.27 ·Tells·the·program·to·read·in·generators·for·a·group·of·symmetries·(subgroup·of·$S_n$)·after·having·read·in·the·ideal.·The·program·checks·that·the·ideal·stays·fixed·when·permuting·the·variables·with·respect·to·elements·in·the·group.·The·program·uses·breadth·first·search·to·compute·the·set·of·reduced·Groebner·bases·up·to·symmetry·with·respect·to·the·specified·subgroup.
Offset 36, 16 lines modifiedOffset 36, 16 lines modified
36 --disableSymmetryTest:36 --disableSymmetryTest:
37 ·When·using·--symmetry·this·option·will·disable·the·check·that·the·group·read·off·from·the·input·actually·is·a·symmetry·group·with·respect·to·the·input·ideal.37 ·When·using·--symmetry·this·option·will·disable·the·check·that·the·group·read·off·from·the·input·actually·is·a·symmetry·group·with·respect·to·the·input·ideal.
  
38 --parameters·value:38 --parameters·value:
39 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.39 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
40 --interrupt·value:40 --interrupt·value:
41 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been·computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for·non-symmetric·traversals·or·for·traversals·restricted·to·fans).41 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been·computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for·non-symmetric·traversals·or·for·traversals·restricted·to·fans).
42 using·temporary·file·/tmp/M2-3757455-0/17242 using·temporary·file·/tmp/M2-2737956-0/172
43 ·--·running:·/usr/bin/gfan·_buchberger·--help·<·/tmp/M2-3757455-0/17443 ·--·running:·/usr/bin/gfan·_buchberger·--help·<·/tmp/M2-2737956-0/174
44 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial·ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.·The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q[a,b,c,...]).44 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial·ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.·The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q[a,b,c,...]).
45 Options:45 Options:
46 -w:46 -w:
47 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with·$a>b>c...$·instead.·The·degrees·are·given·by·a·weight·vector·which·is·read·from·the·input·after·the·generating·set·has·been·read.47 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with·$a>b>c...$·instead.·The·degrees·are·given·by·a·weight·vector·which·is·read·from·the·input·after·the·generating·set·has·been·read.
  
48 -r:48 -r:
49 ·Use·the·reverse·lexicographic·order·(or·the·reverse·lexicographic·order·as·a·tie·breaker·if·-w·is·used).·The·input·must·be·homogeneous·if·the·pure·reverse·lexicographic·order·is·chosen.·Ignored·if·-W·is·used.49 ·Use·the·reverse·lexicographic·order·(or·the·reverse·lexicographic·order·as·a·tie·breaker·if·-w·is·used).·The·input·must·be·homogeneous·if·the·pure·reverse·lexicographic·order·is·chosen.·Ignored·if·-W·is·used.
Offset 54, 69 lines modifiedOffset 54, 69 lines modified
54 ·Do·a·Groebner·walk.·The·input·must·be·a·minimal·Groebner·basis.·If·-W·is·used·-w·is·ignored.54 ·Do·a·Groebner·walk.·The·input·must·be·a·minimal·Groebner·basis.·If·-W·is·used·-w·is·ignored.
  
55 -g:55 -g:
56 ·Do·a·generic·Groebner·walk.·The·input·must·be·homogeneous·and·must·be·a·minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.·The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.56 ·Do·a·generic·Groebner·walk.·The·input·must·be·homogeneous·and·must·be·a·minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.·The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.
  
57 --parameters·value:57 --parameters·value:
58 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.58 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
59 using·temporary·file·/tmp/M2-3757455-0/17459 using·temporary·file·/tmp/M2-2737956-0/174
60 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·<·/tmp/M2-3757455-0/17660 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·<·/tmp/M2-2737956-0/176
61 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of·polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by·applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and·0·for·false.61 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of·polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by·applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and·0·for·false.
62 Options:62 Options:
63 --remainder:63 --remainder:
64 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than·outputting·0·or·1.64 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than·outputting·0·or·1.
65 --multiplier:65 --multiplier:
66 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided·before·doing·the·division.66 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided·before·doing·the·division.
67 using·temporary·file·/tmp/M2-3757455-0/17667 using·temporary·file·/tmp/M2-2737956-0/176
68 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·<·/tmp/M2-3757455-0/17868 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·<·/tmp/M2-2737956-0/178
69 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.69 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.
70 Options:70 Options:
71 -i1·value:71 -i1·value:
72 ·Specify·the·name·of·the·first·input·file.72 ·Specify·the·name·of·the·first·input·file.
73 -i2·value:73 -i2·value:
74 ·Specify·the·name·of·the·second·input·file.74 ·Specify·the·name·of·the·second·input·file.
75 --stable:75 --stable:
76 ·Compute·the·stable·intersection.76 ·Compute·the·stable·intersection.
77 using·temporary·file·/tmp/M2-3757455-0/17877 using·temporary·file·/tmp/M2-2737956-0/178
78 ·--·running:·/usr/bin/gfan·_fanlink·--help·<·/tmp/M2-3757455-0/18078 ·--·running:·/usr/bin/gfan·_fanlink·--help·<·/tmp/M2-2737956-0/180
79 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the·polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension·equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan·containing·the·vector.79 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the·polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension·equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan·containing·the·vector.
80 Options:80 Options:
81 -i·value:81 -i·value:
82 ·Specify·the·name·of·the·input·file.82 ·Specify·the·name·of·the·input·file.
83 --symmetry:83 --symmetry:
84 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must·be·given·on·the·standard·input.84 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must·be·given·on·the·standard·input.
  
85 --star:85 --star:
86 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan·containing·all·cones·of·the·original·fan·containing·the·vector.86 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan·containing·all·cones·of·the·original·fan·containing·the·vector.
87 using·temporary·file·/tmp/M2-3757455-0/18087 using·temporary·file·/tmp/M2-2737956-0/180
88 ·--·running:·/usr/bin/gfan·_fanproduct·--help·<·/tmp/M2-3757455-0/18288 ·--·running:·/usr/bin/gfan·_fanproduct·--help·<·/tmp/M2-2737956-0/182
89 This·program·takes·two·polyhedral·fans·and·computes·their·product.89 This·program·takes·two·polyhedral·fans·and·computes·their·product.
90 Options:90 Options:
91 -i1·value:91 -i1·value:
92 ·Specify·the·name·of·the·first·input·file.92 ·Specify·the·name·of·the·first·input·file.
93 -i2·value:93 -i2·value:
94 ·Specify·the·name·of·the·second·input·file.94 ·Specify·the·name·of·the·second·input·file.
95 using·temporary·file·/tmp/M2-3757455-0/18295 using·temporary·file·/tmp/M2-2737956-0/182
96 ·--·running:·/usr/bin/gfan·_groebnercone·--help·<·/tmp/M2-3757455-0/18496 ·--·running:·/usr/bin/gfan·_groebnercone·--help·<·/tmp/M2-2737956-0/184
97 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The·input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is·computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone·computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones·are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case·is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of·Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.·The·facets·of·the·cone·can·be·read·off·in·section·FACETS·and·the·equations·in·section·IMPLIED_EQUATIONS.97 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The·input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is·computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone·computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones·are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case·is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of·Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.·The·facets·of·the·cone·can·be·read·off·in·section·FACETS·and·the·equations·in·section·IMPLIED_EQUATIONS.
98 Options:98 Options:
99 --restrict:99 --restrict:
100 ·Add·an·inequality·for·each·coordinate,·so·that·the·the·cone·is·restricted·to·the·non-negative·orthant.100 ·Add·an·inequality·for·each·coordinate,·so·that·the·the·cone·is·restricted·to·the·non-negative·orthant.
101 --pair:101 --pair:
102 ·The·Groebner·cone·is·given·by·a·pair·of·compatible·Groebner·bases.·The·first·basis·is·for·the·initial·ideal·and·the·second·for·the·ideal·itself.·See·the·tropical·section·of·the·manual.102 ·The·Groebner·cone·is·given·by·a·pair·of·compatible·Groebner·bases.·The·first·basis·is·for·the·initial·ideal·and·the·second·for·the·ideal·itself.·See·the·tropical·section·of·the·manual.
103 --asfan:103 --asfan:
104 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way·the·extreme·rays·of·the·cone·are·also·computed.104 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way·the·extreme·rays·of·the·cone·are·also·computed.
105 --vectorinput:105 --vectorinput:
106 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The·input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list·of·inequalities·given·as·vectors·and·a·list·of·equations.106 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The·input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list·of·inequalities·given·as·vectors·and·a·list·of·equations.
107 using·temporary·file·/tmp/M2-3757455-0/184107 using·temporary·file·/tmp/M2-2737956-0/184
108 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·<·/tmp/M2-3757455-0/186108 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·<·/tmp/M2-2737956-0/186
109 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a·cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section·LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first·compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A·reduced·Groebner·basis·will·always·suffice·for·this·purpose.109 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a·cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section·LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first·compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A·reduced·Groebner·basis·will·always·suffice·for·this·purpose.
110 Options:110 Options:
111 using·temporary·file·/tmp/M2-3757455-0/186111 using·temporary·file·/tmp/M2-2737956-0/186
112 ·--·running:·/usr/bin/gfan·_homogenize·--help·<·/tmp/M2-3757455-0/188112 ·--·running:·/usr/bin/gfan·_homogenize·--help·<·/tmp/M2-2737956-0/188
113 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra·variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input·after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done·with·respect·to·total·degree.113 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra·variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input·after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done·with·respect·to·total·degree.
114 Example:114 Example:
115 Input:115 Input:
116 Q[x,y]{y-1}116 Q[x,y]{y-1}
117 z117 z
118 Output:118 Output:
119 Q[x,y,z]{y-z}119 Q[x,y,z]{y-z}
Offset 124, 30 lines modifiedOffset 124, 30 lines modified
124 -i:124 -i:
125 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·homogenisation·of·the·input·ideal.·This·is·done·by·computing·a·degree·Groebner·basis·and·homogenising·it.125 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·homogenisation·of·the·input·ideal.·This·is·done·by·computing·a·degree·Groebner·basis·and·homogenising·it.
126 -w:126 -w:
127 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as·the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after·the·list·of·polynomials.127 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as·the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after·the·list·of·polynomials.
  
128 -H:128 -H:
129 ·Let·the·name·of·the·new·variable·be·H·rather·than·reading·in·a·name·from·the·input.129 ·Let·the·name·of·the·new·variable·be·H·rather·than·reading·in·a·name·from·the·input.
130 using·temporary·file·/tmp/M2-3757455-0/188130 using·temporary·file·/tmp/M2-2737956-0/188
131 ·--·running:·/usr/bin/gfan·_initialforms·--help·<·/tmp/M2-3757455-0/190131 ·--·running:·/usr/bin/gfan·_initialforms·--help·<·/tmp/M2-2737956-0/190
132 This·program·converts·a·list·of·polynomials·to·a·list·of·their·initial·forms·with·respect·to·the·vector·given·after·the·list.132 This·program·converts·a·list·of·polynomials·to·a·list·of·their·initial·forms·with·respect·to·the·vector·given·after·the·list.
133 Options:133 Options:
134 --ideal:134 --ideal:
135 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·initial·ideal·of·the·ideal·generated·by·the·input·polynomials.·The·computation·is·done·by·computing·a·Groebner·basis·with·respect·to·the·given·vector.·The·vector·must·be·positive·or·the·input·polynomials·must·be·homogeneous·in·a·positive·grading.·None·of·these·conditions·are·checked·by·the·program.135 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·initial·ideal·of·the·ideal·generated·by·the·input·polynomials.·The·computation·is·done·by·computing·a·Groebner·basis·with·respect·to·the·given·vector.·The·vector·must·be·positive·or·the·input·polynomials·must·be·homogeneous·in·a·positive·grading.·None·of·these·conditions·are·checked·by·the·program.
  
136 --pair:136 --pair:
137 ·Produce·a·pair·of·polynomial·lists.·Used·together·with·--ideal·this·option·will·also·write·a·compatible·reduced·Groebner·basis·for·the·input·ideal·to·the·output.·This·is·useful·for·finding·the·Groebner·cone·of·a·non-monomial·initial·ideal.137 ·Produce·a·pair·of·polynomial·lists.·Used·together·with·--ideal·this·option·will·also·write·a·compatible·reduced·Groebner·basis·for·the·input·ideal·to·the·output.·This·is·useful·for·finding·the·Groebner·cone·of·a·non-monomial·initial·ideal.
  
138 --mark:138 --mark:
139 ·If·the·--pair·option·is·and·the·--ideal·option·is·not·used·this·option·will·still·make·sure·that·the·second·output·basis·is·marked·consistently·with·the·vector.139 ·If·the·--pair·option·is·and·the·--ideal·option·is·not·used·this·option·will·still·make·sure·that·the·second·output·basis·is·marked·consistently·with·the·vector.
140 --list:140 --list:
141 ·Read·in·a·list·of·vectors·instead·of·a·single·vector·and·produce·a·list·of·polynomial·sets·as·output.141 ·Read·in·a·list·of·vectors·instead·of·a·single·vector·and·produce·a·list·of·polynomial·sets·as·output.
142 using·temporary·file·/tmp/M2-3757455-0/190142 using·temporary·file·/tmp/M2-2737956-0/190
143 ·--·running:·/usr/bin/gfan·_interactive·--help·<·/tmp/M2-3757455-0/192143 ·--·running:·/usr/bin/gfan·_interactive·--help·<·/tmp/M2-2737956-0/192
144 This·is·a·program·for·doing·interactive·walks·in·the·Groebner·fan·of·an·ideal.·The·input·is·a·Groebner·basis·defining·the·starting·Groebner·cone·of·the·walk.·The·program·will·list·all·flippable·facets·of·the·Groebner·cone·and·ask·the·user·to·choose·one.·The·user·types·in·the·index·(number)·of·the·facet·in·the·list.·The·program·will·walk·through·the·selected·facet·and·display·the·new·Groebner·basis·and·a·list·of·new·facet·normals·for·the·user·to·choose·from.·Since·the·program·reads·the·user's·choices·through·the·the·standard·input·it·is·recommended·not·to·redirect·the·standard·input·for·this·program.144 This·is·a·program·for·doing·interactive·walks·in·the·Groebner·fan·of·an·ideal.·The·input·is·a·Groebner·basis·defining·the·starting·Groebner·cone·of·the·walk.·The·program·will·list·all·flippable·facets·of·the·Groebner·cone·and·ask·the·user·to·choose·one.·The·user·types·in·the·index·(number)·of·the·facet·in·the·list.·The·program·will·walk·through·the·selected·facet·and·display·the·new·Groebner·basis·and·a·list·of·new·facet·normals·for·the·user·to·choose·from.·Since·the·program·reads·the·user's·choices·through·the·the·standard·input·it·is·recommended·not·to·redirect·the·standard·input·for·this·program.
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./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html
    
Offset 91, 15 lines modifiedOffset 91, 15 lines modified
91 ········<div>91 ········<div>
92 ··········<p>If·you·would·like·to·see·the·input·and·output·files·used·to·communicate·with·<span·class="tt">gfan</span>·you·can·set·the·<span·class="tt">&quot;keepfiles&quot;</span>·configuration·option·to·<span·class="tt">true</span>.·If·<span·class="tt">&quot;verbose&quot;</span>·is·set·to·<span·class="tt">true</span>,·<span·class="tt">gfanInterface</span>·will·output·the·names·of·the·temporary·files·used.</p>92 ··········<p>If·you·would·like·to·see·the·input·and·output·files·used·to·communicate·with·<span·class="tt">gfan</span>·you·can·set·the·<span·class="tt">&quot;keepfiles&quot;</span>·configuration·option·to·<span·class="tt">true</span>.·If·<span·class="tt">&quot;verbose&quot;</span>·is·set·to·<span·class="tt">true</span>,·<span·class="tt">gfanInterface</span>·will·output·the·names·of·the·temporary·files·used.</p>
93 ··········<p></p>93 ··········<p></p>
94 ········</div>94 ········</div>
95 ········<table·class="examples">95 ········<table·class="examples">
96 ··········<tr>96 ··········<tr>
97 <td>··············<pre><code·class="language-macaulay2">i5·:·loadPackage(&quot;gfanInterface&quot;,·Configuration·=>·{·&quot;keepfiles&quot;·=>·true,·&quot;verbose&quot;·=>·true},·Reload·=>·true);97 <td>··············<pre><code·class="language-macaulay2">i5·:·loadPackage(&quot;gfanInterface&quot;,·Configuration·=>·{·&quot;keepfiles&quot;·=>·true,·&quot;verbose&quot;·=>·true},·Reload·=>·true);
98 ·--·running:·/usr/bin/gfan·gfan·--help·&lt;·/tmp/M2-3757455-0/17298 ·--·running:·/usr/bin/gfan·gfan·--help·&lt;·/tmp/M2-2737956-0/172
99 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial·ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By·default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the·ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done·up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting·Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it·will·compute·one·using·Buchberger's·algorithm.99 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial·ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By·default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the·ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done·up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting·Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it·will·compute·one·using·Buchberger's·algorithm.
100 Options:100 Options:
101 -g:101 -g:
102 ·Tells·the·program·that·the·input·is·already·a·Groebner·basis·(with·the·initial·term·of·each·polynomial·being·the·first·ones·listed).·Use·this·option·if·it·takes·too·much·time·to·compute·the·starting·(standard·degree·lexicographic)·Groebner·basis·and·the·input·is·already·a·Groebner·basis.102 ·Tells·the·program·that·the·input·is·already·a·Groebner·basis·(with·the·initial·term·of·each·polynomial·being·the·first·ones·listed).·Use·this·option·if·it·takes·too·much·time·to·compute·the·starting·(standard·degree·lexicographic)·Groebner·basis·and·the·input·is·already·a·Groebner·basis.
  
103 --symmetry:103 --symmetry:
104 ·Tells·the·program·to·read·in·generators·for·a·group·of·symmetries·(subgroup·of·$S_n$)·after·having·read·in·the·ideal.·The·program·checks·that·the·ideal·stays·fixed·when·permuting·the·variables·with·respect·to·elements·in·the·group.·The·program·uses·breadth·first·search·to·compute·the·set·of·reduced·Groebner·bases·up·to·symmetry·with·respect·to·the·specified·subgroup.104 ·Tells·the·program·to·read·in·generators·for·a·group·of·symmetries·(subgroup·of·$S_n$)·after·having·read·in·the·ideal.·The·program·checks·that·the·ideal·stays·fixed·when·permuting·the·variables·with·respect·to·elements·in·the·group.·The·program·uses·breadth·first·search·to·compute·the·set·of·reduced·Groebner·bases·up·to·symmetry·with·respect·to·the·specified·subgroup.
Offset 110, 16 lines modifiedOffset 110, 16 lines modified
110 --disableSymmetryTest:110 --disableSymmetryTest:
111 ·When·using·--symmetry·this·option·will·disable·the·check·that·the·group·read·off·from·the·input·actually·is·a·symmetry·group·with·respect·to·the·input·ideal.111 ·When·using·--symmetry·this·option·will·disable·the·check·that·the·group·read·off·from·the·input·actually·is·a·symmetry·group·with·respect·to·the·input·ideal.
  
112 --parameters·value:112 --parameters·value:
113 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.113 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
114 --interrupt·value:114 --interrupt·value:
115 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been·computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for·non-symmetric·traversals·or·for·traversals·restricted·to·fans).115 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been·computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for·non-symmetric·traversals·or·for·traversals·restricted·to·fans).
116 using·temporary·file·/tmp/M2-3757455-0/172116 using·temporary·file·/tmp/M2-2737956-0/172
117 ·--·running:·/usr/bin/gfan·_buchberger·--help·&lt;·/tmp/M2-3757455-0/174117 ·--·running:·/usr/bin/gfan·_buchberger·--help·&lt;·/tmp/M2-2737956-0/174
118 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial·ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.·The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q[a,b,c,...]).118 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial·ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.·The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q[a,b,c,...]).
119 Options:119 Options:
120 -w:120 -w:
121 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with·$a>b>c...$·instead.·The·degrees·are·given·by·a·weight·vector·which·is·read·from·the·input·after·the·generating·set·has·been·read.121 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with·$a>b>c...$·instead.·The·degrees·are·given·by·a·weight·vector·which·is·read·from·the·input·after·the·generating·set·has·been·read.
  
122 -r:122 -r:
123 ·Use·the·reverse·lexicographic·order·(or·the·reverse·lexicographic·order·as·a·tie·breaker·if·-w·is·used).·The·input·must·be·homogeneous·if·the·pure·reverse·lexicographic·order·is·chosen.·Ignored·if·-W·is·used.123 ·Use·the·reverse·lexicographic·order·(or·the·reverse·lexicographic·order·as·a·tie·breaker·if·-w·is·used).·The·input·must·be·homogeneous·if·the·pure·reverse·lexicographic·order·is·chosen.·Ignored·if·-W·is·used.
Offset 128, 69 lines modifiedOffset 128, 69 lines modified
128 ·Do·a·Groebner·walk.·The·input·must·be·a·minimal·Groebner·basis.·If·-W·is·used·-w·is·ignored.128 ·Do·a·Groebner·walk.·The·input·must·be·a·minimal·Groebner·basis.·If·-W·is·used·-w·is·ignored.
  
129 -g:129 -g:
130 ·Do·a·generic·Groebner·walk.·The·input·must·be·homogeneous·and·must·be·a·minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.·The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.130 ·Do·a·generic·Groebner·walk.·The·input·must·be·homogeneous·and·must·be·a·minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.·The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.
  
131 --parameters·value:131 --parameters·value:
132 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.132 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters·instead·of·variables.·This·makes·it·possible·to·do·computations·where·the·coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
133 using·temporary·file·/tmp/M2-3757455-0/174133 using·temporary·file·/tmp/M2-2737956-0/174
134 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·&lt;·/tmp/M2-3757455-0/176134 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·&lt;·/tmp/M2-2737956-0/176
135 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of·polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by·applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and·0·for·false.135 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of·polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by·applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and·0·for·false.
136 Options:136 Options:
137 --remainder:137 --remainder:
138 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than·outputting·0·or·1.138 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than·outputting·0·or·1.
139 --multiplier:139 --multiplier:
140 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided·before·doing·the·division.140 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided·before·doing·the·division.
141 using·temporary·file·/tmp/M2-3757455-0/176141 using·temporary·file·/tmp/M2-2737956-0/176
142 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·&lt;·/tmp/M2-3757455-0/178142 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·&lt;·/tmp/M2-2737956-0/178
143 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.143 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.
144 Options:144 Options:
145 -i1·value:145 -i1·value:
146 ·Specify·the·name·of·the·first·input·file.146 ·Specify·the·name·of·the·first·input·file.
147 -i2·value:147 -i2·value:
148 ·Specify·the·name·of·the·second·input·file.148 ·Specify·the·name·of·the·second·input·file.
149 --stable:149 --stable:
150 ·Compute·the·stable·intersection.150 ·Compute·the·stable·intersection.
151 using·temporary·file·/tmp/M2-3757455-0/178151 using·temporary·file·/tmp/M2-2737956-0/178
152 ·--·running:·/usr/bin/gfan·_fanlink·--help·&lt;·/tmp/M2-3757455-0/180152 ·--·running:·/usr/bin/gfan·_fanlink·--help·&lt;·/tmp/M2-2737956-0/180
153 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the·polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension·equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan·containing·the·vector.153 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the·polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension·equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan·containing·the·vector.
154 Options:154 Options:
155 -i·value:155 -i·value:
156 ·Specify·the·name·of·the·input·file.156 ·Specify·the·name·of·the·input·file.
157 --symmetry:157 --symmetry:
158 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must·be·given·on·the·standard·input.158 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must·be·given·on·the·standard·input.
  
159 --star:159 --star:
160 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan·containing·all·cones·of·the·original·fan·containing·the·vector.160 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan·containing·all·cones·of·the·original·fan·containing·the·vector.
161 using·temporary·file·/tmp/M2-3757455-0/180161 using·temporary·file·/tmp/M2-2737956-0/180
162 ·--·running:·/usr/bin/gfan·_fanproduct·--help·&lt;·/tmp/M2-3757455-0/182162 ·--·running:·/usr/bin/gfan·_fanproduct·--help·&lt;·/tmp/M2-2737956-0/182
163 This·program·takes·two·polyhedral·fans·and·computes·their·product.163 This·program·takes·two·polyhedral·fans·and·computes·their·product.
164 Options:164 Options:
165 -i1·value:165 -i1·value:
166 ·Specify·the·name·of·the·first·input·file.166 ·Specify·the·name·of·the·first·input·file.
167 -i2·value:167 -i2·value:
168 ·Specify·the·name·of·the·second·input·file.168 ·Specify·the·name·of·the·second·input·file.
169 using·temporary·file·/tmp/M2-3757455-0/182169 using·temporary·file·/tmp/M2-2737956-0/182
170 ·--·running:·/usr/bin/gfan·_groebnercone·--help·&lt;·/tmp/M2-3757455-0/184170 ·--·running:·/usr/bin/gfan·_groebnercone·--help·&lt;·/tmp/M2-2737956-0/184
171 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The·input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is·computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone·computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones·are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case·is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of·Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.·The·facets·of·the·cone·can·be·read·off·in·section·FACETS·and·the·equations·in·section·IMPLIED_EQUATIONS.171 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The·input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is·computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone·computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones·are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case·is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of·Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.·The·facets·of·the·cone·can·be·read·off·in·section·FACETS·and·the·equations·in·section·IMPLIED_EQUATIONS.
172 Options:172 Options:
173 --restrict:173 --restrict:
174 ·Add·an·inequality·for·each·coordinate,·so·that·the·the·cone·is·restricted·to·the·non-negative·orthant.174 ·Add·an·inequality·for·each·coordinate,·so·that·the·the·cone·is·restricted·to·the·non-negative·orthant.
175 --pair:175 --pair:
176 ·The·Groebner·cone·is·given·by·a·pair·of·compatible·Groebner·bases.·The·first·basis·is·for·the·initial·ideal·and·the·second·for·the·ideal·itself.·See·the·tropical·section·of·the·manual.176 ·The·Groebner·cone·is·given·by·a·pair·of·compatible·Groebner·bases.·The·first·basis·is·for·the·initial·ideal·and·the·second·for·the·ideal·itself.·See·the·tropical·section·of·the·manual.
177 --asfan:177 --asfan:
178 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way·the·extreme·rays·of·the·cone·are·also·computed.178 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way·the·extreme·rays·of·the·cone·are·also·computed.
179 --vectorinput:179 --vectorinput:
180 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The·input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list·of·inequalities·given·as·vectors·and·a·list·of·equations.180 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The·input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list·of·inequalities·given·as·vectors·and·a·list·of·equations.
181 using·temporary·file·/tmp/M2-3757455-0/184181 using·temporary·file·/tmp/M2-2737956-0/184
182 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·&lt;·/tmp/M2-3757455-0/186182 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·&lt;·/tmp/M2-2737956-0/186
183 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a·cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section·LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first·compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A·reduced·Groebner·basis·will·always·suffice·for·this·purpose.183 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a·cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section·LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first·compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A·reduced·Groebner·basis·will·always·suffice·for·this·purpose.
184 Options:184 Options:
185 using·temporary·file·/tmp/M2-3757455-0/186185 using·temporary·file·/tmp/M2-2737956-0/186
186 ·--·running:·/usr/bin/gfan·_homogenize·--help·&lt;·/tmp/M2-3757455-0/188186 ·--·running:·/usr/bin/gfan·_homogenize·--help·&lt;·/tmp/M2-2737956-0/188
187 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra·variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input·after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done·with·respect·to·total·degree.187 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra·variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input·after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done·with·respect·to·total·degree.
188 Example:188 Example:
189 Input:189 Input:
190 Q[x,y]{y-1}190 Q[x,y]{y-1}
191 z191 z
192 Output:192 Output:
193 Q[x,y,z]{y-z}193 Q[x,y,z]{y-z}
Offset 198, 30 lines modifiedOffset 198, 30 lines modified
198 -i:198 -i:
199 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·homogenisation·of·the·input·ideal.·This·is·done·by·computing·a·degree·Groebner·basis·and·homogenising·it.199 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·homogenisation·of·the·input·ideal.·This·is·done·by·computing·a·degree·Groebner·basis·and·homogenising·it.
200 -w:200 -w:
201 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as·the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after·the·list·of·polynomials.201 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as·the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after·the·list·of·polynomials.
  
202 -H:202 -H:
203 ·Let·the·name·of·the·new·variable·be·H·rather·than·reading·in·a·name·from·the·input.203 ·Let·the·name·of·the·new·variable·be·H·rather·than·reading·in·a·name·from·the·input.
204 using·temporary·file·/tmp/M2-3757455-0/188204 using·temporary·file·/tmp/M2-2737956-0/188
205 ·--·running:·/usr/bin/gfan·_initialforms·--help·&lt;·/tmp/M2-3757455-0/190205 ·--·running:·/usr/bin/gfan·_initialforms·--help·&lt;·/tmp/M2-2737956-0/190
206 This·program·converts·a·list·of·polynomials·to·a·list·of·their·initial·forms·with·respect·to·the·vector·given·after·the·list.206 This·program·converts·a·list·of·polynomials·to·a·list·of·their·initial·forms·with·respect·to·the·vector·given·after·the·list.
207 Options:207 Options:
208 --ideal:208 --ideal:
209 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·initial·ideal·of·the·ideal·generated·by·the·input·polynomials.·The·computation·is·done·by·computing·a·Groebner·basis·with·respect·to·the·given·vector.·The·vector·must·be·positive·or·the·input·polynomials·must·be·homogeneous·in·a·positive·grading.·None·of·these·conditions·are·checked·by·the·program.209 ·Treat·input·as·an·ideal.·This·will·make·the·program·compute·the·initial·ideal·of·the·ideal·generated·by·the·input·polynomials.·The·computation·is·done·by·computing·a·Groebner·basis·with·respect·to·the·given·vector.·The·vector·must·be·positive·or·the·input·polynomials·must·be·homogeneous·in·a·positive·grading.·None·of·these·conditions·are·checked·by·the·program.
  
210 --pair:210 --pair:
211 ·Produce·a·pair·of·polynomial·lists.·Used·together·with·--ideal·this·option·will·also·write·a·compatible·reduced·Groebner·basis·for·the·input·ideal·to·the·output.·This·is·useful·for·finding·the·Groebner·cone·of·a·non-monomial·initial·ideal.211 ·Produce·a·pair·of·polynomial·lists.·Used·together·with·--ideal·this·option·will·also·write·a·compatible·reduced·Groebner·basis·for·the·input·ideal·to·the·output.·This·is·useful·for·finding·the·Groebner·cone·of·a·non-monomial·initial·ideal.
  
212 --mark:212 --mark:
213 ·If·the·--pair·option·is·and·the·--ideal·option·is·not·used·this·option·will·still·make·sure·that·the·second·output·basis·is·marked·consistently·with·the·vector.213 ·If·the·--pair·option·is·and·the·--ideal·option·is·not·used·this·option·will·still·make·sure·that·the·second·output·basis·is·marked·consistently·with·the·vector.
214 --list:214 --list:
215 ·Read·in·a·list·of·vectors·instead·of·a·single·vector·and·produce·a·list·of·polynomial·sets·as·output.215 ·Read·in·a·list·of·vectors·instead·of·a·single·vector·and·produce·a·list·of·polynomial·sets·as·output.
216 using·temporary·file·/tmp/M2-3757455-0/190216 using·temporary·file·/tmp/M2-2737956-0/190
217 ·--·running:·/usr/bin/gfan·_interactive·--help·&lt;·/tmp/M2-3757455-0/192217 ·--·running:·/usr/bin/gfan·_interactive·--help·&lt;·/tmp/M2-2737956-0/192
218 This·is·a·program·for·doing·interactive·walks·in·the·Groebner·fan·of·an·ideal.·The·input·is·a·Groebner·basis·defining·the·starting·Groebner·cone·of·the·walk.·The·program·will·list·all·flippable·facets·of·the·Groebner·cone·and·ask·the·user·to·choose·one.·The·user·types·in·the·index·(number)·of·the·facet·in·the·list.·The·program·will·walk·through·the·selected·facet·and·display·the·new·Groebner·basis·and·a·list·of·new·facet·normals·for·the·user·to·choose·from.·Since·the·program·reads·the·user's·choices·through·the·the·standard·input·it·is·recommended·not·to·redirect·the·standard·input·for·this·program.218 This·is·a·program·for·doing·interactive·walks·in·the·Groebner·fan·of·an·ideal.·The·input·is·a·Groebner·basis·defining·the·starting·Groebner·cone·of·the·walk.·The·program·will·list·all·flippable·facets·of·the·Groebner·cone·and·ask·the·user·to·choose·one.·The·user·types·in·the·index·(number)·of·the·facet·in·the·list.·The·program·will·walk·through·the·selected·facet·and·display·the·new·Groebner·basis·and·a·list·of·new·facet·normals·for·the·user·to·choose·from.·Since·the·program·reads·the·user's·choices·through·the·the·standard·input·it·is·recommended·not·to·redirect·the·standard·input·for·this·program.
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39 o4·=·/usr/x86_64-Linux-39 o4·=·/usr/x86_64-Linux-
40 ·····Debian-13/libexec/Macaulay2/bin/40 ·····Debian-13/libexec/Macaulay2/bin/
41 If·you·would·like·to·see·the·input·and·output·files·used·to·communicate·with41 If·you·would·like·to·see·the·input·and·output·files·used·to·communicate·with
42 gfan·you·can·set·the·"keepfiles"·configuration·option·to·true.·If·"verbose"·is42 gfan·you·can·set·the·"keepfiles"·configuration·option·to·true.·If·"verbose"·is
43 set·to·true,·gfanInterface·will·output·the·names·of·the·temporary·files·used.43 set·to·true,·gfanInterface·will·output·the·names·of·the·temporary·files·used.
44 i5·:·loadPackage("gfanInterface",·Configuration·=>·{·"keepfiles"·=>·true,44 i5·:·loadPackage("gfanInterface",·Configuration·=>·{·"keepfiles"·=>·true,
45 "verbose"·=>·true},·Reload·=>·true);45 "verbose"·=>·true},·Reload·=>·true);
46 ·--·running:·/usr/bin/gfan·gfan·--help·<·/tmp/M2-3757455-0/17246 ·--·running:·/usr/bin/gfan·gfan·--help·<·/tmp/M2-2737956-0/172
47 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial47 This·is·a·program·for·computing·all·reduced·Groebner·bases·of·a·polynomial
48 ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By48 ideal.·It·takes·the·ring·and·a·generating·set·for·the·ideal·as·input.·By
49 default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the49 default·the·enumeration·is·done·by·an·almost·memoryless·reverse·search.·If·the
50 ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done50 ideal·is·symmetric·the·symmetry·option·is·useful·and·enumeration·will·be·done
51 up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting51 up·to·symmetry·using·a·breadth·first·search.·The·program·needs·a·starting
52 Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it52 Groebner·basis·to·do·its·computations.·If·the·-g·option·is·not·specified·it
53 will·compute·one·using·Buchberger's·algorithm.53 will·compute·one·using·Buchberger's·algorithm.
Offset 77, 16 lines modifiedOffset 77, 16 lines modified
77 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters77 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters
78 instead·of·variables.·This·makes·it·possible·to·do·computations·where·the78 instead·of·variables.·This·makes·it·possible·to·do·computations·where·the
79 coefficient·field·is·the·field·of·rational·functions·in·the·parameters.79 coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
80 --interrupt·value:80 --interrupt·value:
81 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been81 ·Interrupt·the·enumeration·after·a·specified·number·of·facets·have·been
82 computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for82 computed·(works·for·usual·symmetric·traversals,·but·may·not·work·in·general·for
83 non-symmetric·traversals·or·for·traversals·restricted·to·fans).83 non-symmetric·traversals·or·for·traversals·restricted·to·fans).
84 using·temporary·file·/tmp/M2-3757455-0/17284 using·temporary·file·/tmp/M2-2737956-0/172
85 ·--·running:·/usr/bin/gfan·_buchberger·--help·<·/tmp/M2-3757455-0/17485 ·--·running:·/usr/bin/gfan·_buchberger·--help·<·/tmp/M2-2737956-0/174
86 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial86 This·program·computes·a·reduced·lexicographic·Groebner·basis·of·the·polynomial
87 ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.87 ideal·given·as·input.·The·default·behavior·is·to·use·Buchberger's·algorithm.
88 The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q88 The·ordering·of·the·variables·is·$a>b>c...$·(assuming·that·the·ring·is·Q
89 [a,b,c,...]).89 [a,b,c,...]).
90 Options:90 Options:
91 -w:91 -w:
92 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with92 ·Compute·a·Groebner·basis·with·respect·to·a·degree·lexicographic·order·with
Offset 107, 63 lines modifiedOffset 107, 63 lines modified
107 minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.107 minimal·Groebner·basis·with·respect·to·the·reverse·lexicographic·term·order.
108 The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.108 The·target·term·order·is·always·lexicographic.·The·-W·option·must·be·used.
  
109 --parameters·value:109 --parameters·value:
110 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters110 ·With·this·option·you·can·specify·how·many·variables·to·treat·as·parameters
111 instead·of·variables.·This·makes·it·possible·to·do·computations·where·the111 instead·of·variables.·This·makes·it·possible·to·do·computations·where·the
112 coefficient·field·is·the·field·of·rational·functions·in·the·parameters.112 coefficient·field·is·the·field·of·rational·functions·in·the·parameters.
113 using·temporary·file·/tmp/M2-3757455-0/174113 using·temporary·file·/tmp/M2-2737956-0/174
114 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·<·/tmp/M2-3757455-0/176114 ·--·running:·/usr/bin/gfan·_doesidealcontain·--help·<·/tmp/M2-2737956-0/176
115 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of115 This·program·takes·a·marked·Groebner·basis·of·an·ideal·I·and·a·set·of
116 polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by116 polynomials·on·its·input·and·tests·if·the·polynomial·set·is·contained·in·I·by
117 applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and117 applying·the·division·algorithm·for·each·element.·The·output·is·1·for·true·and
118 0·for·false.118 0·for·false.
119 Options:119 Options:
120 --remainder:120 --remainder:
121 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than121 ·Tell·the·program·to·output·the·remainders·of·the·divisions·rather·than
122 outputting·0·or·1.122 outputting·0·or·1.
123 --multiplier:123 --multiplier:
124 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided124 ·Reads·in·a·polynomial·that·will·be·multiplied·to·the·polynomial·to·be·divided
125 before·doing·the·division.125 before·doing·the·division.
126 using·temporary·file·/tmp/M2-3757455-0/176126 using·temporary·file·/tmp/M2-2737956-0/176
127 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·<·/tmp/M2-3757455-0/178127 ·--·running:·/usr/bin/gfan·_fancommonrefinement·--help·<·/tmp/M2-2737956-0/178
128 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.128 This·program·takes·two·polyhedral·fans·and·computes·their·common·refinement.
129 Options:129 Options:
130 -i1·value:130 -i1·value:
131 ·Specify·the·name·of·the·first·input·file.131 ·Specify·the·name·of·the·first·input·file.
132 -i2·value:132 -i2·value:
133 ·Specify·the·name·of·the·second·input·file.133 ·Specify·the·name·of·the·second·input·file.
134 --stable:134 --stable:
135 ·Compute·the·stable·intersection.135 ·Compute·the·stable·intersection.
136 using·temporary·file·/tmp/M2-3757455-0/178136 using·temporary·file·/tmp/M2-2737956-0/178
137 ·--·running:·/usr/bin/gfan·_fanlink·--help·<·/tmp/M2-3757455-0/180137 ·--·running:·/usr/bin/gfan·_fanlink·--help·<·/tmp/M2-2737956-0/180
138 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the138 This·program·takes·a·polyhedral·fan·and·a·vector·and·computes·the·link·of·the
139 polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension139 polyhedral·fan·around·that·vertex.·The·link·will·have·lineality·space·dimension
140 equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan140 equal·to·the·dimension·of·the·relative·open·polyhedral·cone·of·the·original·fan
141 containing·the·vector.141 containing·the·vector.
142 Options:142 Options:
143 -i·value:143 -i·value:
144 ·Specify·the·name·of·the·input·file.144 ·Specify·the·name·of·the·input·file.
145 --symmetry:145 --symmetry:
146 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must146 ·Reads·in·a·fan·stored·with·symmetry.·The·generators·of·the·symmetry·group·must
147 be·given·on·the·standard·input.147 be·given·on·the·standard·input.
  
148 --star:148 --star:
149 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan149 ·Computes·the·star·instead.·The·star·is·defined·as·the·smallest·polyhedral·fan
150 containing·all·cones·of·the·original·fan·containing·the·vector.150 containing·all·cones·of·the·original·fan·containing·the·vector.
151 using·temporary·file·/tmp/M2-3757455-0/180151 using·temporary·file·/tmp/M2-2737956-0/180
152 ·--·running:·/usr/bin/gfan·_fanproduct·--help·<·/tmp/M2-3757455-0/182152 ·--·running:·/usr/bin/gfan·_fanproduct·--help·<·/tmp/M2-2737956-0/182
153 This·program·takes·two·polyhedral·fans·and·computes·their·product.153 This·program·takes·two·polyhedral·fans·and·computes·their·product.
154 Options:154 Options:
155 -i1·value:155 -i1·value:
156 ·Specify·the·name·of·the·first·input·file.156 ·Specify·the·name·of·the·first·input·file.
157 -i2·value:157 -i2·value:
158 ·Specify·the·name·of·the·second·input·file.158 ·Specify·the·name·of·the·second·input·file.
159 using·temporary·file·/tmp/M2-3757455-0/182159 using·temporary·file·/tmp/M2-2737956-0/182
160 ·--·running:·/usr/bin/gfan·_groebnercone·--help·<·/tmp/M2-3757455-0/184160 ·--·running:·/usr/bin/gfan·_groebnercone·--help·<·/tmp/M2-2737956-0/184
161 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The161 This·program·computes·a·Groebner·cone.·Three·different·cases·are·handled.·The
162 input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is162 input·may·be·a·marked·reduced·Groebner·basis·in·which·case·its·Groebner·cone·is
163 computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone163 computed.·The·input·may·be·just·a·marked·minimal·basis·in·which·case·the·cone
164 computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones164 computed·is·not·a·Groebner·cone·in·the·usual·sense·but·smaller.·(These·cones
165 are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case165 are·described·in·[Fukuda,·Jensen,·Lauritzen,·Thomas]).·The·third·possible·case
166 is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of166 is·that·the·Groebner·cone·is·possibly·lower·dimensional·and·given·by·a·pair·of
167 Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.167 Groebner·bases·as·it·is·useful·to·do·for·tropical·varieties,·see·option·--pair.
Offset 180, 24 lines modifiedOffset 180, 24 lines modified
180 --asfan:180 --asfan:
181 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way181 ·Writes·the·cone·as·a·polyhedral·fan·with·all·its·faces·instead.·In·this·way
182 the·extreme·rays·of·the·cone·are·also·computed.182 the·extreme·rays·of·the·cone·are·also·computed.
183 --vectorinput:183 --vectorinput:
184 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The184 ·Compute·a·cone·given·list·of·inequalities·rather·than·a·Groebner·cone.·The
185 input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list185 input·is·an·integer·which·specifies·the·dimension·of·the·ambient·space,·a·list
186 of·inequalities·given·as·vectors·and·a·list·of·equations.186 of·inequalities·given·as·vectors·and·a·list·of·equations.
187 using·temporary·file·/tmp/M2-3757455-0/184187 using·temporary·file·/tmp/M2-2737956-0/184
188 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·<·/tmp/M2-3757455-0/186188 ·--·running:·/usr/bin/gfan·_homogeneityspace·--help·<·/tmp/M2-2737956-0/186
189 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a189 This·program·computes·the·homogeneity·space·of·a·list·of·polynomials·-·as·a
190 cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section190 cone.·Thus·generators·for·the·homogeneity·space·are·found·in·the·section
191 LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first191 LINEALITY_SPACE.·If·you·wish·the·homogeneity·space·of·an·ideal·you·should·first
192 compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A192 compute·a·set·of·homogeneous·generators·and·call·the·program·on·these.·A
193 reduced·Groebner·basis·will·always·suffice·for·this·purpose.193 reduced·Groebner·basis·will·always·suffice·for·this·purpose.
194 Options:194 Options:
195 using·temporary·file·/tmp/M2-3757455-0/186195 using·temporary·file·/tmp/M2-2737956-0/186
196 ·--·running:·/usr/bin/gfan·_homogenize·--help·<·/tmp/M2-3757455-0/188196 ·--·running:·/usr/bin/gfan·_homogenize·--help·<·/tmp/M2-2737956-0/188
197 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra197 This·program·homogenises·a·list·of·polynomials·by·introducing·an·extra
198 variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input198 variable.·The·name·of·the·variable·to·be·introduced·is·read·from·the·input
199 after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done199 after·the·list·of·polynomials.·Without·the·-w·option·the·homogenisation·is·done
200 with·respect·to·total·degree.200 with·respect·to·total·degree.
201 Example:201 Example:
202 Input:202 Input:
203 Q[x,y]{y-1}203 Q[x,y]{y-1}
Offset 213, 16 lines modifiedOffset 213, 16 lines modified
213 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as213 ·Specify·a·homogenisation·vector.·The·length·of·the·vector·must·be·the·same·as
214 the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after214 the·number·of·variables·in·the·ring.·The·vector·is·read·from·the·input·after
215 the·list·of·polynomials.215 the·list·of·polynomials.
  
216 -H:216 -H:
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00012b90:·6f74·6f70·795f·6c70·5f70·645f·7064·5f70··otopy_lp_pd_pd_p00012b90:·6f74·6f70·795f·6c70·5f70·645f·7064·5f70··otopy_lp_pd_pd_p
00012ba0:·645f·636d·5665·7262·6f73·653d·3e5f·7064··d_cmVerbose=>_pd00012ba0:·645f·636d·5665·7262·6f73·653d·3e5f·7064··d_cmVerbose=>_pd
Offset 5271, 15 lines modifiedOffset 5271, 15 lines modified
00014960:·2020·2020·2020·6f72·2072·616e·646f·6d20········or·random·00014960:·2020·2020·2020·6f72·2072·616e·646f·6d20········or·random·
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000149a0:·7265·6374·6f72·792c·203d·3e20·2e2e·2e2c··rectory,·=>·...,000149a0:·7265·6374·6f72·792c·203d·3e20·2e2e·2e2c··rectory,·=>·...,
000149b0:·2064·6566·6175·6c74·2076·616c·7565·0a20···default·value.·000149b0:·2064·6566·6175·6c74·2076·616c·7565·0a20···default·value.·
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000149d0:·3338·3136·3934·322d·302f·3022·2c20·4f70··3816942-0/0",·Op000149d0:·3237·3839·3033·372d·302f·3022·2c20·4f70··2789037-0/0",·Op
000149e0:·7469·6f6e·2074·6f20·6368·616e·6765·2064··tion·to·change·d000149e0:·7469·6f6e·2074·6f20·6368·616e·6765·2064··tion·to·change·d
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00014a10:·202a·2055·7365·5265·6765·6e65·7261·7469···*·UseRegenerati00014a10:·202a·2055·7365·5265·6765·6e65·7261·7469···*·UseRegenerati
00014a20:·6f6e·2028·6d69·7373·696e·6720·646f·6375··on·(missing·docu00014a20:·6f6e·2028·6d69·7373·696e·6720·646f·6375··on·(missing·docu
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00014a40:·2e2c·2064·6566·6175·6c74·2076·616c·7565··.,·default·value00014a40:·2e2c·2064·6566·6175·6c74·2076·616c·7565··.,·default·value
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BettiCharacters.info
    
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000307a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000307a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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00030810:·202d·2d20·2e33·3435·3835·3973·2065·6c61···--·.345859s·ela00030810:·202d·2d20·2e38·3036·3136·3573·2065·6c61···--·.806165s·ela
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00030880:·2020·2020·2020·2020·2020·2020·2020·2020··················00030880:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 13607, 16 lines modifiedOffset 13607, 16 lines modified
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00035280:·2d2b·0a7c·6937·203a·2065·6c61·7073·6564··-+.|i7·:·elapsed00035280:·2d2b·0a7c·6937·203a·2065·6c61·7073·6564··-+.|i7·:·elapsed
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00035330:·2020·2020·2020·2020·2020·2020·2020·2020··················00035330:·2020·2020·2020·2020·2020·2020·2020·2020··················
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00035350:·2020·2020·2020·2020·2020·2020·2020·2020··················00035350:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 15025, 16 lines modifiedOffset 15025, 16 lines modified
0003ab00:·2020·2020·2020·2020·207c·0a2b·2d2d·2d2d···········|.+----0003ab00:·2020·2020·2020·2020·207c·0a2b·2d2d·2d2d···········|.+----
0003ab10:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0003ab10:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0003ab20:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0003ab20:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0003ab30:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a··--------------+.0003ab30:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a··--------------+.
0003ab40:·7c69·3230·203a·2065·6c61·7073·6564·5469··|i20·:·elapsedTi0003ab40:·7c69·3230·203a·2065·6c61·7073·6564·5469··|i20·:·elapsedTi
0003ab50:·6d65·2061·3120·3d20·6368·6172·6163·7465··me·a1·=·characte0003ab50:·6d65·2061·3120·3d20·6368·6172·6163·7465··me·a1·=·characte
0003ab60:·7220·4131·2020·2020·2020·2020·2020·2020··r·A1············0003ab60:·7220·4131·2020·2020·2020·2020·2020·2020··r·A1············
0003ab70:·2020·207c·0a7c·202d·2d20·2e37·3134·3337·····|.|·--·.714370003ab70:·2020·207c·0a7c·202d·2d20·312e·3230·3736·····|.|·--·1.2076
0003ab80:·3673·2065·6c61·7073·6564·2020·2020·2020··6s·elapsed······0003ab80:·3173·2065·6c61·7073·6564·2020·2020·2020··1s·elapsed······
0003ab90:·2020·2020·2020·2020·2020·2020·2020·2020··················0003ab90:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003aba0:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····0003aba0:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····
0003abb0:·2020·2020·2020·2020·2020·2020·2020·2020··················0003abb0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003abc0:·2020·2020·2020·2020·2020·2020·2020·2020··················0003abc0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003abd0:·2020·2020·2020·2020·2020·2020·207c·0a7c···············|.|0003abd0:·2020·2020·2020·2020·2020·2020·207c·0a7c···············|.|
0003abe0:·6f32·3020·3d20·4368·6172·6163·7465·7220··o20·=·Character·0003abe0:·6f32·3020·3d20·4368·6172·6163·7465·7220··o20·=·Character·
0003abf0:·6f76·6572·2052·2020·2020·2020·2020·2020··over·R··········0003abf0:·6f76·6572·2052·2020·2020·2020·2020·2020··over·R··········
Offset 15065, 15 lines modifiedOffset 15065, 15 lines modified
0003ad80:·2020·2020·207c·0a2b·2d2d·2d2d·2d2d·2d2d·······|.+--------0003ad80:·2020·2020·207c·0a2b·2d2d·2d2d·2d2d·2d2d·······|.+--------
0003ad90:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0003ad90:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0003ada0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0003ada0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0003adb0:·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69·3231··----------+.|i210003adb0:·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69·3231··----------+.|i21
0003adc0:·203a·2065·6c61·7073·6564·5469·6d65·2061···:·elapsedTime·a0003adc0:·203a·2065·6c61·7073·6564·5469·6d65·2061···:·elapsedTime·a
0003add0:·3220·3d20·6368·6172·6163·7465·7220·4132··2·=·character·A20003add0:·3220·3d20·6368·6172·6163·7465·7220·4132··2·=·character·A2
0003ade0:·2020·2020·2020·2020·2020·2020·2020·207c·················|0003ade0:·2020·2020·2020·2020·2020·2020·2020·207c·················|
0003adf0:·0a7c·202d·2d20·3238·2e37·3437·3573·2065··.|·--·28.7475s·e0003adf0:·0a7c·202d·2d20·3336·2e39·3536·3573·2065··.|·--·36.9565s·e
0003ae00:·6c61·7073·6564·2020·2020·2020·2020·2020··lapsed··········0003ae00:·6c61·7073·6564·2020·2020·2020·2020·2020··lapsed··········
0003ae10:·2020·2020·2020·2020·2020·2020·2020·2020··················0003ae10:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003ae20:·2020·2020·7c0a·7c20·2020·2020·2020·2020······|.|·········0003ae20:·2020·2020·7c0a·7c20·2020·2020·2020·2020······|.|·········
0003ae30:·2020·2020·2020·2020·2020·2020·2020·2020··················0003ae30:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003ae40:·2020·2020·2020·2020·2020·2020·2020·2020··················0003ae40:·2020·2020·2020·2020·2020·2020·2020·2020··················
0003ae50:·2020·2020·2020·2020·207c·0a7c·6f32·3120···········|.|o21·0003ae50:·2020·2020·2020·2020·207c·0a7c·6f32·3120···········|.|o21·
0003ae60:·3d20·4368·6172·6163·7465·7220·6f76·6572··=·Character·over0003ae60:·3d20·4368·6172·6163·7465·7220·6f76·6572··=·Character·over
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0003ca20:·203a·2041·6374·696f·6e4f·6e47·7261·6465···:·ActionOnGrade0003ca20:·203a·2041·6374·696f·6e4f·6e47·7261·6465···:·ActionOnGrade
0003ca30:·644d·6f64·756c·6520·2020·2020·2020·2020··dModule·········0003ca30:·644d·6f64·756c·6520·2020·2020·2020·2020··dModule·········
0003ca40:·2020·7c0a·2b2d·2d2d·2d2d·2d2d·2d2d·2d2d····|.+-----------0003ca40:·2020·7c0a·2b2d·2d2d·2d2d·2d2d·2d2d·2d2d····|.+-----------
0003ca50:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0003ca50:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0003ca60:·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69·3332··----------+.|i320003ca60:·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69·3332··----------+.|i32
0003ca70:·203a·2065·6c61·7073·6564·5469·6d65·2062···:·elapsedTime·b0003ca70:·203a·2065·6c61·7073·6564·5469·6d65·2062···:·elapsedTime·b
0003ca80:·203d·2063·6861·7261·6374·6572·2842·2c32···=·character(B,20003ca80:·203d·2063·6861·7261·6374·6572·2842·2c32···=·character(B,2
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0003caa0:·7320·656c·6170·7365·6420·2020·2020·2020··s·elapsed·······0003caa0:·7320·656c·6170·7365·6420·2020·2020·2020··s·elapsed·······
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0003cac0:·2020·2020·2020·2020·2020·2020·2020·2020··················0003cac0:·2020·2020·2020·2020·2020·2020·2020·2020··················
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0003cae0:·2020·7c0a·7c6f·3332·203d·2043·6861·7261····|.|o32·=·Chara0003cae0:·2020·7c0a·7c6f·3332·203d·2043·6861·7261····|.|o32·=·Chara
0003caf0:·6374·6572·206f·7665·7220·5220·2020·2020··cter·over·R·····0003caf0:·6374·6572·206f·7665·7220·5220·2020·2020··cter·over·R·····
0003cb00:·2020·2020·2020·2020·2020·7c0a·7c20·2020············|.|···0003cb00:·2020·2020·2020·2020·2020·7c0a·7c20·2020············|.|···
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00003fa0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00003fa0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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CompleteIntersectionResolutions.info
    
Offset 4102, 16 lines modifiedOffset 4102, 16 lines modified
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Offset 4642, 16 lines modifiedOffset 4642, 16 lines modified
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Offset 4683, 16 lines modifiedOffset 4683, 16 lines modified
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000124f0:·2920·2020·2020·2020·2020·2020·2020·2020··)···············000124f0:·2920·2020·2020·2020·2020·2020·2020·2020··)···············
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00012830:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00012830:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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00012860:·2d2b·0a7c·6932·3220·3a20·4747·203d·2074··-+.|i22·:·GG·=·t00012860:·2d2b·0a7c·6932·3220·3a20·4747·203d·2074··-+.|i22·:·GG·=·t
00012870:·696d·6520·4569·7365·6e62·7564·5368·616d··ime·EisenbudSham00012870:·696d·6520·4569·7365·6e62·7564·5368·616d··ime·EisenbudSham
00012880:·6173·6828·5231·2c46·5b32·5d2c·3429·2020··ash(R1,F[2],4)··00012880:·6173·6828·5231·2c46·5b32·5d2c·3429·2020··ash(R1,F[2],4)··
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000128d0:·0a7c·2020·2020·2020·2020·2020·2020·2020··.|··············000128d0:·0a7c·2020·2020·2020·2020·2020·2020·2020··.|··············
000128e0:·2020·2020·2020·2020·2020·2020·2020·2020··················000128e0:·2020·2020·2020·2020·2020·2020·2020·2020··················
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00012900:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······00012900:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
00012910:·2031·2020·2020·2020·2036·2020·2020·2020···1·······6······00012910:·2031·2020·2020·2020·2036·2020·2020·2020···1·······6······
00012920:·2031·3820·2020·2020·2020·3338·2020·2020···18·······38····00012920:·2031·3820·2020·2020·2020·3338·2020·2020···18·······38····
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0006bf40:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0006bf40:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0006bf50:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0006bf50:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0006bf60:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69··------------+.|i0006bf60:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69··------------+.|i
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0006bf80:·616c·7328·322c·3329·2020·2020·2020·2020··als(2,3)········0006bf80:·616c·7328·322c·3329·2020·2020·2020·2020··als(2,3)········
0006bf90:·2020·2020·2020·2020·2020·2020·2020·2020··················0006bf90:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006bfa0:·2020·2020·2020·2020·7c0a·7c20·2d2d·2075··········|.|·--·u0006bfa0:·2020·2020·2020·2020·7c0a·7c20·2d2d·2075··········|.|·--·u
0006bfb0:·7365·6420·312e·3330·3237·3973·2028·6370··sed·1.30279s·(cp0006bfb0:·7365·6420·312e·3230·3137·7320·2863·7075··sed·1.2017s·(cpu
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0006c020:·7c0a·7c20·2d2d·2075·7365·6420·302e·3030··|.|·--·used·0.000006c020:·7c0a·7c20·2d2d·2075·7365·6420·302e·3030··|.|·--·used·0.00
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0006c040:·312e·3034·3165·2d30·3673·2028·7468·7265··1.041e-06s·(thre0006c040:·322e·3934·3665·2d30·3673·2028·7468·7265··2.946e-06s·(thre
0006c050:·6164·293b·2030·7320·2867·6329·7c0a·7c32··ad);·0s·(gc)|.|20006c050:·6164·293b·2030·7320·2867·6329·7c0a·7c32··ad);·0s·(gc)|.|2
0006c060:·2020·2020·2020·2020·2020·2020·2020·2020··················0006c060:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006c070:·2020·2020·2020·2020·2020·2020·2020·2020··················0006c070:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006c080:·2020·2020·2020·2020·2020·2020·2020·2020··················0006c080:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006c090:·2020·2020·2020·2020·7c0a·7c54·616c·6c79··········|.|Tally0006c090:·2020·2020·2020·2020·7c0a·7c54·616c·6c79··········|.|Tally
0006c0a0:·7b7b·7b32·2c20·327d·2c20·7b31·2c20·327d··{{{2,·2},·{1,·2}0006c0a0:·7b7b·7b32·2c20·327d·2c20·7b31·2c20·327d··{{{2,·2},·{1,·2}
0006c0b0:·7d20·3d3e·2033·7d20·2020·2020·2020·2020··}·=>·3}·········0006c0b0:·7d20·3d3e·2033·7d20·2020·2020·2020·2020··}·=>·3}·········
Offset 28166, 17 lines modifiedOffset 28166, 17 lines modified
0006e050:·2b2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··+---------------0006e050:·2b2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··+---------------
0006e060:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0006e060:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0006e070:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0006e070:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0006e080:·2d2d·2d2d·2d2d·2b0a·7c69·3220·3a20·7477··------+.|i2·:·tw0006e080:·2d2d·2d2d·2d2d·2b0a·7c69·3220·3a20·7477··------+.|i2·:·tw
0006e090:·6f4d·6f6e·6f6d·6961·6c73·2832·2c33·2920··oMonomials(2,3)·0006e090:·6f4d·6f6e·6f6d·6961·6c73·2832·2c33·2920··oMonomials(2,3)·
0006e0a0:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e0a0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e0b0:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.0006e0b0:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.
0006e0c0:·7c20·2d2d·2075·7365·6420·322e·3330·3636··|·--·used·2.30660006e0c0:·7c20·2d2d·2075·7365·6420·322e·3232·3133··|·--·used·2.2213
0006e0d0:·3173·2028·6370·7529·3b20·302e·3730·3939··1s·(cpu);·0.70990006e0d0:·3373·2028·6370·7529·3b20·302e·3738·3631··3s·(cpu);·0.7861
0006e0e0:·3134·7320·2874·6872·6561·6429·3b20·3073··14s·(thread);·0s0006e0e0:·3633·7320·2874·6872·6561·6429·3b20·3073··63s·(thread);·0s
0006e0f0:·2028·6763·2920·7c0a·7c32·2020·2020·2020···(gc)·|.|2······0006e0f0:·2028·6763·2920·7c0a·7c32·2020·2020·2020···(gc)·|.|2······
0006e100:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e100:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e110:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e110:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e120:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.0006e120:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.
0006e130:·7c54·616c·6c79·7b7b·7b31·2c20·317d·7d20··|Tally{{{1,·1}}·0006e130:·7c54·616c·6c79·7b7b·7b31·2c20·317d·7d20··|Tally{{{1,·1}}·
0006e140:·3d3e·2032·2020·2020·2020·2020·7d20·2020··=>·2········}···0006e140:·3d3e·2032·2020·2020·2020·2020·7d20·2020··=>·2········}···
0006e150:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e150:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 28184, 16 lines modifiedOffset 28184, 16 lines modified
0006e170:·7b32·2c20·327d·2c20·7b31·2c20·327d·7d20··{2,·2},·{1,·2}}·0006e170:·7b32·2c20·327d·2c20·7b31·2c20·327d·7d20··{2,·2},·{1,·2}}·
0006e180:·3d3e·2034·2020·2020·2020·2020·2020·2020··=>·4············0006e180:·3d3e·2034·2020·2020·2020·2020·2020·2020··=>·4············
0006e190:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.0006e190:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.
0006e1a0:·7c20·2020·2020·2020·2020·2020·2020·2020··|···············0006e1a0:·7c20·2020·2020·2020·2020·2020·2020·2020··|···············
0006e1b0:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e1b0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e1c0:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e1c0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e1d0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use0006e1d0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
0006e1e0:·6420·312e·3232·3730·3673·2028·6370·7529··d·1.22706s·(cpu)0006e1e0:·6420·312e·3231·3437·3373·2028·6370·7529··d·1.21473s·(cpu)
0006e1f0:·3b20·302e·3431·3739·3035·7320·2874·6872··;·0.417905s·(thr0006e1f0:·3b20·302e·3434·3237·3032·7320·2874·6872··;·0.442702s·(thr
0006e200:·6561·6429·3b20·3073·2028·6763·2920·7c0a··ead);·0s·(gc)·|.0006e200:·6561·6429·3b20·3073·2028·6763·2920·7c0a··ead);·0s·(gc)·|.
0006e210:·7c33·2020·2020·2020·2020·2020·2020·2020··|3··············0006e210:·7c33·2020·2020·2020·2020·2020·2020·2020··|3··············
0006e220:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e220:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e230:·2020·2020·2020·2020·2020·2020·2020·2020··················0006e230:·2020·2020·2020·2020·2020·2020·2020·2020··················
0006e240:·2020·2020·2020·7c0a·7c54·616c·6c79·7b7b········|.|Tally{{0006e240:·2020·2020·2020·7c0a·7c54·616c·6c79·7b7b········|.|Tally{{
0006e250:·7b32·2c20·327d·2c20·7b31·2c20·327d·7d20··{2,·2},·{1,·2}}·0006e250:·7b32·2c20·327d·2c20·7b31·2c20·327d·7d20··{2,·2},·{1,·2}}·
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Offset 1169, 18 lines modifiedOffset 1169, 18 lines modified
00004900:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00004900:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00004910:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2b··---------------+00004910:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2b··---------------+
00004920:·0a7c·6935·203a·2048·4220·3d20·4848·2042··.|i5·:·HB·=·HH·B00004920:·0a7c·6935·203a·2048·4220·3d20·4848·2042··.|i5·:·HB·=·HH·B
00004930:·2020·2020·2020·2020·2020·2020·2020·2020··················00004930:·2020·2020·2020·2020·2020·2020·2020·2020··················
00004940:·2020·2020·2020·2020·2020·2020·2020·2020··················00004940:·2020·2020·2020·2020·2020·2020·2020·2020··················
00004950:·2020·2020·2020·2020·2020·2020·2020·2020··················00004950:·2020·2020·2020·2020·2020·2020·2020·2020··················
00004960:·2020·2020·2020·2020·2020·2020·2020·207c·················|00004960:·2020·2020·2020·2020·2020·2020·2020·207c·················|
00004970:·0a7c·202d·2d20·7573·6564·2030·2e36·3631··.|·--·used·0.66100004970:·0a7c·202d·2d20·7573·6564·2030·2e35·3430··.|·--·used·0.540
00004980:·3134·3173·2028·6370·7529·3b20·302e·3039··141s·(cpu);·0.0900004980:·3739·3573·2028·6370·7529·3b20·302e·3130··795s·(cpu);·0.10
00004990:·3934·3039·3273·2028·7468·7265·6164·293b··94092s·(thread);00004990:·3132·3437·7320·2874·6872·6561·6429·3b20··1247s·(thread);·
000049a0:·2030·7320·2867·6329·2020·2020·2020·2020···0s·(gc)········000049a0:·3073·2028·6763·2920·2020·2020·2020·2020··0s·(gc)·········
000049b0:·2020·2020·2020·2020·2020·2020·2020·207c·················|000049b0:·2020·2020·2020·2020·2020·2020·2020·207c·················|
000049c0:·0a7c·4669·6e64·696e·6720·6561·7379·2072··.|Finding·easy·r000049c0:·0a7c·4669·6e64·696e·6720·6561·7379·2072··.|Finding·easy·r
000049d0:·656c·6174·696f·6e73·2020·2020·2020·2020··elations········000049d0:·656c·6174·696f·6e73·2020·2020·2020·2020··elations········
000049e0:·2020·203a·2020·2020·2020·2020·2020·2020·····:············000049e0:·2020·203a·2020·2020·2020·2020·2020·2020·····:············
000049f0:·2020·2020·2020·2020·2020·2020·2020·2020··················000049f0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00004a00:·2020·2020·2020·2020·2020·2020·2020·207c·················|00004a00:·2020·2020·2020·2020·2020·2020·2020·207c·················|
00004a10:·0a7c·6f35·203d·2048·4220·2020·2020·2020··.|o5·=·HB·······00004a10:·0a7c·6f35·203d·2048·4220·2020·2020·2020··.|o5·=·HB·······
Offset 1400, 17 lines modifiedOffset 1400, 17 lines modified
00005770:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00005770:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00005780:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00005780:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00005790:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00005790:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000057a0:·2d2d·2b0a·7c69·3130·203a·2068·6f6d·6f6c··--+.|i10·:·homol000057a0:·2d2d·2b0a·7c69·3130·203a·2068·6f6d·6f6c··--+.|i10·:·homol
000057b0:·6f67·7941·6c67·6562·7261·2843·2c47·656e··ogyAlgebra(C,Gen000057b0:·6f67·7941·6c67·6562·7261·2843·2c47·656e··ogyAlgebra(C,Gen
000057c0:·4465·6772·6565·4c69·6d69·743d·3e34·2c52··DegreeLimit=>4,R000057c0:·4465·6772·6565·4c69·6d69·743d·3e34·2c52··DegreeLimit=>4,R
000057d0:·656c·4465·6772·6565·4c69·6d69·743d·3e34··elDegreeLimit=>4000057d0:·656c·4465·6772·6565·4c69·6d69·743d·3e34··elDegreeLimit=>4
000057e0:·297c·0a7c·202d·2d20·7573·6564·2030·2e39··)|.|·--·used·0.9000057e0:·297c·0a7c·202d·2d20·7573·6564·2030·2e38··)|.|·--·used·0.8
000057f0:·3938·3639·3273·2028·6370·7529·3b20·302e··98692s·(cpu);·0.000057f0:·3736·3539·3773·2028·6370·7529·3b20·302e··76597s·(cpu);·0.
00005800:·3133·3533·3235·7320·2874·6872·6561·6429··135325s·(thread)00005800:·3134·3830·3739·7320·2874·6872·6561·6429··148079s·(thread)
00005810:·3b20·3073·2028·6763·2920·2020·2020·2020··;·0s·(gc)·······00005810:·3b20·3073·2028·6763·2920·2020·2020·2020··;·0s·(gc)·······
00005820:·7c0a·7c46·696e·6469·6e67·2065·6173·7920··|.|Finding·easy·00005820:·7c0a·7c46·696e·6469·6e67·2065·6173·7920··|.|Finding·easy·
00005830:·7265·6c61·7469·6f6e·7320·2020·2020·2020··relations·······00005830:·7265·6c61·7469·6f6e·7320·2020·2020·2020··relations·······
00005840:·2020·2020·3a20·2020·2020·2020·2020·2020······:···········00005840:·2020·2020·3a20·2020·2020·2020·2020·2020······:···········
00005850:·2020·2020·2020·2020·2020·2020·2020·207c·················|00005850:·2020·2020·2020·2020·2020·2020·2020·207c·················|
00005860:·0a7c·2020·2020·2020·205a·5a20·2020·2020··.|·······ZZ·····00005860:·0a7c·2020·2020·2020·205a·5a20·2020·2020··.|·······ZZ·····
00005870:·2020·2020·2020·2020·2020·2020·2020·2020··················00005870:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 2539, 17 lines modifiedOffset 2539, 17 lines modified
00009ea0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00009ea0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00009eb0:·2d2d·2d2d·2d2d·2b0a·7c69·3720·3a20·484b··------+.|i7·:·HK00009eb0:·2d2d·2d2d·2d2d·2b0a·7c69·3720·3a20·484b··------+.|i7·:·HK
00009ec0:·5220·3d20·4848·204b·5220·2020·2020·2020··R·=·HH·KR·······00009ec0:·5220·3d20·4848·204b·5220·2020·2020·2020··R·=·HH·KR·······
00009ed0:·2020·2020·2020·2020·2020·2020·2020·2020··················00009ed0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009ee0:·2020·2020·2020·2020·2020·2020·2020·2020··················00009ee0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009ef0:·2020·2020·2020·2020·2020·2020·2020·2020··················00009ef0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009f00:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use00009f00:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
00009f10:·6420·302e·3730·3235·3639·7320·2863·7075··d·0.702569s·(cpu00009f10:·6420·302e·3638·3138·3836·7320·2863·7075··d·0.681886s·(cpu
00009f20:·293b·2030·2e31·3231·3839·3773·2028·7468··);·0.121897s·(th 
00009f30:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··00009f20:·293b·2030·2e30·3935·3436·3738·7320·2874··);·0.0954678s·(t
 00009f30:·6872·6561·6429·3b20·3073·2028·6763·2920··hread);·0s·(gc)·
00009f40:·2020·2020·2020·2020·2020·2020·2020·2020··················00009f40:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009f50:·2020·2020·2020·7c0a·7c46·696e·6469·6e67········|.|Finding00009f50:·2020·2020·2020·7c0a·7c46·696e·6469·6e67········|.|Finding
00009f60:·2065·6173·7920·7265·6c61·7469·6f6e·7320···easy·relations·00009f60:·2065·6173·7920·7265·6c61·7469·6f6e·7320···easy·relations·
00009f70:·2020·2020·2020·2020·2020·3a20·2020·2020············:·····00009f70:·2020·2020·2020·2020·2020·3a20·2020·2020············:·····
00009f80:·2020·2020·2020·2020·2020·2020·2020·2020··················00009f80:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009f90:·2020·2020·2020·2020·2020·2020·2020·2020··················00009f90:·2020·2020·2020·2020·2020·2020·2020·2020··················
00009fa0:·2020·2020·2020·7c0a·7c6f·3720·3d20·484b········|.|o7·=·HK00009fa0:·2020·2020·2020·7c0a·7c6f·3720·3d20·484b········|.|o7·=·HK
Offset 2644, 16 lines modifiedOffset 2644, 16 lines modified
0000a530:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0000a530:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0000a540:·2d2d·2d2d·2d2d·2b0a·7c69·3130·203a·2048··------+.|i10·:·H0000a540:·2d2d·2d2d·2d2d·2b0a·7c69·3130·203a·2048··------+.|i10·:·H
0000a550:·4b52·2720·3d20·4848·206b·6f73·7a75·6c43··KR'·=·HH·koszulC0000a550:·4b52·2720·3d20·4848·206b·6f73·7a75·6c43··KR'·=·HH·koszulC
0000a560:·6f6d·706c·6578·4447·4120·5227·2020·2020··omplexDGA·R'····0000a560:·6f6d·706c·6578·4447·4120·5227·2020·2020··omplexDGA·R'····
0000a570:·2020·2020·2020·2020·2020·2020·2020·2020··················0000a570:·2020·2020·2020·2020·2020·2020·2020·2020··················
0000a580:·2020·2020·2020·2020·2020·2020·2020·2020··················0000a580:·2020·2020·2020·2020·2020·2020·2020·2020··················
0000a590:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use0000a590:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
0000a5a0:·6420·322e·3534·3736·3773·2028·6370·7529··d·2.54767s·(cpu)0000a5a0:·6420·322e·3032·3138·3773·2028·6370·7529··d·2.02187s·(cpu)
0000a5b0:·3b20·302e·3730·3936·3935·7320·2874·6872··;·0.709695s·(thr0000a5b0:·3b20·302e·3734·3834·3732·7320·2874·6872··;·0.748472s·(thr
0000a5c0:·6561·6429·3b20·3073·2028·6763·2920·2020··ead);·0s·(gc)···0000a5c0:·6561·6429·3b20·3073·2028·6763·2920·2020··ead);·0s·(gc)···
0000a5d0:·2020·2020·2020·2020·2020·2020·2020·2020··················0000a5d0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0000a5e0:·2020·2020·2020·7c0a·7c46·696e·6469·6e67········|.|Finding0000a5e0:·2020·2020·2020·7c0a·7c46·696e·6469·6e67········|.|Finding
0000a5f0:·2065·6173·7920·7265·6c61·7469·6f6e·7320···easy·relations·0000a5f0:·2065·6173·7920·7265·6c61·7469·6f6e·7320···easy·relations·
0000a600:·2020·2020·2020·2020·2020·3a20·2020·2020············:·····0000a600:·2020·2020·2020·2020·2020·3a20·2020·2020············:·····
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Divisor.info
    
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Offset 2199, 17 lines modifiedOffset 2199, 17 lines modified
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Offset 2248, 16 lines modifiedOffset 2248, 16 lines modified
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Offset 3635, 17 lines modifiedOffset 3635, 17 lines modified
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00011630:·2020·2020·2020·2020·2020·2020·2020·2020··················00011630:·2020·2020·2020·2020·2020·2020·2020·2020··················
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00011650:·2020·2020·2020·2020·2020·2020·2020·2020··················00011650:·2020·2020·2020·2020·2020·2020·2020·2020··················
00011660:·2020·2020·2020·2020·2020·2020·2020·2020··················00011660:·2020·2020·2020·2020·2020·2020·2020·2020··················
00011670:·2020·2020·2020·2020·2020·2020·2020·2020··················00011670:·2020·2020·2020·2020·2020·2020·2020·2020··················
00011680:·2020·2020·2020·2020·2020·2020·2020·2020··················00011680:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 4932, 16 lines modifiedOffset 4932, 16 lines modified
00013430:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00013430:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00013440:·2d2d·2d2d·2d2d·2b0a·7c69·3130·203a·2074··------+.|i10·:·t00013440:·2d2d·2d2d·2d2d·2b0a·7c69·3130·203a·2074··------+.|i10·:·t
00013450:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra00013450:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra
00013460:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra00013460:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra
00013470:·7465·6779·203d·3e20·5261·6469·6361·6c29··tegy·=>·Radical)00013470:·7465·6779·203d·3e20·5261·6469·6361·6c29··tegy·=>·Radical)
00013480:·2020·2020·2020·2020·2020·2020·2020·2020··················00013480:·2020·2020·2020·2020·2020·2020·2020·2020··················
00013490:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use00013490:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
000134a0:·6420·302e·3539·3635·3534·7320·2863·7075··d·0.596554s·(cpu000134a0:·6420·302e·3636·3930·3432·7320·2863·7075··d·0.669042s·(cpu
000134b0:·293b·2030·2e33·3637·3731·3773·2028·7468··);·0.367717s·(th000134b0:·293b·2030·2e33·3635·3538·3673·2028·7468··);·0.365586s·(th
000134c0:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··000134c0:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··
000134d0:·2020·2020·2020·2020·2020·2020·2020·2020··················000134d0:·2020·2020·2020·2020·2020·2020·2020·2020··················
000134e0:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······000134e0:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
000134f0:·2020·2020·2020·2020·2020·2020·2020·2020··················000134f0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00013500:·2020·2020·2020·2020·2020·2020·2020·2020··················00013500:·2020·2020·2020·2020·2020·2020·2020·2020··················
00013510:·2020·2020·2020·2020·2020·2020·2020·2020··················00013510:·2020·2020·2020·2020·2020·2020·2020·2020··················
00013520:·2020·2020·2020·2020·2020·2020·2020·2020··················00013520:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 5422, 16 lines modifiedOffset 5422, 16 lines modified
000152d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000152d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000152e0:·2d2d·2d2d·2d2d·2b0a·7c69·3136·203a·2074··------+.|i16·:·t000152e0:·2d2d·2d2d·2d2d·2b0a·7c69·3136·203a·2074··------+.|i16·:·t
000152f0:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra000152f0:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra
00015300:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra00015300:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra
00015310:·7465·6779·203d·3e20·416c·6c43·6f64·696d··tegy·=>·AllCodim00015310:·7465·6779·203d·3e20·416c·6c43·6f64·696d··tegy·=>·AllCodim
00015320:·656e·7369·6f6e·7329·2020·2020·2020·2020··ensions)········00015320:·656e·7369·6f6e·7329·2020·2020·2020·2020··ensions)········
00015330:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use00015330:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
00015340:·6420·302e·3539·3531·3138·7320·2863·7075··d·0.595118s·(cpu00015340:·6420·302e·3638·3633·3839·7320·2863·7075··d·0.686389s·(cpu
00015350:·293b·2030·2e33·3732·3930·3973·2028·7468··);·0.372909s·(th00015350:·293b·2030·2e33·3634·3236·3873·2028·7468··);·0.364268s·(th
00015360:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··00015360:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··
00015370:·2020·2020·2020·2020·2020·2020·2020·2020··················00015370:·2020·2020·2020·2020·2020·2020·2020·2020··················
00015380:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······00015380:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
00015390:·2020·2020·2020·2020·2020·2020·2020·2020··················00015390:·2020·2020·2020·2020·2020·2020·2020·2020··················
000153a0:·2020·2020·2020·2020·2020·2020·2020·2020··················000153a0:·2020·2020·2020·2020·2020·2020·2020·2020··················
000153b0:·2020·2020·2020·2020·2020·2020·2020·2020··················000153b0:·2020·2020·2020·2020·2020·2020·2020·2020··················
000153c0:·2020·2020·2020·2020·2020·2020·2020·2020··················000153c0:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 5867, 16 lines modifiedOffset 5867, 16 lines modified
00016ea0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00016ea0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00016eb0:·2d2d·2d2d·2d2d·2b0a·7c69·3231·203a·2074··------+.|i21·:·t00016eb0:·2d2d·2d2d·2d2d·2b0a·7c69·3231·203a·2074··------+.|i21·:·t
00016ec0:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra00016ec0:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra
00016ed0:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra00016ed0:·6c43·6c6f·7375·7265·2852·2c20·5374·7261··lClosure(R,·Stra
00016ee0:·7465·6779·203d·3e20·5369·6d70·6c69·6679··tegy·=>·Simplify00016ee0:·7465·6779·203d·3e20·5369·6d70·6c69·6679··tegy·=>·Simplify
00016ef0:·4672·6163·7469·6f6e·7329·2020·2020·2020··Fractions)······00016ef0:·4672·6163·7469·6f6e·7329·2020·2020·2020··Fractions)······
00016f00:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use00016f00:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
00016f10:·6420·302e·3630·3736·3035·7320·2863·7075··d·0.607605s·(cpu00016f10:·6420·302e·3833·3236·3531·7320·2863·7075··d·0.832651s·(cpu
00016f20:·293b·2030·2e33·3732·3031·3773·2028·7468··);·0.372017s·(th00016f20:·293b·2030·2e34·3232·3639·3773·2028·7468··);·0.422697s·(th
00016f30:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··00016f30:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··
00016f40:·2020·2020·2020·2020·2020·2020·2020·2020··················00016f40:·2020·2020·2020·2020·2020·2020·2020·2020··················
00016f50:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······00016f50:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
00016f60:·2020·2020·2020·2020·2020·2020·2020·2020··················00016f60:·2020·2020·2020·2020·2020·2020·2020·2020··················
00016f70:·2020·2020·2020·2020·2020·2020·2020·2020··················00016f70:·2020·2020·2020·2020·2020·2020·2020·2020··················
00016f80:·2020·2020·2020·2020·2020·2020·2020·2020··················00016f80:·2020·2020·2020·2020·2020·2020·2020·2020··················
00016f90:·2020·2020·2020·2020·2020·2020·2020·2020··················00016f90:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 6312, 16 lines modifiedOffset 6312, 16 lines modified
00018a70:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00018a70:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00018a80:·2d2d·2d2d·2d2d·2b0a·7c69·3236·203a·2074··------+.|i26·:·t00018a80:·2d2d·2d2d·2d2d·2b0a·7c69·3236·203a·2074··------+.|i26·:·t
00018a90:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra00018a90:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra
00018aa0:·6c43·6c6f·7375·7265·2028·522c·2053·7472··lClosure·(R,·Str00018aa0:·6c43·6c6f·7375·7265·2028·522c·2053·7472··lClosure·(R,·Str
00018ab0:·6174·6567·7920·3d3e·2052·6164·6963·616c··ategy·=>·Radical00018ab0:·6174·6567·7920·3d3e·2052·6164·6963·616c··ategy·=>·Radical
00018ac0:·436f·6469·6d31·2920·2020·2020·2020·2020··Codim1)·········00018ac0:·436f·6469·6d31·2920·2020·2020·2020·2020··Codim1)·········
00018ad0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use00018ad0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
00018ae0:·6420·312e·3132·3439·3473·2028·6370·7529··d·1.12494s·(cpu)00018ae0:·6420·312e·3533·3831·3773·2028·6370·7529··d·1.53817s·(cpu)
00018af0:·3b20·302e·3636·3531·3336·7320·2874·6872··;·0.665136s·(thr00018af0:·3b20·302e·3733·3133·3438·7320·2874·6872··;·0.731348s·(thr
00018b00:·6561·6429·3b20·3073·2028·6763·2920·2020··ead);·0s·(gc)···00018b00:·6561·6429·3b20·3073·2028·6763·2920·2020··ead);·0s·(gc)···
00018b10:·2020·2020·2020·2020·2020·2020·2020·2020··················00018b10:·2020·2020·2020·2020·2020·2020·2020·2020··················
00018b20:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······00018b20:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
00018b30:·2020·2020·2020·2020·2020·2020·2020·2020··················00018b30:·2020·2020·2020·2020·2020·2020·2020·2020··················
00018b40:·2020·2020·2020·2020·2020·2020·2020·2020··················00018b40:·2020·2020·2020·2020·2020·2020·2020·2020··················
00018b50:·2020·2020·2020·2020·2020·2020·2020·2020··················00018b50:·2020·2020·2020·2020·2020·2020·2020·2020··················
00018b60:·2020·2020·2020·2020·2020·2020·2020·2020··················00018b60:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 6757, 16 lines modifiedOffset 6757, 16 lines modified
0001a640:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0001a640:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0001a650:·2d2d·2d2d·2d2d·2b0a·7c69·3331·203a·2074··------+.|i31·:·t0001a650:·2d2d·2d2d·2d2d·2b0a·7c69·3331·203a·2074··------+.|i31·:·t
0001a660:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra0001a660:·696d·6520·5227·203d·2069·6e74·6567·7261··ime·R'·=·integra
0001a670:·6c43·6c6f·7375·7265·2028·522c·2053·7472··lClosure·(R,·Str0001a670:·6c43·6c6f·7375·7265·2028·522c·2053·7472··lClosure·(R,·Str
0001a680:·6174·6567·7920·3d3e·2056·6173·636f·6e63··ategy·=>·Vasconc0001a680:·6174·6567·7920·3d3e·2056·6173·636f·6e63··ategy·=>·Vasconc
0001a690:·656c·6f73·2920·2020·2020·2020·2020·2020··elos)···········0001a690:·656c·6f73·2920·2020·2020·2020·2020·2020··elos)···········
0001a6a0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use0001a6a0:·2020·2020·2020·7c0a·7c20·2d2d·2075·7365········|.|·--·use
0001a6b0:·6420·302e·3637·3836·3634·7320·2863·7075··d·0.678664s·(cpu0001a6b0:·6420·302e·3839·3333·3539·7320·2863·7075··d·0.893359s·(cpu
0001a6c0:·293b·2030·2e33·3936·3630·3673·2028·7468··);·0.396606s·(th0001a6c0:·293b·2030·2e34·3430·3337·3173·2028·7468··);·0.440371s·(th
0001a6d0:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··0001a6d0:·7265·6164·293b·2030·7320·2867·6329·2020··read);·0s·(gc)··
0001a6e0:·2020·2020·2020·2020·2020·2020·2020·2020··················0001a6e0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001a6f0:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······0001a6f0:·2020·2020·2020·7c0a·7c20·2020·2020·2020········|.|·······
0001a700:·2020·2020·2020·2020·2020·2020·2020·2020··················0001a700:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001a710:·2020·2020·2020·2020·2020·2020·2020·2020··················0001a710:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001a720:·2020·2020·2020·2020·2020·2020·2020·2020··················0001a720:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001a730:·2020·2020·2020·2020·2020·2020·2020·2020··················0001a730:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 7174, 18 lines modifiedOffset 7174, 18 lines modified
0001c050:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0001c050:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0001c060:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0001c060:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0001c070:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0001c070:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0001c080:·2d2d·2b0a·7c69·3336·203a·2074·696d·6520··--+.|i36·:·time·0001c080:·2d2d·2b0a·7c69·3336·203a·2074·696d·6520··--+.|i36·:·time·
0001c090:·5227·203d·2069·6e74·6567·7261·6c43·6c6f··R'·=·integralClo0001c090:·5227·203d·2069·6e74·6567·7261·6c43·6c6f··R'·=·integralClo
0001c0a0:·7375·7265·2052·2020·2020·2020·2020·2020··sure·R··········0001c0a0:·7375·7265·2052·2020·2020·2020·2020·2020··sure·R··········
0001c0b0:·2020·2020·2020·2020·2020·2020·207c·0a7c···············|.|0001c0b0:·2020·2020·2020·2020·2020·2020·207c·0a7c···············|.|
0001c0c0:·202d·2d20·7573·6564·2030·2e30·3938·3133···--·used·0.098130001c0c0:·202d·2d20·7573·6564·2030·2e31·3434·3631···--·used·0.14461
0001c0d0:·3036·7320·2863·7075·293b·2030·2e30·3531··06s·(cpu);·0.051 
0001c0e0:·3230·3232·7320·2874·6872·6561·6429·3b20··2022s·(thread);·0001c0d0:·3573·2028·6370·7529·3b20·302e·3035·3830··5s·(cpu);·0.0580
 0001c0e0:·3935·3373·2028·7468·7265·6164·293b·2030··953s·(thread);·0
0001c0f0:·3073·2028·6763·2920·7c0a·7c20·2020·2020··0s·(gc)·|.|·····0001c0f0:·7320·2867·6329·2020·7c0a·7c20·2020·2020··s·(gc)··|.|·····
0001c100:·2020·2020·2020·2020·2020·2020·2020·2020··················0001c100:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001c110:·2020·2020·2020·2020·2020·2020·2020·2020··················0001c110:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001c120:·2020·2020·2020·2020·2020·2020·2020·2020··················0001c120:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001c130:·2020·207c·0a7c·6f33·3620·3d20·5227·2020·····|.|o36·=·R'··0001c130:·2020·207c·0a7c·6f33·3620·3d20·5227·2020·····|.|o36·=·R'··
0001c140:·2020·2020·2020·2020·2020·2020·2020·2020··················0001c140:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001c150:·2020·2020·2020·2020·2020·2020·2020·2020··················0001c150:·2020·2020·2020·2020·2020·2020·2020·2020··················
0001c160:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.0001c160:·2020·2020·2020·2020·2020·2020·2020·7c0a················|.
Offset 7364, 16 lines modifiedOffset 7364, 16 lines modified
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65.7 KB
MonodromySolver.info
    
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Offset 9504, 16 lines modifiedOffset 9504, 16 lines modified
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Offset 10141, 16 lines modifiedOffset 10141, 16 lines modified
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Offset 10206, 16 lines modifiedOffset 10206, 16 lines modified
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Offset 3599, 16 lines modifiedOffset 3599, 16 lines modified
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Offset 4112, 28 lines modifiedOffset 4112, 28 lines modified
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NoetherianOperators.info
    
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NormalToricVarieties.info
    
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Offset 8203, 41 lines modifiedOffset 8203, 41 lines modified
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Offset 26336, 15 lines modifiedOffset 26336, 15 lines modified
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00066e80:·2020·2020·2020·2020·2020·2020·2020·207c·················|00066e80:·2020·2020·2020·2020·2020·2020·2020·207c·················|
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Offset 26399, 15 lines modifiedOffset 26399, 15 lines modified
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0007b720:·3120·2020·2020·2020·2020·2020·2020·2020··1···············0007b720:·3120·2020·2020·2020·2020·2020·2020·2020··1···············
0007b730:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b730:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b740:·2020·2020·2020·2020·2020·2020·7c0a·7c20··············|.|·0007b740:·2020·2020·2020·2020·2020·2020·7c0a·7c20··············|.|·
0007b750:·2d2d·202e·3134·3939·3032·7320·656c·6170··--·.149902s·elap0007b750:·2d2d·202e·3135·3430·3173·2065·6c61·7073··--·.15401s·elaps
0007b760:·7365·6420·2020·2020·2020·2020·2020·2020··sed·············0007b760:·6564·2020·2020·2020·2020·2020·2020·2020··ed··············
0007b770:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b770:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b780:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b780:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b790:·2020·2020·2020·2020·2020·2020·7c0a·7c20··············|.|·0007b790:·2020·2020·2020·2020·2020·2020·7c0a·7c20··············|.|·
0007b7a0:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b7a0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b7b0:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b7b0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b7c0:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b7c0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007b7d0:·2020·2020·2020·2020·2020·2020·2020·2020··················0007b7d0:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 31664, 16 lines modifiedOffset 31664, 16 lines modified
0007baf0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007baf0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0007bb00:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69··------------+.|i0007bb00:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2b0a·7c69··------------+.|i
0007bb10:·3420·3a20·656c·6170·7365·6454·696d·6520··4·:·elapsedTime·0007bb10:·3420·3a20·656c·6170·7365·6454·696d·6520··4·:·elapsedTime·
0007bb20:·6173·7365·7274·2028·7365·7420·4d31·203d··assert·(set·M1·=0007bb20:·6173·7365·7274·2028·7365·7420·4d31·203d··assert·(set·M1·=
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0007bb40:·7269·6573·2062·6173·6973·2864·6567·7265··ries·basis(degre0007bb40:·7269·6573·2062·6173·6973·2864·6567·7265··ries·basis(degre
0007bb50:·6520·4431·2c20·7269·6e67·2020·7c0a·7c20··e·D1,·ring··|.|·0007bb50:·6520·4431·2c20·7269·6e67·2020·7c0a·7c20··e·D1,·ring··|.|·
0007bb60:·2d2d·202e·3030·3130·3735·3973·2065·6c61··--·.0010759s·ela0007bb60:·2d2d·202e·3030·3435·3537·3536·7320·656c··--·.00455756s·el
0007bb70:·7073·6564·2020·2020·2020·2020·2020·2020··psed············0007bb70:·6170·7365·6420·2020·2020·2020·2020·2020··apsed···········
0007bb80:·2020·2020·2020·2020·2020·2020·2020·2020··················0007bb80:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007bb90:·2020·2020·2020·2020·2020·2020·2020·2020··················0007bb90:·2020·2020·2020·2020·2020·2020·2020·2020··················
0007bba0:·2020·2020·2020·2020·2020·2020·7c0a·7c2d··············|.|-0007bba0:·2020·2020·2020·2020·2020·2020·7c0a·7c2d··············|.|-
0007bbb0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007bbb0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0007bbc0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007bbc0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0007bbd0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007bbd0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
0007bbe0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007bbe0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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0007bfe0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------0007bfe0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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0001e540:·2b0a·7c69·3420·3a20·7472·7920·7061·7261··+.|i4·:·try·para0001e540:·2b0a·7c69·3420·3a20·7472·7920·7061·7261··+.|i4·:·try·para
0001e550:·6d65·7465·7248·6f6d·6f74·6f70·7928·7b66··meterHomotopy({f0001e550:·6d65·7465·7248·6f6d·6f74·6f70·7928·7b66··meterHomotopy({f
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Offset 1698, 24 lines modifiedOffset 1698, 24 lines modified
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Offset 2659, 35 lines modifiedOffset 2659, 35 lines modified
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Offset 7659, 15 lines modifiedOffset 7659, 15 lines modified
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00001ff0:·2020·2020·2020·2020·2020·2020·2020·2020··················00001ff0:·2020·2020·2020·2020·2020·2020·2020·2020··················
00002000:·2020·2020·2020·2020·7c0a·7c6f·3420·3d20··········|.|o4·=·00002000:·2020·2020·2020·2020·7c0a·7c6f·3420·3d20··········|.|o4·=·
00002010:·6964·6561·6c20·2878·2020·2078·2020·2020··ideal·(x···x····00002010:·6964·6561·6c20·2878·2020·2078·2020·2020··ideal·(x···x····
00002020:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···00002020:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···
00002030:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,00002030:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,
00002040:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···00002040:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···
00002050:·7820·2020·2c20·2020·7c0a·7c20·2020·2020··x···,···|.|·····00002050:·7820·2020·2c20·2020·7c0a·7c20·2020·2020··x···,···|.|·····
00002060:·2020·2020·2020·2020·322c·3420·312c·3320··········2,4·1,3·00002060:·2020·2020·2020·2020·322c·3320·312c·3120··········2,3·1,1·
00002070:·2020·2032·2c33·2031·2c34·2020·2034·2c34·····2,3·1,4···4,400002070:·2020·2032·2c31·2031·2c33·2020·2034·2c33·····2,1·1,3···4,3
00002080:·2033·2c33·2020·2020·342c·3320·332c·3420···3,3····4,3·3,4·00002080:·2033·2c31·2020·2020·342c·3120·332c·3320···3,1····4,1·3,3·
00002090:·2020·332c·3220·312c·3120·2020·2033·2c31····3,2·1,1····3,100002090:·2020·342c·3420·312c·3320·2020·2034·2c33····4,4·1,3····4,3
000020a0:·2031·2c32·2020·2020·7c0a·7c20·2020·2020···1,2····|.|·····000020a0:·2031·2c34·2020·2020·7c0a·7c20·2020·2020···1,4····|.|·····
000020b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000020b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000020c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000020c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000020d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000020d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000020e0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000020e0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000020f0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000020f0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
00002100:·7820·2020·7820·2020·202d·2078·2020·2078··x···x····-·x···x00002100:·7820·2020·7820·2020·202d·2078·2020·2078··x···x····-·x···x
00002110:·2020·202c·2078·2020·2078·2020·2020·2d20·····,·x···x····-·00002110:·2020·202c·2078·2020·2078·2020·2020·2d20·····,·x···x····-·
00002120:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·00002120:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·
00002130:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x00002130:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x
00002140:·2020·2078·2020·2020·7c0a·7c20·2020·2020·····x····|.|·····00002140:·2020·2078·2020·2020·7c0a·7c20·2020·2020·····x····|.|·····
00002150:·2034·2c34·2032·2c33·2020·2020·342c·3320···4,4·2,3····4,3·00002150:·2033·2c34·2032·2c33·2020·2020·332c·3320···3,4·2,3····3,3·
00002160:·322c·3420·2020·322c·3220·312c·3120·2020··2,4···2,2·1,1···00002160:·322c·3420·2020·342c·3220·322c·3120·2020··2,4···4,2·2,1···
00002170:·2032·2c31·2031·2c32·2020·2034·2c32·2033···2,1·1,2···4,2·300002170:·2034·2c31·2032·2c32·2020·2032·2c33·2031···4,1·2,2···2,3·1
00002180:·2c31·2020·2020·342c·3120·332c·3220·2020··,1····4,1·3,2···00002180:·2c32·2020·2020·322c·3220·312c·3320·2020··,2····2,2·1,3···
00002190:·322c·3320·312c·3120·7c0a·7c20·2020·2020··2,3·1,1·|.|·····00002190:·322c·3420·312c·3120·7c0a·7c20·2020·2020··2,4·1,1·|.|·····
000021a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000021a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000021b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000021b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000021c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000021c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000021d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000021d0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000021e0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000021e0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
000021f0:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···000021f0:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···
00002200:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,00002200:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,
00002210:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···00002210:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···
00002220:·7820·2020·2c20·7820·2020·7820·2020·202d··x···,·x···x····-00002220:·7820·2020·2c20·7820·2020·7820·2020·202d··x···,·x···x····-
00002230:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····00002230:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····
00002240:·2020·2032·2c31·2031·2c33·2020·2034·2c33·····2,1·1,3···4,300002240:·2020·2032·2c31·2031·2c34·2020·2034·2c34·····2,1·1,4···4,4
00002250:·2033·2c31·2020·2020·342c·3120·332c·3320···3,1····4,1·3,3·00002250:·2033·2c31·2020·2020·342c·3120·332c·3420···3,1····4,1·3,4·
00002260:·2020·342c·3420·312c·3320·2020·2034·2c33····4,4·1,3····4,300002260:·2020·342c·3320·332c·3220·2020·2034·2c32····4,3·3,2····4,2
00002270:·2031·2c34·2020·2033·2c34·2032·2c33·2020···1,4···3,4·2,3··00002270:·2033·2c33·2020·2033·2c34·2031·2c33·2020···3,3···3,4·1,3··
00002280:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····00002280:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····
00002290:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002290:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000022a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000022a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000022b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000022b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000022c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000022c0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000022d0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000022d0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
000022e0:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·000022e0:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·
000022f0:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x000022f0:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x
00002300:·2020·2078·2020·2020·2d20·7820·2020·7820·····x····-·x···x·00002300:·2020·2078·2020·2020·2d20·7820·2020·7820·····x····-·x···x·
00002310:·2020·2c20·7820·2020·7820·2020·202d·2078····,·x···x····-·x00002310:·2020·2c20·7820·2020·7820·2020·202d·2078····,·x···x····-·x
00002320:·2020·2078·2020·202c·7c0a·7c20·2020·2020·····x···,|.|·····00002320:·2020·2078·2020·202c·7c0a·7c20·2020·2020·····x···,|.|·····
00002330:·2033·2c33·2032·2c34·2020·2034·2c32·2032···3,3·2,4···4,2·200002330:·2033·2c33·2031·2c34·2020·2033·2c32·2032···3,3·1,4···3,2·2
00002340:·2c31·2020·2020·342c·3120·322c·3220·2020··,1····4,1·2,2···00002340:·2c31·2020·2020·332c·3120·322c·3220·2020··,1····3,1·2,2···
00002350:·322c·3320·312c·3220·2020·2032·2c32·2031··2,3·1,2····2,2·100002350:·342c·3220·312c·3120·2020·2034·2c31·2031··4,2·1,1····4,1·1
00002360:·2c33·2020·2032·2c34·2031·2c31·2020·2020··,3···2,4·1,1····00002360:·2c32·2020·2032·2c34·2031·2c32·2020·2020··,2···2,4·1,2····
00002370:·322c·3120·312c·3420·7c0a·7c20·2020·2020··2,1·1,4·|.|·····00002370:·322c·3220·312c·3420·7c0a·7c20·2020·2020··2,2·1,4·|.|·····
00002380:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002380:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002390:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002390:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000023a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000023a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000023b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000023b0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000023c0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000023c0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
000023d0:·7820·2020·7820·2020·202d·2078·2020·2078··x···x····-·x···x000023d0:·7820·2020·7820·2020·202d·2078·2020·2078··x···x····-·x···x
000023e0:·2020·202c·2078·2020·2078·2020·2020·2d20·····,·x···x····-·000023e0:·2020·202c·2078·2020·2078·2020·2020·2d20·····,·x···x····-·
000023f0:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·000023f0:·7820·2020·7820·2020·2c20·7820·2020·7820··x···x···,·x···x·
00002400:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x00002400:·2020·202d·2078·2020·2078·2020·202c·2078·····-·x···x···,·x
00002410:·2020·2078·2020·2020·7c0a·7c20·2020·2020·····x····|.|·····00002410:·2020·2078·2020·2020·7c0a·7c20·2020·2020·····x····|.|·····
00002420:·2034·2c34·2033·2c31·2020·2020·342c·3120···4,4·3,1····4,1·00002420:·2034·2c34·2033·2c32·2020·2020·342c·3220···4,4·3,2····4,2·
00002430:·332c·3420·2020·342c·3320·332c·3220·2020··3,4···4,3·3,2···00002430:·332c·3420·2020·322c·3420·312c·3320·2020··3,4···2,4·1,3···
00002440:·2034·2c32·2033·2c33·2020·2033·2c34·2031···4,2·3,3···3,4·100002440:·2032·2c33·2031·2c34·2020·2034·2c34·2033···2,3·1,4···4,4·3
00002450:·2c33·2020·2020·332c·3320·312c·3420·2020··,3····3,3·1,4···00002450:·2c33·2020·2020·342c·3320·332c·3420·2020··,3····4,3·3,4···
00002460:·332c·3220·322c·3120·7c0a·7c20·2020·2020··3,2·2,1·|.|·····00002460:·332c·3220·312c·3120·7c0a·7c20·2020·2020··3,2·1,1·|.|·····
00002470:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002470:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002480:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002480:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002490:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002490:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000024a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------000024a0:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000024b0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000024b0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
000024c0:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···000024c0:·2d20·7820·2020·7820·2020·2c20·7820·2020··-·x···x···,·x···
000024d0:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,000024d0:·7820·2020·202d·2078·2020·2078·2020·202c··x····-·x···x···,
000024e0:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···000024e0:·2078·2020·2078·2020·2020·2d20·7820·2020···x···x····-·x···
000024f0:·7820·2020·2c20·7820·2020·7820·2020·202d··x···,·x···x····-000024f0:·7820·2020·2c20·7820·2020·7820·2020·202d··x···,·x···x····-
00002500:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····00002500:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····
00002510:·2020·2033·2c31·2032·2c32·2020·2034·2c32·····3,1·2,2···4,200002510:·2020·2033·2c31·2031·2c32·2020·2034·2c34·····3,1·1,2···4,4
00002520:·2031·2c31·2020·2020·342c·3120·312c·3220···1,1····4,1·1,2·00002520:·2032·2c33·2020·2020·342c·3320·322c·3420···2,3····4,3·2,4·
00002530:·2020·322c·3420·312c·3220·2020·2032·2c32····2,4·1,2····2,200002530:·2020·322c·3220·312c·3120·2020·2032·2c31····2,2·1,1····2,1
00002540:·2031·2c34·2020·2034·2c34·2033·2c32·2020···1,4···4,4·3,2··00002540:·2031·2c32·2020·2034·2c32·2033·2c31·2020···1,2···4,2·3,1··
00002550:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····00002550:·2020·2020·2020·2020·7c0a·7c20·2020·2020··········|.|·····
00002560:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002560:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002570:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002570:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002580:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002580:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
00002590:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00002590:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
000025a0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····000025a0:·2d2d·2d2d·2d2d·2d2d·7c0a·7c20·2020·2020··--------|.|·····
000025b0:·7820·2020·7820·2020·2920·2020·2020·2020··x···x···)·······000025b0:·7820·2020·7820·2020·2920·2020·2020·2020··x···x···)·······
000025c0:·2020·2020·2020·2020·2020·2020·2020·2020··················000025c0:·2020·2020·2020·2020·2020·2020·2020·2020··················
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00022380:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------00022380:·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d·2d2d··----------------
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RandomComplexes.info
    
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00019d00:·2020·2020·207c·0a7c·202d·2d20·2e30·3036·······|.|·--·.00600019d00:·2020·2020·207c·0a7c·202d·2d20·2e30·3032·······|.|·--·.002
00019d10:·3333·3131·3273·2065·6c61·7073·6564·2020··33112s·elapsed··00019d10:·3534·3037·3173·2065·6c61·7073·6564·2020··54071s·elapsed··
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Offset 9798, 15 lines modifiedOffset 9798, 15 lines modified
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Offset 10208, 15 lines modifiedOffset 10208, 15 lines modified
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00027e40:·793d·3e4c·6170·6c61·6369·616e·293b·2020··y=>Laplacian);··00027e40:·793d·3e4c·6170·6c61·6369·616e·293b·2020··y=>Laplacian);··
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Offset 10569, 16 lines modifiedOffset 10569, 16 lines modified
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./usr/share/info/Schubert2.info.gz
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Schubert2.info
    
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SpecialFanoFourfolds.info
    
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gfanInterface.info
    
Offset 11570, 16 lines modifiedOffset 11570, 16 lines modified
0002d310:·6f61·6450·6163·6b61·6765·2822·6766·616e··oadPackage("gfan0002d310:·6f61·6450·6163·6b61·6765·2822·6766·616e··oadPackage("gfan
0002d320:·496e·7465·7266·6163·6522·2c20·436f·6e66··Interface",·Conf0002d320:·496e·7465·7266·6163·6522·2c20·436f·6e66··Interface",·Conf
0002d330:·6967·7572·6174·696f·6e20·3d3e·207b·2022··iguration·=>·{·"0002d330:·6967·7572·6174·696f·6e20·3d3e·207b·2022··iguration·=>·{·"
0002d340:·6b65·6570·6669·6c65·7322·203d·3e20·7472··keepfiles"·=>·tr0002d340:·6b65·6570·6669·6c65·7322·203d·3e20·7472··keepfiles"·=>·tr
0002d350:·7565·2c20·2276·657c·0a7c·202d·2d20·7275··ue,·"ve|.|·--·ru0002d350:·7565·2c20·2276·657c·0a7c·202d·2d20·7275··ue,·"ve|.|·--·ru
0002d360:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002d360:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002d370:·6766·616e·2067·6661·6e20·2d2d·6865·6c70··gfan·gfan·--help0002d370:·6766·616e·2067·6661·6e20·2d2d·6865·6c70··gfan·gfan·--help
0002d380:·203c·202f·746d·702f·4d32·2d33·3735·3734···<·/tmp/M2-375740002d380:·203c·202f·746d·702f·4d32·2d32·3733·3739···<·/tmp/M2-27379
0002d390:·3535·2d30·2f31·3732·2020·2020·2020·2020··55-0/172········0002d390:·3536·2d30·2f31·3732·2020·2020·2020·2020··56-0/172········
0002d3a0:·2020·2020·2020·207c·0a7c·5468·6973·2069·········|.|This·i0002d3a0:·2020·2020·2020·207c·0a7c·5468·6973·2069·········|.|This·i
0002d3b0:·7320·6120·7072·6f67·7261·6d20·666f·7220··s·a·program·for·0002d3b0:·7320·6120·7072·6f67·7261·6d20·666f·7220··s·a·program·for·
0002d3c0:·636f·6d70·7574·696e·6720·616c·6c20·7265··computing·all·re0002d3c0:·636f·6d70·7574·696e·6720·616c·6c20·7265··computing·all·re
0002d3d0:·6475·6365·6420·4772·6f65·626e·6572·2062··duced·Groebner·b0002d3d0:·6475·6365·6420·4772·6f65·626e·6572·2062··duced·Groebner·b
0002d3e0:·6173·6573·206f·6620·6120·706f·6c79·6e6f··ases·of·a·polyno0002d3e0:·6173·6573·206f·6620·6120·706f·6c79·6e6f··ases·of·a·polyno
0002d3f0:·6d69·616c·2069·647c·0a7c·4f70·7469·6f6e··mial·id|.|Option0002d3f0:·6d69·616c·2069·647c·0a7c·4f70·7469·6f6e··mial·id|.|Option
0002d400:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············0002d400:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············
Offset 11664, 22 lines modifiedOffset 11664, 22 lines modified
0002d8f0:·2020·2020·2020·207c·0a7c·2049·6e74·6572·········|.|·Inter0002d8f0:·2020·2020·2020·207c·0a7c·2049·6e74·6572·········|.|·Inter
0002d900:·7275·7074·2074·6865·2065·6e75·6d65·7261··rupt·the·enumera0002d900:·7275·7074·2074·6865·2065·6e75·6d65·7261··rupt·the·enumera
0002d910:·7469·6f6e·2061·6674·6572·2061·2073·7065··tion·after·a·spe0002d910:·7469·6f6e·2061·6674·6572·2061·2073·7065··tion·after·a·spe
0002d920:·6369·6669·6564·206e·756d·6265·7220·6f66··cified·number·of0002d920:·6369·6669·6564·206e·756d·6265·7220·6f66··cified·number·of
0002d930:·2066·6163·6574·7320·6861·7665·2062·6565···facets·have·bee0002d930:·2066·6163·6574·7320·6861·7665·2062·6565···facets·have·bee
0002d940:·6e20·636f·6d70·757c·0a7c·7573·696e·6720··n·compu|.|using·0002d940:·6e20·636f·6d70·757c·0a7c·7573·696e·6720··n·compu|.|using·
0002d950:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/0002d950:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/
0002d960:·746d·702f·4d32·2d33·3735·3734·3535·2d30··tmp/M2-3757455-00002d960:·746d·702f·4d32·2d32·3733·3739·3536·2d30··tmp/M2-2737956-0
0002d970:·2f31·3732·2020·2020·2020·2020·2020·2020··/172············0002d970:·2f31·3732·2020·2020·2020·2020·2020·2020··/172············
0002d980:·2020·2020·2020·2020·2020·2020·2020·2020··················0002d980:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002d990:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru0002d990:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru
0002d9a0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002d9a0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002d9b0:·6766·616e·205f·6275·6368·6265·7267·6572··gfan·_buchberger0002d9b0:·6766·616e·205f·6275·6368·6265·7267·6572··gfan·_buchberger
0002d9c0:·202d·2d68·656c·7020·3c20·2f74·6d70·2f4d···--help·<·/tmp/M0002d9c0:·202d·2d68·656c·7020·3c20·2f74·6d70·2f4d···--help·<·/tmp/M
0002d9d0:·322d·3337·3537·3435·352d·302f·3137·3420··2-3757455-0/174·0002d9d0:·322d·3237·3337·3935·362d·302f·3137·3420··2-2737956-0/174·
0002d9e0:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p0002d9e0:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p
0002d9f0:·726f·6772·616d·2063·6f6d·7075·7465·7320··rogram·computes·0002d9f0:·726f·6772·616d·2063·6f6d·7075·7465·7320··rogram·computes·
0002da00:·6120·7265·6475·6365·6420·6c65·7869·636f··a·reduced·lexico0002da00:·6120·7265·6475·6365·6420·6c65·7869·636f··a·reduced·lexico
0002da10:·6772·6170·6869·6320·4772·6f65·626e·6572··graphic·Groebner0002da10:·6772·6170·6869·6320·4772·6f65·626e·6572··graphic·Groebner
0002da20:·2062·6173·6973·206f·6620·7468·6520·706f···basis·of·the·po0002da20:·2062·6173·6973·206f·6620·7468·6520·706f···basis·of·the·po
0002da30:·6c79·6e6f·6d69·617c·0a7c·4f70·7469·6f6e··lynomia|.|Option0002da30:·6c79·6e6f·6d69·617c·0a7c·4f70·7469·6f6e··lynomia|.|Option
0002da40:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············0002da40:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············
Offset 11754, 22 lines modifiedOffset 11754, 22 lines modified
0002de90:·2020·2020·2020·207c·0a7c·2057·6974·6820·········|.|·With·0002de90:·2020·2020·2020·207c·0a7c·2057·6974·6820·········|.|·With·
0002dea0:·7468·6973·206f·7074·696f·6e20·796f·7520··this·option·you·0002dea0:·7468·6973·206f·7074·696f·6e20·796f·7520··this·option·you·
0002deb0:·6361·6e20·7370·6563·6966·7920·686f·7720··can·specify·how·0002deb0:·6361·6e20·7370·6563·6966·7920·686f·7720··can·specify·how·
0002dec0:·6d61·6e79·2076·6172·6961·626c·6573·2074··many·variables·t0002dec0:·6d61·6e79·2076·6172·6961·626c·6573·2074··many·variables·t
0002ded0:·6f20·7472·6561·7420·6173·2070·6172·616d··o·treat·as·param0002ded0:·6f20·7472·6561·7420·6173·2070·6172·616d··o·treat·as·param
0002dee0:·6574·6572·7320·697c·0a7c·7573·696e·6720··eters·i|.|using·0002dee0:·6574·6572·7320·697c·0a7c·7573·696e·6720··eters·i|.|using·
0002def0:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/0002def0:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/
0002df00:·746d·702f·4d32·2d33·3735·3734·3535·2d30··tmp/M2-3757455-00002df00:·746d·702f·4d32·2d32·3733·3739·3536·2d30··tmp/M2-2737956-0
0002df10:·2f31·3734·2020·2020·2020·2020·2020·2020··/174············0002df10:·2f31·3734·2020·2020·2020·2020·2020·2020··/174············
0002df20:·2020·2020·2020·2020·2020·2020·2020·2020··················0002df20:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002df30:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru0002df30:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru
0002df40:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002df40:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002df50:·6766·616e·205f·646f·6573·6964·6561·6c63··gfan·_doesidealc0002df50:·6766·616e·205f·646f·6573·6964·6561·6c63··gfan·_doesidealc
0002df60:·6f6e·7461·696e·202d·2d68·656c·7020·3c20··ontain·--help·<·0002df60:·6f6e·7461·696e·202d·2d68·656c·7020·3c20··ontain·--help·<·
0002df70:·2f74·6d70·2f4d·322d·3337·3537·3435·352d··/tmp/M2-3757455-0002df70:·2f74·6d70·2f4d·322d·3237·3337·3935·362d··/tmp/M2-2737956-
0002df80:·302f·3137·3620·207c·0a7c·5468·6973·2070··0/176··|.|This·p0002df80:·302f·3137·3620·207c·0a7c·5468·6973·2070··0/176··|.|This·p
0002df90:·726f·6772·616d·2074·616b·6573·2061·206d··rogram·takes·a·m0002df90:·726f·6772·616d·2074·616b·6573·2061·206d··rogram·takes·a·m
0002dfa0:·6172·6b65·6420·4772·6f65·626e·6572·2062··arked·Groebner·b0002dfa0:·6172·6b65·6420·4772·6f65·626e·6572·2062··arked·Groebner·b
0002dfb0:·6173·6973·206f·6620·616e·2069·6465·616c··asis·of·an·ideal0002dfb0:·6173·6973·206f·6620·616e·2069·6465·616c··asis·of·an·ideal
0002dfc0:·2049·2061·6e64·2061·2073·6574·206f·6620···I·and·a·set·of·0002dfc0:·2049·2061·6e64·2061·2073·6574·206f·6620···I·and·a·set·of·
0002dfd0:·706f·6c79·6e6f·6d7c·0a7c·4f70·7469·6f6e··polynom|.|Option0002dfd0:·706f·6c79·6e6f·6d7c·0a7c·4f70·7469·6f6e··polynom|.|Option
0002dfe0:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············0002dfe0:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············
Offset 11794, 23 lines modifiedOffset 11794, 23 lines modified
0002e110:·2020·2020·2020·207c·0a7c·2052·6561·6473·········|.|·Reads0002e110:·2020·2020·2020·207c·0a7c·2052·6561·6473·········|.|·Reads
0002e120:·2069·6e20·6120·706f·6c79·6e6f·6d69·616c···in·a·polynomial0002e120:·2069·6e20·6120·706f·6c79·6e6f·6d69·616c···in·a·polynomial
0002e130:·2074·6861·7420·7769·6c6c·2062·6520·6d75···that·will·be·mu0002e130:·2074·6861·7420·7769·6c6c·2062·6520·6d75···that·will·be·mu
0002e140:·6c74·6970·6c69·6564·2074·6f20·7468·6520··ltiplied·to·the·0002e140:·6c74·6970·6c69·6564·2074·6f20·7468·6520··ltiplied·to·the·
0002e150:·706f·6c79·6e6f·6d69·616c·2074·6f20·6265··polynomial·to·be0002e150:·706f·6c79·6e6f·6d69·616c·2074·6f20·6265··polynomial·to·be
0002e160:·2064·6976·6964·657c·0a7c·7573·696e·6720···divide|.|using·0002e160:·2064·6976·6964·657c·0a7c·7573·696e·6720···divide|.|using·
0002e170:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/0002e170:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/
0002e180:·746d·702f·4d32·2d33·3735·3734·3535·2d30··tmp/M2-3757455-00002e180:·746d·702f·4d32·2d32·3733·3739·3536·2d30··tmp/M2-2737956-0
0002e190:·2f31·3736·2020·2020·2020·2020·2020·2020··/176············0002e190:·2f31·3736·2020·2020·2020·2020·2020·2020··/176············
0002e1a0:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e1a0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002e1b0:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru0002e1b0:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru
0002e1c0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002e1c0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002e1d0:·6766·616e·205f·6661·6e63·6f6d·6d6f·6e72··gfan·_fancommonr0002e1d0:·6766·616e·205f·6661·6e63·6f6d·6d6f·6e72··gfan·_fancommonr
0002e1e0:·6566·696e·656d·656e·7420·2d2d·6865·6c70··efinement·--help0002e1e0:·6566·696e·656d·656e·7420·2d2d·6865·6c70··efinement·--help
0002e1f0:·203c·202f·746d·702f·4d32·2d33·3735·3734···<·/tmp/M2-375740002e1f0:·203c·202f·746d·702f·4d32·2d32·3733·3739···<·/tmp/M2-27379
0002e200:·3535·2d30·2f31·377c·0a7c·5468·6973·2070··55-0/17|.|This·p0002e200:·3536·2d30·2f31·377c·0a7c·5468·6973·2070··56-0/17|.|This·p
0002e210:·726f·6772·616d·2074·616b·6573·2074·776f··rogram·takes·two0002e210:·726f·6772·616d·2074·616b·6573·2074·776f··rogram·takes·two
0002e220:·2070·6f6c·7968·6564·7261·6c20·6661·6e73···polyhedral·fans0002e220:·2070·6f6c·7968·6564·7261·6c20·6661·6e73···polyhedral·fans
0002e230:·2061·6e64·2063·6f6d·7075·7465·7320·7468···and·computes·th0002e230:·2061·6e64·2063·6f6d·7075·7465·7320·7468···and·computes·th
0002e240:·6569·7220·636f·6d6d·6f6e·2072·6566·696e··eir·common·refin0002e240:·6569·7220·636f·6d6d·6f6e·2072·6566·696e··eir·common·refin
0002e250:·656d·656e·742e·207c·0a7c·4f70·7469·6f6e··ement.·|.|Option0002e250:·656d·656e·742e·207c·0a7c·4f70·7469·6f6e··ement.·|.|Option
0002e260:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············0002e260:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············
0002e270:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e270:·2020·2020·2020·2020·2020·2020·2020·2020··················
Offset 11844, 22 lines modifiedOffset 11844, 22 lines modified
0002e430:·2020·2020·2020·207c·0a7c·2043·6f6d·7075·········|.|·Compu0002e430:·2020·2020·2020·207c·0a7c·2043·6f6d·7075·········|.|·Compu
0002e440:·7465·2074·6865·2073·7461·626c·6520·696e··te·the·stable·in0002e440:·7465·2074·6865·2073·7461·626c·6520·696e··te·the·stable·in
0002e450:·7465·7273·6563·7469·6f6e·2e20·2020·2020··tersection.·····0002e450:·7465·7273·6563·7469·6f6e·2e20·2020·2020··tersection.·····
0002e460:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e460:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002e470:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e470:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002e480:·2020·2020·2020·207c·0a7c·7573·696e·6720·········|.|using·0002e480:·2020·2020·2020·207c·0a7c·7573·696e·6720·········|.|using·
0002e490:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/0002e490:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/
0002e4a0:·746d·702f·4d32·2d33·3735·3734·3535·2d30··tmp/M2-3757455-00002e4a0:·746d·702f·4d32·2d32·3733·3739·3536·2d30··tmp/M2-2737956-0
0002e4b0:·2f31·3738·2020·2020·2020·2020·2020·2020··/178············0002e4b0:·2f31·3738·2020·2020·2020·2020·2020·2020··/178············
0002e4c0:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e4c0:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002e4d0:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru0002e4d0:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru
0002e4e0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002e4e0:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002e4f0:·6766·616e·205f·6661·6e6c·696e·6b20·2d2d··gfan·_fanlink·--0002e4f0:·6766·616e·205f·6661·6e6c·696e·6b20·2d2d··gfan·_fanlink·--
0002e500:·6865·6c70·203c·202f·746d·702f·4d32·2d33··help·<·/tmp/M2-30002e500:·6865·6c70·203c·202f·746d·702f·4d32·2d32··help·<·/tmp/M2-2
0002e510:·3735·3734·3535·2d30·2f31·3830·2020·2020··757455-0/180····0002e510:·3733·3739·3536·2d30·2f31·3830·2020·2020··737956-0/180····
0002e520:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p0002e520:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p
0002e530:·726f·6772·616d·2074·616b·6573·2061·2070··rogram·takes·a·p0002e530:·726f·6772·616d·2074·616b·6573·2061·2070··rogram·takes·a·p
0002e540:·6f6c·7968·6564·7261·6c20·6661·6e20·616e··olyhedral·fan·an0002e540:·6f6c·7968·6564·7261·6c20·6661·6e20·616e··olyhedral·fan·an
0002e550:·6420·6120·7665·6374·6f72·2061·6e64·2063··d·a·vector·and·c0002e550:·6420·6120·7665·6374·6f72·2061·6e64·2063··d·a·vector·and·c
0002e560:·6f6d·7075·7465·7320·7468·6520·6c69·6e6b··omputes·the·link0002e560:·6f6d·7075·7465·7320·7468·6520·6c69·6e6b··omputes·the·link
0002e570:·206f·6620·7468·657c·0a7c·4f70·7469·6f6e···of·the|.|Option0002e570:·206f·6620·7468·657c·0a7c·4f70·7469·6f6e···of·the|.|Option
0002e580:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············0002e580:·733a·2020·2020·2020·2020·2020·2020·2020··s:··············
Offset 11899, 22 lines modifiedOffset 11899, 22 lines modified
0002e7a0:·2020·2020·2020·207c·0a7c·2043·6f6d·7075·········|.|·Compu0002e7a0:·2020·2020·2020·207c·0a7c·2043·6f6d·7075·········|.|·Compu
0002e7b0:·7465·7320·7468·6520·7374·6172·2069·6e73··tes·the·star·ins0002e7b0:·7465·7320·7468·6520·7374·6172·2069·6e73··tes·the·star·ins
0002e7c0:·7465·6164·2e20·5468·6520·7374·6172·2069··tead.·The·star·i0002e7c0:·7465·6164·2e20·5468·6520·7374·6172·2069··tead.·The·star·i
0002e7d0:·7320·6465·6669·6e65·6420·6173·2074·6865··s·defined·as·the0002e7d0:·7320·6465·6669·6e65·6420·6173·2074·6865··s·defined·as·the
0002e7e0:·2073·6d61·6c6c·6573·7420·706f·6c79·6865···smallest·polyhe0002e7e0:·2073·6d61·6c6c·6573·7420·706f·6c79·6865···smallest·polyhe
0002e7f0:·6472·616c·2066·617c·0a7c·7573·696e·6720··dral·fa|.|using·0002e7f0:·6472·616c·2066·617c·0a7c·7573·696e·6720··dral·fa|.|using·
0002e800:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/0002e800:·7465·6d70·6f72·6172·7920·6669·6c65·202f··temporary·file·/
0002e810:·746d·702f·4d32·2d33·3735·3734·3535·2d30··tmp/M2-3757455-00002e810:·746d·702f·4d32·2d32·3733·3739·3536·2d30··tmp/M2-2737956-0
0002e820:·2f31·3830·2020·2020·2020·2020·2020·2020··/180············0002e820:·2f31·3830·2020·2020·2020·2020·2020·2020··/180············
0002e830:·2020·2020·2020·2020·2020·2020·2020·2020··················0002e830:·2020·2020·2020·2020·2020·2020·2020·2020··················
0002e840:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru0002e840:·2020·2020·2020·207c·0a7c·202d·2d20·7275·········|.|·--·ru
0002e850:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/0002e850:·6e6e·696e·673a·202f·7573·722f·6269·6e2f··nning:·/usr/bin/
0002e860:·6766·616e·205f·6661·6e70·726f·6475·6374··gfan·_fanproduct0002e860:·6766·616e·205f·6661·6e70·726f·6475·6374··gfan·_fanproduct
0002e870:·202d·2d68·656c·7020·3c20·2f74·6d70·2f4d···--help·<·/tmp/M0002e870:·202d·2d68·656c·7020·3c20·2f74·6d70·2f4d···--help·<·/tmp/M
0002e880:·322d·3337·3537·3435·352d·302f·3138·3220··2-3757455-0/182·0002e880:·322d·3237·3337·3935·362d·302f·3138·3220··2-2737956-0/182·
0002e890:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p0002e890:·2020·2020·2020·207c·0a7c·5468·6973·2070·········|.|This·p
0002e8a0:·726f·6772·616d·2074·616b·6573·2074·776f··rogram·takes·two0002e8a0:·726f·6772·616d·2074·616b·6573·2074·776f··rogram·takes·two
Max diff block lines reached; 65554/75163 bytes (87.22%) of diff not shown.
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20.0 MB
macaulay2_1.24.11+ds-5_amd64.deb
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367 B
file list
    
Offset 1, 3 lines modifiedOffset 1, 3 lines modified
1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary
2 -rw-r--r--···0········0········0·····2112·2025-02-09·22:54:37.000000·control.tar.xz2 -rw-r--r--···0········0········0·····2112·2025-02-09·22:54:37.000000·control.tar.xz
3 -rw-r--r--···0········0········0··2578580·2025-02-09·22:54:37.000000·data.tar.xz3 -rw-r--r--···0········0········0··2577588·2025-02-09·22:54:37.000000·data.tar.xz
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98.0 B
control.tar.xz
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70.0 B
control.tar
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48.0 B
./md5sums
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30.0 B
./md5sums
Files differ
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20.0 MB
data.tar.xz
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20.0 MB
data.tar
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20.0 MB
./usr/bin/M2-binary
File has been modified after NT_GNU_BUILD_ID has been applied.
⊟
1.96 KB
readelf --wide --program-header {}
    
Offset 6, 23 lines modifiedOffset 6, 23 lines modified
6 Program·Headers:6 Program·Headers:
7 ··Type···········Offset···VirtAddr···········PhysAddr···········FileSiz··MemSiz···Flg·Align7 ··Type···········Offset···VirtAddr···········PhysAddr···········FileSiz··MemSiz···Flg·Align
8 ··PHDR···········0x000040·0x0000000000000040·0x0000000000000040·0x000348·0x000348·R···0x88 ··PHDR···········0x000040·0x0000000000000040·0x0000000000000040·0x000348·0x000348·R···0x8
9 ··INTERP·········0x0003cc·0x00000000000003cc·0x00000000000003cc·0x00001c·0x00001c·R···0x19 ··INTERP·········0x0003cc·0x00000000000003cc·0x00000000000003cc·0x00001c·0x00001c·R···0x1
10 ······[Requesting·program·interpreter:·/lib64/ld-linux-x86-64.so.2]10 ······[Requesting·program·interpreter:·/lib64/ld-linux-x86-64.so.2]
11 ··LOAD···········0x000000·0x0000000000000000·0x0000000000000000·0x0579a8·0x0579a8·R···0x100011 ··LOAD···········0x000000·0x0000000000000000·0x0000000000000000·0x0579a8·0x0579a8·R···0x1000
12 ··LOAD···········0x058000·0x0000000000058000·0x0000000000058000·0x67aef1·0x67aef1·R·E·0x100012 ··LOAD···········0x058000·0x0000000000058000·0x0000000000058000·0x67aef1·0x67aef1·R·E·0x1000
13 ··LOAD···········0x6d3000·0x00000000006d3000·0x00000000006d3000·0x16e8ec·0x16e8ec·R···0x100013 ··LOAD···········0x6d3000·0x00000000006d3000·0x00000000006d3000·0x16e8cc·0x16e8cc·R···0x1000
14 ··LOAD···········0x8420c0·0x00000000008420c0·0x00000000008420c0·0x018aa0·0x408c50·RW··0x100014 ··LOAD···········0x8420c0·0x00000000008420c0·0x00000000008420c0·0x018aa0·0x408c50·RW··0x1000
15 ··DYNAMIC········0x857700·0x0000000000857700·0x0000000000857700·0x0003c0·0x0003c0·RW··0x815 ··DYNAMIC········0x857700·0x0000000000857700·0x0000000000857700·0x0003c0·0x0003c0·RW··0x8
16 ··NOTE···········0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x816 ··NOTE···········0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x8
17 ··NOTE···········0x0003a8·0x00000000000003a8·0x00000000000003a8·0x000024·0x000024·R···0x417 ··NOTE···········0x0003a8·0x00000000000003a8·0x00000000000003a8·0x000024·0x000024·R···0x4
18 ··NOTE···········0x8418cc·0x00000000008418cc·0x00000000008418cc·0x000020·0x000020·R···0x418 ··NOTE···········0x8418ac·0x00000000008418ac·0x00000000008418ac·0x000020·0x000020·R···0x4
19 ··TLS············0x8420c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x00166c·R···0x1019 ··TLS············0x8420c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x00166c·R···0x10
20 ··GNU_PROPERTY···0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x820 ··GNU_PROPERTY···0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x8
21 ··GNU_EH_FRAME···0x6ffc1c·0x00000000006ffc1c·0x00000000006ffc1c·0x020c84·0x020c84·R···0x421 ··GNU_EH_FRAME···0x6ffbfc·0x00000000006ffbfc·0x00000000006ffbfc·0x020c84·0x020c84·R···0x4
22 ··GNU_STACK······0x000000·0x0000000000000000·0x0000000000000000·0x000000·0x000000·RW··0x1022 ··GNU_STACK······0x000000·0x0000000000000000·0x0000000000000000·0x000000·0x000000·RW··0x10
23 ··GNU_RELRO······0x8420c0·0x00000000008420c0·0x00000000008420c0·0x017f40·0x017f40·R···0x123 ··GNU_RELRO······0x8420c0·0x00000000008420c0·0x00000000008420c0·0x017f40·0x017f40·R···0x1
  
24 ·Section·to·Segment·mapping:24 ·Section·to·Segment·mapping:
25 ··Segment·Sections...25 ··Segment·Sections...
26 ···00·····26 ···00·····
27 ···01·····.interp·27 ···01·····.interp·
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2.18 KB
readelf --wide --sections {}
    
Offset 14, 19 lines modifiedOffset 14, 19 lines modified
14 ··[·9]·.rela.dyn·········RELA············0000000000012378·012378·03e910·18···A··5···0··814 ··[·9]·.rela.dyn·········RELA············0000000000012378·012378·03e910·18···A··5···0··8
15 ··[10]·.rela.plt·········RELA············0000000000050c88·050c88·006d20·18··AI··5··26··815 ··[10]·.rela.plt·········RELA············0000000000050c88·050c88·006d20·18··AI··5··26··8
16 ··[11]·.init·············PROGBITS········0000000000058000·058000·000017·00··AX··0···0··416 ··[11]·.init·············PROGBITS········0000000000058000·058000·000017·00··AX··0···0··4
17 ··[12]·.plt··············PROGBITS········0000000000058020·058020·0048d0·10··AX··0···0·1617 ··[12]·.plt··············PROGBITS········0000000000058020·058020·0048d0·10··AX··0···0·16
18 ··[13]·.plt.got··········PROGBITS········000000000005c8f0·05c8f0·000020·08··AX··0···0··818 ··[13]·.plt.got··········PROGBITS········000000000005c8f0·05c8f0·000020·08··AX··0···0··8
19 ··[14]·.text·············PROGBITS········000000000005c940·05c940·6765a5·00··AX··0···0·6419 ··[14]·.text·············PROGBITS········000000000005c940·05c940·6765a5·00··AX··0···0·64
20 ··[15]·.fini·············PROGBITS········00000000006d2ee8·6d2ee8·000009·00··AX··0···0··420 ··[15]·.fini·············PROGBITS········00000000006d2ee8·6d2ee8·000009·00··AX··0···0··4
21 ··[16]·.rodata···········PROGBITS········00000000006d3000·6d3000·02cc1c·00···A··0···0·3221 ··[16]·.rodata···········PROGBITS········00000000006d3000·6d3000·02cbfc·00···A··0···0·32
22 ··[17]·.eh_frame_hdr·····PROGBITS········00000000006ffc1c·6ffc1c·020c84·00···A··0···0··422 ··[17]·.eh_frame_hdr·····PROGBITS········00000000006ffbfc·6ffbfc·020c84·00···A··0···0··4
23 ··[18]·.eh_frame·········PROGBITS········00000000007208a0·7208a0·0db320·00···A··0···0··823 ··[18]·.eh_frame·········PROGBITS········0000000000720880·720880·0db320·00···A··0···0··8
24 ··[19]·.gcc_except_table·PROGBITS········00000000007fbbc0·7fbbc0·045d0c·00···A··0···0··424 ··[19]·.gcc_except_table·PROGBITS········00000000007fbba0·7fbba0·045d0c·00···A··0···0··4
25 ··[20]·.note.ABI-tag·····NOTE············00000000008418cc·8418cc·000020·00···A··0···0··425 ··[20]·.note.ABI-tag·····NOTE············00000000008418ac·8418ac·000020·00···A··0···0··4
26 ··[21]·.tbss·············NOBITS··········00000000008420c0·8420c0·00166c·00·WAT··0···0·1626 ··[21]·.tbss·············NOBITS··········00000000008420c0·8420c0·00166c·00·WAT··0···0·16
27 ··[22]·.init_array·······INIT_ARRAY······00000000008420c0·8420c0·000378·08··WA··0···0··827 ··[22]·.init_array·······INIT_ARRAY······00000000008420c0·8420c0·000378·08··WA··0···0··8
28 ··[23]·.fini_array·······FINI_ARRAY······0000000000842438·842438·000008·08··WA··0···0··828 ··[23]·.fini_array·······FINI_ARRAY······0000000000842438·842438·000008·08··WA··0···0··8
29 ··[24]·.data.rel.ro······PROGBITS········0000000000842440·842440·0152c0·00··WA··0···0·3229 ··[24]·.data.rel.ro······PROGBITS········0000000000842440·842440·0152c0·00··WA··0···0·32
30 ··[25]·.dynamic··········DYNAMIC·········0000000000857700·857700·0003c0·10··WA··6···0··830 ··[25]·.dynamic··········DYNAMIC·········0000000000857700·857700·0003c0·10··WA··6···0··8
31 ··[26]·.got··············PROGBITS········0000000000857ac0·857ac0·002538·08··WA··0···0··831 ··[26]·.got··············PROGBITS········0000000000857ac0·857ac0·002538·08··WA··0···0··8
32 ··[27]·.data·············PROGBITS········000000000085a000·85a000·000b60·00··WA··0···0·3232 ··[27]·.data·············PROGBITS········000000000085a000·85a000·000b60·00··WA··0···0·32
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12.0 KB
readelf --wide --symbols {}
    
Offset 1209, 15 lines modifiedOffset 1209, 15 lines modified
1209 ··1205:·000000000085ad18·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_TypeError1209 ··1205:·000000000085ad18·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_TypeError
1210 ··1206:·000000000048f5d0···557·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz25Sublattice_RepresentationI10__gmp_exprIA1_12__mpz_structS3_EED1Ev1210 ··1206:·000000000048f5d0···557·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz25Sublattice_RepresentationI10__gmp_exprIA1_12__mpz_structS3_EED1Ev
1211 ··1207:·0000000000849cb0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint8@LIBFFI_BASE_8.0·(28)1211 ··1207:·0000000000849cb0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint8@LIBFFI_BASE_8.0·(28)
1212 ··1208:·000000000084b260····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d113delegate_baseE1212 ··1208:·000000000084b260····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d113delegate_baseE
1213 ··1209:·000000000085ae80·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt5ctypeIcE2idE@GLIBCXX_3.4·(5)1213 ··1209:·000000000085ae80·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt5ctypeIcE2idE@GLIBCXX_3.4·(5)
1214 ··1210:·00000000000f7520···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EED1Ev1214 ··1210:·00000000000f7520···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EED1Ev
1215 ··1211:·0000000000849b80····24·OBJECT··WEAK···DEFAULT···24·_ZTISt13runtime_error@GLIBCXX_3.4·(5)1215 ··1211:·0000000000849b80····24·OBJECT··WEAK···DEFAULT···24·_ZTISt13runtime_error@GLIBCXX_3.4·(5)
1216 ··1212:·00000000006f7030····25·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d07no_copyE1216 ··1212:·00000000006f7010····25·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d07no_copyE
1217 ··1213:·0000000000491810··6178·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz4ConeI10__gmp_exprIA1_12__mpz_structS3_EEC2IS4_EENS_4Type9InputTypeERKNS_6MatrixIT_EE1217 ··1213:·0000000000491810··6178·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz4ConeI10__gmp_exprIA1_12__mpz_structS3_EEC2IS4_EENS_4Type9InputTypeERKNS_6MatrixIT_EE
1218 ··1214:·000000000048f800···570·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz13HilbertSeriesD2Ev1218 ··1214:·000000000048f800···570·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz13HilbertSeriesD2Ev
1219 ··1215:·0000000000490ad0···281·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_IN11libnormaliz6MatrixI10__gmp_exprIA1_12__mpz_structS4_EEESaIS6_EESaIS8_EED2Ev1219 ··1215:·0000000000490ad0···281·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_IN11libnormaliz6MatrixI10__gmp_exprIA1_12__mpz_structS4_EEESaIS6_EESaIS8_EED2Ev
1220 ··1216:·000000000085ad60···272·OBJECT··GLOBAL·DEFAULT···28·_ZSt4cerr@GLIBCXX_3.4·(5)1220 ··1216:·000000000085ad60···272·OBJECT··GLOBAL·DEFAULT···28·_ZSt4cerr@GLIBCXX_3.4·(5)
1221 ··1217:·0000000000105c00····64·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEED2Ev1221 ··1217:·0000000000105c00····64·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEED2Ev
1222 ··1218:·00000000000f4bd0···387·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIiSaIiEE17_M_default_appendEm1222 ··1218:·00000000000f4bd0···387·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIiSaIiEE17_M_default_appendEm
1223 ··1219:·000000000084b250····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d09no_assignE1223 ··1219:·000000000084b250····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d09no_assignE
Offset 1229, 25 lines modifiedOffset 1229, 25 lines modified
1229 ··1225:·00000000000f4120·····7·FUNC····WEAK···DEFAULT···14·_ZNKSt5ctypeIcE8do_widenEc1229 ··1225:·00000000000f4120·····7·FUNC····WEAK···DEFAULT···14·_ZNKSt5ctypeIcE8do_widenEc
1230 ··1226:·00000000002f0920···137·FUNC····WEAK···DEFAULT···14·_ZN3NTL20default_BlockDestroyINS_3VecINS_2ZZEEEEEvPT_l1230 ··1226:·00000000002f0920···137·FUNC····WEAK···DEFAULT···14·_ZN3NTL20default_BlockDestroyINS_3VecINS_2ZZEEEEEvPT_l
1231 ··1227:·000000000048fcd0···783·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz11FusionBasicD2Ev1231 ··1227:·000000000048fcd0···783·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz11FusionBasicD2Ev
1232 ··1228:·0000000000491210···189·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz15STANLEYDATA_intESaIS2_EE8_M_clearEv1232 ··1228:·0000000000491210···189·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz15STANLEYDATA_intESaIS2_EE8_M_clearEv
1233 ··1229:·000000000048f540···139·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_I10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EESaIS5_EED2Ev1233 ··1229:·000000000048f540···139·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_I10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EESaIS5_EED2Ev
1234 ··1230:·0000000000491810··6178·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz4ConeI10__gmp_exprIA1_12__mpz_structS3_EEC1IS4_EENS_4Type9InputTypeERKNS_6MatrixIT_EE1234 ··1230:·0000000000491810··6178·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz4ConeI10__gmp_exprIA1_12__mpz_structS3_EEC1IS4_EENS_4Type9InputTypeERKNS_6MatrixIT_EE
1235 ··1231:·00000000000f7bb0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIiSaIiEED2Ev1235 ··1231:·00000000000f7bb0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIiSaIiEED2Ev
1236 ··1232:·00000000006f7210····22·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d14taskE1236 ··1232:·00000000006f71f0····22·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d14taskE
1237 ··1233:·000000000085af90·····8·OBJECT··GLOBAL·DEFAULT···28·stdin@GLIBC_2.2.5·(10)1237 ··1233:·000000000085af90·····8·OBJECT··GLOBAL·DEFAULT···28·stdin@GLIBC_2.2.5·(10)
1238 ··1234:·0000000000849c50····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint16@LIBFFI_BASE_8.0·(28)1238 ··1234:·0000000000849c50····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint16@LIBFFI_BASE_8.0·(28)
1239 ··1235:·00000000004913c0···153·FUNC····WEAK···DEFAULT···14·_ZSt16__do_uninit_copyIN9__gnu_cxx17__normal_iteratorIPK10__gmp_exprIA1_12__mpz_structS4_ESt6vectorIS5_SaIS5_EEEEPS5_ET0_T_SE_SD_1239 ··1235:·00000000004913c0···153·FUNC····WEAK···DEFAULT···14·_ZSt16__do_uninit_copyIN9__gnu_cxx17__normal_iteratorIPK10__gmp_exprIA1_12__mpz_structS4_ESt6vectorIS5_SaIS5_EEEEPS5_ET0_T_SE_SD_
1240 ··1236:·00000000005a46d0·····7·FUNC····GLOBAL·DEFAULT···14·GC_get_full_gc_total_time1240 ··1236:·00000000005a46d0·····7·FUNC····GLOBAL·DEFAULT···14·GC_get_full_gc_total_time
1241 ··1237:·0000000000435fe0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIdSaIdEED2Ev1241 ··1237:·0000000000435fe0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIdSaIdEED2Ev
1242 ··1238:·000000000085af80·····8·OBJECT··GLOBAL·DEFAULT···28·rl_basic_word_break_characters1242 ··1238:·000000000085af80·····8·OBJECT··GLOBAL·DEFAULT···28·rl_basic_word_break_characters
1243 ··1239:·000000000026c9f0···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_realloc_appendIJmEEEvDpOT_1243 ··1239:·000000000026c9f0···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_realloc_appendIJmEEEvDpOT_
1244 ··1240:·00000000000f7010···674·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx119to_stringEm1244 ··1240:·00000000000f7010···674·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx119to_stringEm
1245 ··1241:·00000000002a9820····33·FUNC····WEAK···DEFAULT···14·_ZNSt14_Function_baseD2Ev1245 ··1241:·00000000002a9820····33·FUNC····WEAK···DEFAULT···14·_ZNSt14_Function_baseD2Ev
1246 ··1242:·00000000006f7280····32·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d113callback_baseE1246 ··1242:·00000000006f7260····32·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d113callback_baseE
1247 ··1243:·000000000084b298····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d14taskE1247 ··1243:·000000000084b298····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d14taskE
1248 ··1244:·000000000085aba0·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt7__cxx118messagesIcE2idE@GLIBCXX_3.4.21·(4)1248 ··1244:·000000000085aba0·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt7__cxx118messagesIcE2idE@GLIBCXX_3.4.21·(4)
1249 ··1245:·00000000008499a8···128·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)1249 ··1245:·00000000008499a8···128·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)
1250 ··1246:·00000000000f7c00···548·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EE17_M_realloc_appendIJRKS5_EEEvDpOT_1250 ··1246:·00000000000f7c00···548·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EE17_M_realloc_appendIJRKS5_EEEvDpOT_
1251 ··1247:·000000000048f4b0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIjSaIjEED1Ev1251 ··1247:·000000000048f4b0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIjSaIjEED1Ev
1252 ··1248:·000000000085acd0·····4·OBJECT··GLOBAL·DEFAULT···28·history_length1252 ··1248:·000000000085acd0·····4·OBJECT··GLOBAL·DEFAULT···28·history_length
1253 ··1249:·0000000000849db8····32·OBJECT··WEAK···DEFAULT···24·_ZTTNSt7__cxx1119basic_ostringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)1253 ··1249:·0000000000849db8····32·OBJECT··WEAK···DEFAULT···24·_ZTTNSt7__cxx1119basic_ostringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)
Offset 1265, 15 lines modifiedOffset 1265, 15 lines modified
1265 ··1261:·0000000000849e68···128·OBJECT··WEAK···DEFAULT···24·_ZTVSt15basic_streambufIcSt11char_traitsIcEE@GLIBCXX_3.4·(5)1265 ··1261:·0000000000849e68···128·OBJECT··WEAK···DEFAULT···24·_ZTVSt15basic_streambufIcSt11char_traitsIcEE@GLIBCXX_3.4·(5)
1266 ··1262:·0000000000133180···103·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d19fold_treeINS1_9tree_nodeEEEvPNS1_4nodeERKNS1_14execution_dataE1266 ··1262:·0000000000133180···103·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d19fold_treeINS1_9tree_nodeEEEvPNS1_4nodeERKNS1_14execution_dataE
1267 ··1263:·0000000000490910····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseI10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EED1Ev1267 ··1263:·0000000000490910····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseI10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EED1Ev
1268 ··1264:·000000000085abb0·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_is_strictly_neg@LIBMPFI_0.0.0·(6)1268 ··1264:·000000000085abb0·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_is_strictly_neg@LIBMPFI_0.0.0·(6)
1269 ··1265:·000000000085af00·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt7__cxx117collateIcE2idE@GLIBCXX_3.4.21·(4)1269 ··1265:·000000000085af00·····8·OBJECT··GLOBAL·DEFAULT···28·_ZNSt7__cxx117collateIcE2idE@GLIBCXX_3.4.21·(4)
1270 ··1266:·000000000048f5d0···557·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz25Sublattice_RepresentationI10__gmp_exprIA1_12__mpz_structS3_EED2Ev1270 ··1266:·000000000048f5d0···557·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz25Sublattice_RepresentationI10__gmp_exprIA1_12__mpz_structS3_EED2Ev
1271 ··1267:·000000000012a0f0···193·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d014atomic_do_onceIZNS0_2d110task_arena10initializeEvEUlvE_EEvRKT_RSt6atomicINS1_13do_once_stateEE1271 ··1267:·000000000012a0f0···193·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d014atomic_do_onceIZNS0_2d110task_arena10initializeEvEUlvE_EEvRKT_RSt6atomicINS1_13do_once_stateEE
1272 ··1268:·00000000006f7240····56·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d18ets_baseILNS1_18ets_key_usage_typeE1EEE1272 ··1268:·00000000006f7220····56·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d18ets_baseILNS1_18ets_key_usage_typeE1EEE
1273 ··1269:·000000000085ad00·····8·OBJECT··GLOBAL·DEFAULT···28·stderr@GLIBC_2.2.5·(10)1273 ··1269:·000000000085ad00·····8·OBJECT··GLOBAL·DEFAULT···28·stderr@GLIBC_2.2.5·(10)
1274 ··1270:·000000000026c9c0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseImSaImEED1Ev1274 ··1270:·000000000026c9c0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseImSaImEED1Ev
1275 ··1271:·00000000002c9220···395·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIlSaIlEE17_M_default_appendEm1275 ··1271:·00000000002c9220···395·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIlSaIlEE17_M_default_appendEm
1276 ··1272:·00000000004912d0···233·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz11STANLEYDATAI10__gmp_exprIA1_12__mpz_structS5_EEESaIS7_EE8_M_clearEv1276 ··1272:·00000000004912d0···233·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz11STANLEYDATAI10__gmp_exprIA1_12__mpz_structS5_EEESaIS7_EE8_M_clearEv
1277 ··1273:·000000000084a010····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_sint8@LIBFFI_BASE_8.0·(28)1277 ··1273:·000000000084a010····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_sint8@LIBFFI_BASE_8.0·(28)
1278 ··1274:·000000000085ab98·····4·OBJECT··GLOBAL·DEFAULT···28·MATHIC_VERSION_STRING1278 ··1274:·000000000085ab98·····4·OBJECT··GLOBAL·DEFAULT···28·MATHIC_VERSION_STRING
1279 ··1275:·00000000000f7520···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EED2Ev1279 ··1275:·00000000000f7520···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorINSt7__cxx1112basic_stringIcSt11char_traitsIcESaIcEEESaIS5_EED2Ev
Offset 1293, 35 lines modifiedOffset 1293, 35 lines modified
1293 ··1289:·000000000085ae70·····8·OBJECT··GLOBAL·DEFAULT···28·xmlFree@LIBXML2_2.4.30·(13)1293 ··1289:·000000000085ae70·····8·OBJECT··GLOBAL·DEFAULT···28·xmlFree@LIBXML2_2.4.30·(13)
1294 ··1290:·000000000084b228····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d07no_copyE1294 ··1290:·000000000084b228····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d07no_copyE
1295 ··1291:·0000000000490e40···166·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz9FACETDATAI10__gmp_exprIA1_12__mpz_structS5_EEESaIS7_EE8_M_clearEv1295 ··1291:·0000000000490e40···166·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1110_List_baseIN11libnormaliz9FACETDATAI10__gmp_exprIA1_12__mpz_structS5_EEESaIS7_EE8_M_clearEv
1296 ··1292:·00000000005347a0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIcSaIcEED1Ev1296 ··1292:·00000000005347a0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIcSaIcEED1Ev
1297 ··1293:·0000000000849980····40·OBJECT··WEAK···DEFAULT···24·_ZTVSt12domain_error@GLIBCXX_3.4·(5)1297 ··1293:·0000000000849980····40·OBJECT··WEAK···DEFAULT···24·_ZTVSt12domain_error@GLIBCXX_3.4·(5)
1298 ··1294:·0000000000105c40····77·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEED0Ev1298 ··1294:·0000000000105c40····77·FUNC····WEAK···DEFAULT···14·_ZNSt7__cxx1115basic_stringbufIcSt11char_traitsIcESaIcEED0Ev
1299 ··1295:·000000000085af60·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_StopIteration1299 ··1295:·000000000085af60·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_StopIteration
1300 ··1296:·00000000006f7180····32·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d113delegate_baseE1300 ··1296:·00000000006f7160····32·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d113delegate_baseE
1301 ··1297:·0000000000849a50····80·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1119basic_ostringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)1301 ··1297:·0000000000849a50····80·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1119basic_ostringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)
1302 ··1298:·0000000000849c90····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_sint64@LIBFFI_BASE_8.0·(28)1302 ··1298:·0000000000849c90····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_sint64@LIBFFI_BASE_8.0·(28)
1303 ··1299:·000000000065de90····29·FUNC····WEAK···DEFAULT···14·_ZNSt5mutex4lockEv1303 ··1299:·000000000065de90····29·FUNC····WEAK···DEFAULT···14·_ZNSt5mutex4lockEv
1304 ··1300:·00000000004909c0···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_IdSaIdEESaIS1_EED1Ev1304 ··1300:·00000000004909c0···101·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIS_IdSaIdEESaIS1_EED1Ev
1305 ··1301:·0000000000849ad0···120·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1118basic_stringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)1305 ··1301:·0000000000849ad0···120·OBJECT··WEAK···DEFAULT···24·_ZTVNSt7__cxx1118basic_stringstreamIcSt11char_traitsIcESaIcEEE@GLIBCXX_3.4.21·(4)
1306 ··1302:·000000000085ae88·····4·OBJECT··GLOBAL·DEFAULT···28·rl_catch_signals1306 ··1302:·000000000085ae88·····4·OBJECT··GLOBAL·DEFAULT···28·rl_catch_signals
1307 ··1303:·000000000085aba8·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_RuntimeError1307 ··1303:·000000000085aba8·····8·OBJECT··GLOBAL·DEFAULT···28·PyExc_RuntimeError
1308 ··1304:·0000000000849be0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_float@LIBFFI_BASE_8.0·(28)1308 ··1304:·0000000000849be0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_float@LIBFFI_BASE_8.0·(28)
1309 ··1305:·000000000085aec0·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_cmp@LIBMPFI_0.0.0·(6)1309 ··1305:·000000000085aec0·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_cmp@LIBMPFI_0.0.0·(6)
1310 ··1306:·00000000006fbb62·····1·OBJECT··UNIQUE·DEFAULT···16·_ZSt19piecewise_construct1310 ··1306:·00000000006fbb42·····1·OBJECT··UNIQUE·DEFAULT···16·_ZSt19piecewise_construct
1311 ··1307:·0000000000490ef0···795·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz14CONVEXHULLDATAI10__gmp_exprIA1_12__mpz_structS3_EED1Ev1311 ··1307:·0000000000490ef0···795·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz14CONVEXHULLDATAI10__gmp_exprIA1_12__mpz_structS3_EED1Ev
1312 ··1308:·0000000000849ee8····32·OBJECT··WEAK···DEFAULT···24·_ZTTSt14basic_ifstreamIcSt11char_traitsIcEE@GLIBCXX_3.4·(5)1312 ··1308:·0000000000849ee8····32·OBJECT··WEAK···DEFAULT···24·_ZTTSt14basic_ifstreamIcSt11char_traitsIcEE@GLIBCXX_3.4·(5)
1313 ··1309:·000000000085ad10·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_is_strictly_pos@LIBMPFI_0.0.0·(6)1313 ··1309:·000000000085ad10·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_is_strictly_pos@LIBMPFI_0.0.0·(6)
1314 ··1310:·000000000085aec8·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_cmp_d@LIBMPFI_0.0.0·(6)1314 ··1310:·000000000085aec8·····8·OBJECT··GLOBAL·DEFAULT···28·mpfi_cmp_d@LIBMPFI_0.0.0·(6)
1315 ··1311:·000000000048f4b0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIjSaIjEED2Ev1315 ··1311:·000000000048f4b0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIjSaIjEED2Ev
1316 ··1312:·0000000000849ab0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_pointer@LIBFFI_BASE_8.0·(28)1316 ··1312:·0000000000849ab0····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_pointer@LIBFFI_BASE_8.0·(28)
1317 ··1313:·0000000000490c60···480·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIN11libnormaliz13OurPolynomialI10__gmp_exprIA1_12__mpz_structS4_EEESaIS6_EED2Ev1317 ··1313:·0000000000490c60···480·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIN11libnormaliz13OurPolynomialI10__gmp_exprIA1_12__mpz_structS4_EEESaIS6_EED2Ev
1318 ··1314:·000000000085af40·····4·OBJECT··GLOBAL·DEFAULT···28·rl_attempted_completion_over1318 ··1314:·000000000085af40·····4·OBJECT··GLOBAL·DEFAULT···28·rl_attempted_completion_over
1319 ··1315:·0000000000335c20···395·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_default_appendEm1319 ··1315:·0000000000335c20···395·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_default_appendEm
1320 ··1316:·00000000006f7010····30·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d111task_traitsE1320 ··1316:·00000000006f6ff0····30·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d111task_traitsE
1321 ··1317:·0000000000129770····30·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d118task_group_contextD2Ev1321 ··1317:·0000000000129770····30·FUNC····WEAK···DEFAULT···14·_ZN3tbb6detail2d118task_group_contextD2Ev
1322 ··1318:·0000000000849c20····40·OBJECT··WEAK···DEFAULT···24·_ZTVSt14overflow_error@GLIBCXX_3.4·(5)1322 ··1318:·0000000000849c20····40·OBJECT··WEAK···DEFAULT···24·_ZTVSt14overflow_error@GLIBCXX_3.4·(5)
1323 ··1319:·00000000000f4d60···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIiSaIiEE17_M_realloc_appendIJRKiEEEvDpOT_1323 ··1319:·00000000000f4d60···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIiSaIiEE17_M_realloc_appendIJRKiEEEvDpOT_
1324 ··1320:·0000000000490910····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseI10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EED2Ev1324 ··1320:·0000000000490910····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseI10__gmp_exprIA1_12__mpz_structS2_ESaIS3_EED2Ev
1325 ··1321:·000000000084aff8····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d111task_traitsE1325 ··1321:·000000000084aff8····16·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d111task_traitsE
1326 ··1322:·0000000000849dd8····80·OBJECT··WEAK···DEFAULT···24·_ZTVSi@GLIBCXX_3.4·(5)1326 ··1322:·0000000000849dd8····80·OBJECT··WEAK···DEFAULT···24·_ZTVSi@GLIBCXX_3.4·(5)
1327 ··1323:·0000000000849e50····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint32@LIBFFI_BASE_8.0·(28)1327 ··1323:·0000000000849e50····24·OBJECT··GLOBAL·DEFAULT···24·ffi_type_uint32@LIBFFI_BASE_8.0·(28)
Offset 1365, 15 lines modifiedOffset 1365, 15 lines modified
1365 ··1361:·00000000004916f0···273·FUNC····WEAK···DEFAULT···14·_ZSt16__do_uninit_copyIN9__gnu_cxx17__normal_iteratorIPKSt6vectorI10__gmp_exprIA1_12__mpz_structS5_ESaIS6_EES2_IS8_SaIS8_EEEEPS8_ET0_T_SG_SF_1365 ··1361:·00000000004916f0···273·FUNC····WEAK···DEFAULT···14·_ZSt16__do_uninit_copyIN9__gnu_cxx17__normal_iteratorIPKSt6vectorI10__gmp_exprIA1_12__mpz_structS5_ESaIS6_EES2_IS8_SaIS8_EEEEPS8_ET0_T_SG_SF_
1366 ··1362:·0000000000849b60····32·OBJECT··WEAK···DEFAULT···24·_ZTIPKc@CXXABI_1.3·(8)1366 ··1362:·0000000000849b60····32·OBJECT··WEAK···DEFAULT···24·_ZTIPKc@CXXABI_1.3·(8)
1367 ··1363:·000000000029cab0···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_realloc_appendIJRKmEEEvDpOT_1367 ··1363:·000000000029cab0···242·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorImSaImEE17_M_realloc_appendIJRKmEEEvDpOT_
1368 ··1364:·0000000000490ef0···795·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz14CONVEXHULLDATAI10__gmp_exprIA1_12__mpz_structS3_EED2Ev1368 ··1364:·0000000000490ef0···795·FUNC····WEAK···DEFAULT···14·_ZN11libnormaliz14CONVEXHULLDATAI10__gmp_exprIA1_12__mpz_structS3_EED2Ev
1369 ··1365:·0000000000849aa0····16·OBJECT··WEAK···DEFAULT···24·_ZTIi@CXXABI_1.3·(8)1369 ··1365:·0000000000849aa0····16·OBJECT··WEAK···DEFAULT···24·_ZTIi@CXXABI_1.3·(8)
1370 ··1366:·00000000003de810···286·FUNC····WEAK···DEFAULT···14·_ZN3NTL3VecINS_2ZZEE9FixLengthEl1370 ··1366:·00000000003de810···286·FUNC····WEAK···DEFAULT···14·_ZN3NTL3VecINS_2ZZEE9FixLengthEl
1371 ··1367:·0000000000849ce0····16·OBJECT··WEAK···DEFAULT···24·_ZTIj@CXXABI_1.3·(8)1371 ··1367:·0000000000849ce0····16·OBJECT··WEAK···DEFAULT···24·_ZTIj@CXXABI_1.3·(8)
1372 ··1368:·00000000006f7050····27·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d09no_assignE1372 ··1368:·00000000006f7030····27·OBJECT··WEAK···DEFAULT···16·_ZTSN3tbb6detail2d09no_assignE
1373 ··1369:·000000000085aee0····16·OBJECT··GLOBAL·DEFAULT···28·_Py_NoneStruct1373 ··1369:·000000000085aee0····16·OBJECT··GLOBAL·DEFAULT···28·_Py_NoneStruct
1374 ··1370:·0000000000435fe0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIdSaIdEED1Ev1374 ··1370:·0000000000435fe0····33·FUNC····WEAK···DEFAULT···14·_ZNSt12_Vector_baseIdSaIdEED1Ev
1375 ··1371:·0000000000849d80····24·OBJECT··WEAK···DEFAULT···24·_ZTISt9bad_alloc@GLIBCXX_3.4·(5)1375 ··1371:·0000000000849d80····24·OBJECT··WEAK···DEFAULT···24·_ZTISt9bad_alloc@GLIBCXX_3.4·(5)
1376 ··1372:·000000000084b2c0····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d18ets_baseILNS1_18ets_key_usage_typeE1EEE1376 ··1372:·000000000084b2c0····40·OBJECT··WEAK···DEFAULT···24·_ZTIN3tbb6detail2d18ets_baseILNS1_18ets_key_usage_typeE1EEE
1377 ··1373:·00000000002e1dc0··1016·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIbSaIbEE13_M_insert_auxESt13_Bit_iteratorb1377 ··1373:·00000000002e1dc0··1016·FUNC····WEAK···DEFAULT···14·_ZNSt6vectorIbSaIbEE13_M_insert_auxESt13_Bit_iteratorb
1378 ··1374:·0000000000000000·····0·FUNC····GLOBAL·DEFAULT··UND·__gmpz_clear1378 ··1374:·0000000000000000·····0·FUNC····GLOBAL·DEFAULT··UND·__gmpz_clear
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174 KB
readelf --wide --relocs {}
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821 B
readelf --wide --notes {}
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
  
1 Displaying·notes·found·in:·.note.gnu.property1 Displaying·notes·found·in:·.note.gnu.property
2 ··Owner················Data·size·»  Description2 ··Owner················Data·size·»  Description
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4 Displaying·notes·found·in:·.note.gnu.build-id4 Displaying·notes·found·in:·.note.gnu.build-id
5 ··Owner················Data·size·»  Description5 ··Owner················Data·size·»  Description
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⊟
2.08 MB
readelf --wide --debug-dump=frames {}
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538 B
strings --all --bytes=8 {}
    
Offset 1, 8 lines modifiedOffset 1, 8 lines modified
1 ;/lib64/ld-linux-x86-64.so.21 /lib64/ld-linux-x86-64.so.2
2 __gmon_start__2 __gmon_start__
3 _ITM_deregisterTMCloneTable3 _ITM_deregisterTMCloneTable
4 _ITM_registerTMCloneTable4 _ITM_registerTMCloneTable
5 cblas_dscal5 cblas_dscal
6 cblas_dtrsm6 cblas_dtrsm
7 cblas_sscal7 cblas_sscal
8 cblas_dgemm8 cblas_dgemm
Offset 12659, 15 lines modifiedOffset 12659, 15 lines modified
12659 evaluate_evalSequenceHadError12659 evaluate_evalSequenceHadError
12660 errorreturn12660 errorreturn
12661 evaluate_backtrace12661 evaluate_backtrace
12662 lastCode12662 lastCode
12663 tryEvalSuccess12663 tryEvalSuccess
12664 gcc·14.2.012664 gcc·14.2.0
12665 x86_64-Linux-Debian-1312665 x86_64-Linux-Debian-13
12666 6.1.0-37-cloud-amd6412666 6.12.22+bpo-amd64
12667 m2-compile-node12667 m2-compile-node
12668 %d.%d.%d12668 %d.%d.%d
12669 not·present12669 not·present
12670 x86_64-pc-linux-gnu12670 x86_64-pc-linux-gnu
12671 1.24.11-bin12671 1.24.11-bin
12672 whichway12672 whichway
12673 sortlist12673 sortlist
⊟
5.69 MB
objdump --line-numbers --disassemble --demangle --reloc --no-show-raw-insn --section=.text {}
Max HTML report size reached
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1.36 MB
readelf --wide --decompress --hex-dump=.rodata {}
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1.07 MB
readelf --wide --decompress --hex-dump=.eh_frame_hdr {}
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7.17 MB
readelf --wide --decompress --hex-dump=.eh_frame {}
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2.28 MB
readelf --wide --decompress --hex-dump=.gcc_except_table {}
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102 KB
readelf --wide --decompress --hex-dump=.data.rel.ro {}
    
Offset 1780, 106 lines modifiedOffset 1780, 106 lines modified
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1862 ··0x00849870·28406f00·00000000·98406f00·00000000·(@o......@o..... 
1863 ··0x00849880·08416f00·00000000·80416f00·00000000·.Ao......Ao.....1862 ··0x00849870·08406f00·00000000·78406f00·00000000·.@o.....x@o.....
 1863 ··0x00849880·e8406f00·00000000·60416f00·00000000·.@o.....`Ao.....
1864 ··0x00849890·f8416f00·00000000·70426f00·00000000·.Ao.....pBo.....1864 ··0x00849890·d8416f00·00000000·50426f00·00000000·.Ao.....PBo.....
1865 ··0x008498a0·e8426f00·00000000·60436f00·00000000·.Bo.....`Co.....1865 ··0x008498a0·c8426f00·00000000·40436f00·00000000·.Bo.....@Co.....
1866 ··0x008498b0·e0436f00·00000000·60446f00·00000000·.Co.....`Do.....1866 ··0x008498b0·c0436f00·00000000·40446f00·00000000·.Co.....@Do.....
1867 ··0x008498c0·e0446f00·00000000·60456f00·00000000·.Do.....`Eo.....1867 ··0x008498c0·c0446f00·00000000·40456f00·00000000·.Do.....@Eo.....
1868 ··0x008498d0·e8456f00·00000000·70466f00·00000000·.Eo.....pFo.....1868 ··0x008498d0·c8456f00·00000000·50466f00·00000000·.Eo.....PFo.....
1869 ··0x008498e0·f8466f00·00000000·80476f00·00000000·.Fo......Go.....1869 ··0x008498e0·d8466f00·00000000·60476f00·00000000·.Fo.....`Go.....
1870 ··0x008498f0·10486f00·00000000·a0486f00·00000000·.Ho......Ho.....1870 ··0x008498f0·f0476f00·00000000·80486f00·00000000·.Go......Ho.....
1871 ··0x00849900·30496f00·00000000·c0496f00·00000000·0Io......Io.....1871 ··0x00849900·10496f00·00000000·a0496f00·00000000·.Io......Io.....
1872 ··0x00849910·584a6f00·00000000·f04a6f00·00000000·XJo......Jo.....1872 ··0x00849910·384a6f00·00000000·d04a6f00·00000000·8Jo......Jo.....
1873 ··0x00849920·884b6f00·00000000·204c6f00·00000000·.Ko.....·Lo.....1873 ··0x00849920·684b6f00·00000000·004c6f00·00000000·hKo......Lo.....
1874 ··0x00849930·c04c6f00·00000000·604d6f00·00000000·.Lo.....`Mo.....1874 ··0x00849930·a04c6f00·00000000·404d6f00·00000000·.Lo.....@Mo.....
1875 ··0x00849940·004e6f00·00000000·00728500·00000000·.No......r......1875 ··0x00849940·e04d6f00·00000000·00728500·00000000·.Mo......r......
1876 ··0x00849950·a0758500·00000000·28758500·00000000·.u......(u......1876 ··0x00849950·a0758500·00000000·28758500·00000000·.u......(u......
1877 ··0x00849960·90768500·00000000·18768500·00000000·.v.......v......1877 ··0x00849960·90768500·00000000·18768500·00000000·.v.......v......
1878 ··0x00849970·40516f00·00000000·00000000·00000000·@Qo.............1878 ··0x00849970·20516f00·00000000·00000000·00000000··Qo.............
1879 ··0x00849980·00000000·00000000·00000000·00000000·................1879 ··0x00849980·00000000·00000000·00000000·00000000·................
1880 ··0x00849990·00000000·00000000·00000000·00000000·................1880 ··0x00849990·00000000·00000000·00000000·00000000·................
1881 ··0x008499a0·00000000·00000000·00000000·00000000·................1881 ··0x008499a0·00000000·00000000·00000000·00000000·................
1882 ··0x008499b0·00000000·00000000·00000000·00000000·................1882 ··0x008499b0·00000000·00000000·00000000·00000000·................
1883 ··0x008499c0·00000000·00000000·00000000·00000000·................1883 ··0x008499c0·00000000·00000000·00000000·00000000·................
1884 ··0x008499d0·00000000·00000000·00000000·00000000·................1884 ··0x008499d0·00000000·00000000·00000000·00000000·................
1885 ··0x008499e0·00000000·00000000·00000000·00000000·................1885 ··0x008499e0·00000000·00000000·00000000·00000000·................
Offset 1980, 27 lines modifiedOffset 1980, 27 lines modified
1980 ··0x00849fd0·00000000·00000000·00000000·00000000·................1980 ··0x00849fd0·00000000·00000000·00000000·00000000·................
1981 ··0x00849fe0·00000000·00000000·00000000·00000000·................1981 ··0x00849fe0·00000000·00000000·00000000·00000000·................
1982 ··0x00849ff0·00000000·00000000·00000000·00000000·................1982 ··0x00849ff0·00000000·00000000·00000000·00000000·................
1983 ··0x0084a000·00000000·00000000·00000000·00000000·................1983 ··0x0084a000·00000000·00000000·00000000·00000000·................
1984 ··0x0084a010·00000000·00000000·00000000·00000000·................1984 ··0x0084a010·00000000·00000000·00000000·00000000·................
1985 ··0x0084a020·00000000·00000000·00000000·00000000·................1985 ··0x0084a020·00000000·00000000·00000000·00000000·................
1986 ··0x0084a030·00000000·00000000·00000000·00000000·................1986 ··0x0084a030·00000000·00000000·00000000·00000000·................
1987 ··0x0084a040·00000000·00000000·30596f00·00000000·........0Yo.....1987 ··0x0084a040·00000000·00000000·10596f00·00000000·.........Yo.....
1988 ··0x0084a050·00000000·00000000·00000000·00000000·................1988 ··0x0084a050·00000000·00000000·00000000·00000000·................
Max diff block lines reached; 90761/104144 bytes (87.15%) of diff not shown.
⊟
10.4 KB
readelf --wide --decompress --hex-dump=.data {}
    
Offset 4, 68 lines modifiedOffset 4, 68 lines modified
4 ··0x0085a010·46656220·20392032·3032352c·2032323a·Feb··9·2025,·22:4 ··0x0085a010·46656220·20392032·3032352c·2032323a·Feb··9·2025,·22:
5 ··0x0085a020·35343a33·37000000·00000000·00000000·54:37...........5 ··0x0085a020·35343a33·37000000·00000000·00000000·54:37...........
6 ··0x0085a030·05000000·00000000·70ba8400·00000000·........p.......6 ··0x0085a030·05000000·00000000·70ba8400·00000000·........p.......
7 ··0x0085a040·40a08400·00000000·05000000·4b000000·@...........K...7 ··0x0085a040·40a08400·00000000·05000000·4b000000·@...........K...
8 ··0x0085a050·0a000000·0f000000·0d000000·00000000·................8 ··0x0085a050·0a000000·0f000000·0d000000·00000000·................
9 ··0x0085a060·00000000·00000000·3d4f5f01·24d95b07·........=O_.$.[.9 ··0x0085a060·00000000·00000000·3d4f5f01·24d95b07·........=O_.$.[.
10 ··0x0085a070·18000000·00000000·00000000·00000000·................10 ··0x0085a070·18000000·00000000·00000000·00000000·................
11 ··0x0085a080·d0046e00·00000000·f8046e00·00000000·..n.......n.....11 ··0x0085a080·b0046e00·00000000·d8046e00·00000000·..n.......n.....
12 ··0x0085a090·40176e00·00000000·68176e00·00000000·@.n.....h.n.....12 ··0x0085a090·20176e00·00000000·48176e00·00000000··.n.....H.n.....
13 ··0x0085a0a0·40186e00·00000000·68186e00·00000000·@.n.....h.n.....13 ··0x0085a0a0·20186e00·00000000·48186e00·00000000··.n.....H.n.....
14 ··0x0085a0b0·48196e00·00000000·70196e00·00000000·H.n.....p.n.....14 ··0x0085a0b0·28196e00·00000000·50196e00·00000000·(.n.....P.n.....
15 ··0x0085a0c0·a81b6e00·00000000·d01b6e00·00000000·..n.......n.....15 ··0x0085a0c0·881b6e00·00000000·b01b6e00·00000000·..n.......n.....
16 ··0x0085a0d0·a0266e00·00000000·c8266e00·00000000·.&n......&n.....16 ··0x0085a0d0·80266e00·00000000·a8266e00·00000000·.&n......&n.....
17 ··0x0085a0e0·80276e00·00000000·a8276e00·00000000·.'n......'n.....17 ··0x0085a0e0·60276e00·00000000·88276e00·00000000·`'n......'n.....
18 ··0x0085a0f0·782b6e00·00000000·a02b6e00·00000000·x+n......+n.....18 ··0x0085a0f0·582b6e00·00000000·802b6e00·00000000·X+n......+n.....
19 ··0x0085a100·d02c6e00·00000000·f82c6e00·00000000·.,n......,n.....19 ··0x0085a100·b02c6e00·00000000·d82c6e00·00000000·.,n......,n.....
20 ··0x0085a110·a02e6e00·00000000·c82e6e00·00000000·..n.......n.....20 ··0x0085a110·802e6e00·00000000·a82e6e00·00000000·..n.......n.....
21 ··0x0085a120·103a6e00·00000000·383a6e00·00000000·.:n.....8:n..... 
22 ··0x0085a130·083f6e00·00000000·303f6e00·00000000·.?n.....0?n.....21 ··0x0085a120·f0396e00·00000000·183a6e00·00000000·.9n......:n.....
 22 ··0x0085a130·e83e6e00·00000000·103f6e00·00000000·.>n......?n.....
23 ··0x0085a140·a8506e00·00000000·d0506e00·00000000·.Pn......Pn.....23 ··0x0085a140·88506e00·00000000·b0506e00·00000000·.Pn......Pn.....
24 ··0x0085a150·98526e00·00000000·c0526e00·00000000·.Rn......Rn.....24 ··0x0085a150·78526e00·00000000·a0526e00·00000000·xRn......Rn.....
25 ··0x0085a160·f8526e00·00000000·20536e00·00000000·.Rn.....·Sn.....25 ··0x0085a160·d8526e00·00000000·00536e00·00000000·.Rn......Sn.....
26 ··0x0085a170·c8536e00·00000000·f0536e00·00000000·.Sn......Sn.....26 ··0x0085a170·a8536e00·00000000·d0536e00·00000000·.Sn......Sn.....
27 ··0x0085a180·f0546e00·00000000·18556e00·00000000·.Tn......Un.....27 ··0x0085a180·d0546e00·00000000·f8546e00·00000000·.Tn......Tn.....
28 ··0x0085a190·f8566e00·00000000·20576e00·00000000·.Vn.....·Wn.....28 ··0x0085a190·d8566e00·00000000·00576e00·00000000·.Vn......Wn.....
29 ··0x0085a1a0·60596e00·00000000·90596e00·00000000·`Yn......Yn.....29 ··0x0085a1a0·40596e00·00000000·70596e00·00000000·@Yn.....pYn.....
30 ··0x0085a1b0·485a6e00·00000000·705a6e00·00000000·HZn.....pZn.....30 ··0x0085a1b0·285a6e00·00000000·505a6e00·00000000·(Zn.....PZn.....
31 ··0x0085a1c0·605b6e00·00000000·885b6e00·00000000·`[n......[n.....31 ··0x0085a1c0·405b6e00·00000000·685b6e00·00000000·@[n.....h[n.....
32 ··0x0085a1d0·d05c6e00·00000000·f85c6e00·00000000·.\n......\n.....32 ··0x0085a1d0·b05c6e00·00000000·d85c6e00·00000000·.\n......\n.....
33 ··0x0085a1e0·20626e00·00000000·48626e00·00000000··bn.....Hbn.....33 ··0x0085a1e0·00626e00·00000000·28626e00·00000000·.bn.....(bn.....
34 ··0x0085a1f0·60636e00·00000000·88636e00·00000000·`cn......cn.....34 ··0x0085a1f0·40636e00·00000000·68636e00·00000000·@cn.....hcn.....
35 ··0x0085a200·28646e00·00000000·50646e00·00000000·(dn.....Pdn.....35 ··0x0085a200·08646e00·00000000·30646e00·00000000·.dn.....0dn.....
36 ··0x0085a210·00000000·00000000·00000000·00000000·................36 ··0x0085a210·00000000·00000000·00000000·00000000·................
37 ··0x0085a220·1b3d6d00·00000000·cc3d6d00·00000000·.=m......=m.....37 ··0x0085a220·1b3d6d00·00000000·cc3d6d00·00000000·.=m......=m.....
38 ··0x0085a230·2c426d00·00000000·005a6d00·00000000·,Bm......Zm.....38 ··0x0085a230·2c426d00·00000000·005a6d00·00000000·,Bm......Zm.....
39 ··0x0085a240·68326d00·00000000·e7306d00·00000000·h2m......0m.....39 ··0x0085a240·68326d00·00000000·e7306d00·00000000·h2m......0m.....
40 ··0x0085a250·025a6d00·00000000·133e6d00·00000000·.Zm......>m.....40 ··0x0085a250·025a6d00·00000000·133e6d00·00000000·.Zm......>m.....
41 ··0x0085a260·50436d00·00000000·d8386d00·00000000·PCm......8m.....41 ··0x0085a260·50436d00·00000000·d8386d00·00000000·PCm......8m.....
42 ··0x0085a270·9c516d00·00000000·045a6d00·00000000·.Qm......Zm.....42 ··0x0085a270·9c516d00·00000000·045a6d00·00000000·.Qm......Zm.....
43 ··0x0085a280·065a6d00·00000000·bd3f6d00·00000000·.Zm......?m.....43 ··0x0085a280·065a6d00·00000000·bd3f6d00·00000000·.Zm......?m.....
44 ··0x0085a290·ea3d6d00·00000000·7d326d00·00000000·.=m.....}2m.....44 ··0x0085a290·ea3d6d00·00000000·7d326d00·00000000·.=m.....}2m.....
45 ··0x0085a2a0·1e566d00·00000000·085a6d00·00000000·.Vm......Zm.....45 ··0x0085a2a0·1e566d00·00000000·085a6d00·00000000·.Vm......Zm.....
46 ··0x0085a2b0·be5a6d00·00000000·ae456d00·00000000·.Zm......Em.....46 ··0x0085a2b0·be5a6d00·00000000·ae456d00·00000000·.Zm......Em.....
47 ··0x0085a2c0·f8446d00·00000000·02626d00·00000000·.Dm......bm.....47 ··0x0085a2c0·f8446d00·00000000·ff616d00·00000000·.Dm......am.....
48 ··0x0085a2d0·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....48 ··0x0085a2d0·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....
49 ··0x0085a2e0·135a6d00·00000000·165a6d00·00000000·.Zm......Zm.....49 ··0x0085a2e0·135a6d00·00000000·165a6d00·00000000·.Zm......Zm.....
50 ··0x0085a2f0·1a3d6d00·00000000·b5316d00·00000000·.=m......1m.....50 ··0x0085a2f0·1a3d6d00·00000000·b5316d00·00000000·.=m......1m.....
51 ··0x0085a300·b35b6d00·00000000·84496d00·00000000·.[m......Im.....51 ··0x0085a300·b35b6d00·00000000·84496d00·00000000·.[m......Im.....
52 ··0x0085a310·2d656d00·00000000·6a466d00·00000000·-em.....jFm.....52 ··0x0085a310·2a656d00·00000000·6a466d00·00000000·*em.....jFm.....
53 ··0x0085a320·10456d00·00000000·195a6d00·00000000·.Em......Zm.....53 ··0x0085a320·10456d00·00000000·195a6d00·00000000·.Em......Zm.....
54 ··0x0085a330·99616d00·00000000·0c5e6d00·00000000·.am......^m.....54 ··0x0085a330·96616d00·00000000·0c5e6d00·00000000·.am......^m.....
55 ··0x0085a340·b44a6d00·00000000·51526d00·00000000·.Jm.....QRm.....55 ··0x0085a340·b44a6d00·00000000·51526d00·00000000·.Jm.....QRm.....
56 ··0x0085a350·08536d00·00000000·1b3d6d00·00000000·.Sm......=m.....56 ··0x0085a350·08536d00·00000000·1b3d6d00·00000000·.Sm......=m.....
57 ··0x0085a360·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....57 ··0x0085a360·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....
58 ··0x0085a370·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....58 ··0x0085a370·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....
59 ··0x0085a380·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....59 ··0x0085a380·1b3d6d00·00000000·1b3d6d00·00000000·.=m......=m.....
60 ··0x0085a390·1b3d6d00·00000000·505a6d00·00000000·.=m.....PZm.....60 ··0x0085a390·1b3d6d00·00000000·505a6d00·00000000·.=m.....PZm.....
61 ··0x0085a3a0·955a6d00·00000000·50466d00·00000000·.Zm.....PFm.....61 ··0x0085a3a0·955a6d00·00000000·50466d00·00000000·.Zm.....PFm.....
62 ··0x0085a3b0·d7586d00·00000000·745a6d00·00000000·.Xm.....tZm.....62 ··0x0085a3b0·d7586d00·00000000·745a6d00·00000000·.Xm.....tZm.....
63 ··0x0085a3c0·1b5a6d00·00000000·9b3e6d00·00000000·.Zm......>m.....63 ··0x0085a3c0·1b5a6d00·00000000·9b3e6d00·00000000·.Zm......>m.....
64 ··0x0085a3d0·80616d00·00000000·d7596d00·00000000·.am......Ym.....64 ··0x0085a3d0·7d616d00·00000000·d7596d00·00000000·}am......Ym.....
65 ··0x0085a3e0·025b6d00·00000000·1d5a6d00·00000000·.[m......Zm.....65 ··0x0085a3e0·025b6d00·00000000·1d5a6d00·00000000·.[m......Zm.....
66 ··0x0085a3f0·f65a6d00·00000000·60466d00·00000000·.Zm.....`Fm.....66 ··0x0085a3f0·f65a6d00·00000000·60466d00·00000000·.Zm.....`Fm.....
67 ··0x0085a400·495a6d00·00000000·4c5a6d00·00000000·IZm.....LZm.....67 ··0x0085a400·495a6d00·00000000·4c5a6d00·00000000·IZm.....LZm.....
68 ··0x0085a410·4f5a6d00·00000000·e65c6d00·00000000·OZm......\m.....68 ··0x0085a410·4f5a6d00·00000000·e65c6d00·00000000·OZm......\m.....
69 ··0x0085a420·525a6d00·00000000·555a6d00·00000000·RZm.....UZm.....69 ··0x0085a420·525a6d00·00000000·555a6d00·00000000·RZm.....UZm.....
70 ··0x0085a430·81386d00·00000000·585a6d00·00000000·.8m.....XZm.....70 ··0x0085a430·81386d00·00000000·585a6d00·00000000·.8m.....XZm.....
71 ··0x0085a440·5b5a6d00·00000000·2d326d00·00000000·[Zm.....-2m.....71 ··0x0085a440·5b5a6d00·00000000·2d326d00·00000000·[Zm.....-2m.....
Offset 127, 20 lines modifiedOffset 127, 20 lines modified
127 ··0x0085a7c0·945c6d00·00000000·b35b6d00·00000000·.\m......[m.....127 ··0x0085a7c0·945c6d00·00000000·b35b6d00·00000000·.\m......[m.....
128 ··0x0085a7d0·be5a6d00·00000000·0c5e6d00·00000000·.Zm......^m.....128 ··0x0085a7d0·be5a6d00·00000000·0c5e6d00·00000000·.Zm......^m.....
129 ··0x0085a7e0·0f366d00·00000000·08536d00·00000000·.6m......Sm.....129 ··0x0085a7e0·0f366d00·00000000·08536d00·00000000·.6m......Sm.....
130 ··0x0085a7f0·84496d00·00000000·7a4c6d00·00000000·.Im.....zLm.....130 ··0x0085a7f0·84496d00·00000000·7a4c6d00·00000000·.Im.....zLm.....
131 ··0x0085a800·535a6d00·00000000·bc586d00·00000000·SZm......Xm.....131 ··0x0085a800·535a6d00·00000000·bc586d00·00000000·SZm......Xm.....
132 ··0x0085a810·7a5a6d00·00000000·1e5a6d00·00000000·zZm......Zm.....132 ··0x0085a810·7a5a6d00·00000000·1e5a6d00·00000000·zZm......Zm.....
133 ··0x0085a820·595a6d00·00000000·2d5e6d00·00000000·YZm.....-^m.....133 ··0x0085a820·595a6d00·00000000·2d5e6d00·00000000·YZm.....-^m.....
134 ··0x0085a830·2d656d00·00000000·71456d00·00000000·-em.....qEm.....134 ··0x0085a830·2a656d00·00000000·71456d00·00000000·*em.....qEm.....
135 ··0x0085a840·80616d00·00000000·87616d00·00000000·.am......am.....135 ··0x0085a840·7d616d00·00000000·84616d00·00000000·}am......am.....
136 ··0x0085a850·6a466d00·00000000·66416d00·00000000·jFm.....fAm.....136 ··0x0085a850·6a466d00·00000000·66416d00·00000000·jFm.....fAm.....
137 ··0x0085a860·10456d00·00000000·8c496d00·00000000·.Em......Im.....137 ··0x0085a860·10456d00·00000000·8c496d00·00000000·.Em......Im.....
138 ··0x0085a870·195a6d00·00000000·7d5d6d00·00000000·.Zm.....}]m.....138 ··0x0085a870·195a6d00·00000000·7d5d6d00·00000000·.Zm.....}]m.....
139 ··0x0085a880·99616d00·00000000·74616d00·00000000·.am.....tam.....139 ··0x0085a880·96616d00·00000000·71616d00·00000000·.am.....qam.....
140 ··0x0085a890·715a6d00·00000000·a15c6d00·00000000·qZm......\m.....140 ··0x0085a890·715a6d00·00000000·a15c6d00·00000000·qZm......\m.....
141 ··0x0085a8a0·b45c6d00·00000000·c25c6d00·00000000·.\m......\m.....141 ··0x0085a8a0·b45c6d00·00000000·c25c6d00·00000000·.\m......\m.....
142 ··0x0085a8b0·d65c6d00·00000000·dc5c6d00·00000000·.\m......\m.....142 ··0x0085a8b0·d65c6d00·00000000·dc5c6d00·00000000·.\m......\m.....
143 ··0x0085a8c0·1b3d6d00·00000000·00000000·00000000·.=m.............143 ··0x0085a8c0·1b3d6d00·00000000·00000000·00000000·.=m.............
144 ··0x0085a8d0·25733a25·643a2065·72726f72·3a202573·%s:%d:·error:·%s144 ··0x0085a8d0·25733a25·643a2065·72726f72·3a202573·%s:%d:·error:·%s
145 ··0x0085a8e0·00000000·00000000·00000000·00000000·................145 ··0x0085a8e0·00000000·00000000·00000000·00000000·................
146 ··0x0085a8f0·25733a25·643a2564·3a206572·726f723a·%s:%d:%d:·error:146 ··0x0085a8f0·25733a25·643a2564·3a206572·726f723a·%s:%d:%d:·error:
Offset 163, 23 lines modifiedOffset 163, 23 lines modified
163 ··0x0085aa00·00000000·00000000·00000000·00000000·................163 ··0x0085aa00·00000000·00000000·00000000·00000000·................
164 ··0x0085aa10·00000000·00000000·00000000·00000000·................164 ··0x0085aa10·00000000·00000000·00000000·00000000·................
165 ··0x0085aa20·2f5e6d00·00000000·402d5400·00000000·/^m.....@-T.....165 ··0x0085aa20·2f5e6d00·00000000·402d5400·00000000·/^m.....@-T.....
166 ··0x0085aa30·01000000·00000000·365e6d00·00000000·........6^m.....166 ··0x0085aa30·01000000·00000000·365e6d00·00000000·........6^m.....
167 ··0x0085aa40·00000000·00000000·00000000·00000000·................167 ··0x0085aa40·00000000·00000000·00000000·00000000·................
168 ··0x0085aa50·00000000·00000000·00000000·00000000·................168 ··0x0085aa50·00000000·00000000·00000000·00000000·................
169 ··0x0085aa60·0a000000·00000000·64636261·00000000·........dcba....169 ··0x0085aa60·0a000000·00000000·64636261·00000000·........dcba....
170 ··0x0085aa70·c8326f00·00000000·f0326f00·00000000·.2o......2o.....170 ··0x0085aa70·a8326f00·00000000·d0326f00·00000000·.2o......2o.....
171 ··0x0085aa80·53656d00·00000000·6c626d00·00000000·Sem.....lbm.....171 ··0x0085aa80·50656d00·00000000·69626d00·00000000·Pem.....ibm.....
172 ··0x0085aa90·59636d00·00000000·f82a6f00·00000000·Ycm......*o.....172 ··0x0085aa90·56636d00·00000000·d82a6f00·00000000·Vcm......*o.....
173 ··0x0085aaa0·18336f00·00000000·40336f00·00000000·.3o.....@3o.....173 ··0x0085aaa0·f8326f00·00000000·20336f00·00000000·.2o.....·3o.....
174 ··0x0085aab0·a04e6f00·00000000·f0326f00·00000000·.No......2o.....174 ··0x0085aab0·804e6f00·00000000·d0326f00·00000000·.No......2o.....
175 ··0x0085aac0·e04e6f00·00000000·e04e6f00·00000000·.No......No.....175 ··0x0085aac0·c04e6f00·00000000·c04e6f00·00000000·.No......No.....
176 ··0x0085aad0·d8646d00·00000000·ee646d00·00000000·.dm......dm.....176 ··0x0085aad0·d5646d00·00000000·eb646d00·00000000·.dm......dm.....
177 ··0x0085aae0·084f6f00·00000000·53656d00·00000000·.Oo.....Sem.....177 ··0x0085aae0·e84e6f00·00000000·50656d00·00000000·.No.....Pem.....
178 ··0x0085aaf0·6c626d00·00000000·88316f00·00000000·lbm......1o.....178 ··0x0085aaf0·69626d00·00000000·68316f00·00000000·ibm.....h1o.....
179 ··0x0085ab00·304f6f00·00000000·584f6f00·00000000·0Oo.....XOo.....179 ··0x0085ab00·104f6f00·00000000·384f6f00·00000000·.Oo.....8Oo.....
180 ··0x0085ab10·804f6f00·00000000·e04e6f00·00000000·.Oo......No.....180 ··0x0085ab10·604f6f00·00000000·c04e6f00·00000000·`Oo......No.....
181 ··0x0085ab20·18336f00·00000000·32626d00·00000000·.3o.....2bm.....181 ··0x0085ab20·f8326f00·00000000·2f626d00·00000000·.2o...../bm.....
182 ··0x0085ab30·e04e6f00·00000000·584f6f00·00000000·.No.....XOo.....182 ··0x0085ab30·c04e6f00·00000000·384f6f00·00000000·.No.....8Oo.....
183 ··0x0085ab40·b04f6f00·00000000·03000000·14000000·.Oo.............183 ··0x0085ab40·904f6f00·00000000·03000000·14000000·.Oo.............
184 ··0x0085ab50·90738500·00000000·00000000·00000000·.s..............184 ··0x0085ab50·90738500·00000000·00000000·00000000·.s..............
  
⊟
866 B
readelf --wide --decompress --hex-dump=.gnu_debuglink {}
error from `readelf --wide --decompress --hex-dump=.gnu_debuglink {}`: readelf: Error: Unable to find program interpreter name readelf: Error: no .dynamic section in the dynamic segment
    
Offset 1, 7 lines modifiedOffset 1, 7 lines modified
  
1 Hex·dump·of·section·'.gnu_debuglink':1 Hex·dump·of·section·'.gnu_debuglink':
2 ··0x00000000·38323865·35356434·63333836·31643761·828e55d4c3861d7a 
3 ··0x00000010·64613536·39636562·61353161·32396664·da569ceba51a29fd2 ··0x00000000·65653537·32393830·39366361·61636462·ee57298096caacdb
 3 ··0x00000010·61626661·37323364·34303864·34616431·abfa723d408d4ad1
4 ··0x00000020·61383066·33622e64·65627567·00000000·a80f3b.debug....4 ··0x00000020·36363438·38342e64·65627567·00000000·664884.debug....
5 ··0x00000030·0c114448····························..DH5 ··0x00000030·56b19d36····························V..6
  
⊟
483 MB
macaulay2-dbgsym_1.24.11+ds-5_amd64.deb
⊟
367 B
file list
    
Offset 1, 3 lines modifiedOffset 1, 3 lines modified
1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary1 -rw-r--r--···0········0········0········4·2025-02-09·22:54:37.000000·debian-binary
2 -rw-r--r--···0········0········0······532·2025-02-09·22:54:37.000000·control.tar.xz2 -rw-r--r--···0········0········0······532·2025-02-09·22:54:37.000000·control.tar.xz
3 -rw-r--r--···0········0········0·48894964·2025-02-09·22:54:37.000000·data.tar.xz3 -rw-r--r--···0········0········0·48894728·2025-02-09·22:54:37.000000·data.tar.xz
⊟
636 B
control.tar.xz
⊟
608 B
control.tar
⊟
360 B
./control
    
Offset 5, 8 lines modifiedOffset 5, 8 lines modified
5 Architecture:·amd645 Architecture:·amd64
6 Maintainer:·Debian·Math·Team·<team+math@tracker.debian.org>6 Maintainer:·Debian·Math·Team·<team+math@tracker.debian.org>
7 Installed-Size:·515987 Installed-Size:·51598
8 Depends:·macaulay2·(=·1.24.11+ds-5)8 Depends:·macaulay2·(=·1.24.11+ds-5)
9 Section:·debug9 Section:·debug
10 Priority:·optional10 Priority:·optional
11 Description:·debug·symbols·for·macaulay211 Description:·debug·symbols·for·macaulay2
12 Build-Ids:·a7828e55d4c3861d7ada569ceba51a29fda80f3b12 Build-Ids:·f3ee57298096caacdbabfa723d408d4ad1664884
⊟
226 B
./md5sums
⊟
30.0 B
./md5sums
Files differ
⊟
178 B
line order
    
Offset 1, 1 lines modifiedOffset 1, 1 lines modified
1 usr/lib/debug/.build-id/a7/828e55d4c3861d7ada569ceba51a29fda80f3b.debug1 usr/lib/debug/.build-id/f3/ee57298096caacdbabfa723d408d4ad1664884.debug
⊟
483 MB
data.tar.xz
⊟
483 MB
data.tar
⊟
1.38 KB
file list
    
Offset 1, 10 lines modifiedOffset 1, 10 lines modified
1 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./1 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./
2 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/2 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/
3 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/3 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/
4 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/4 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/
5 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/5 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/
6 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/a7/6 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/f3/
7 -rw-r--r--···0·root·········(0)·root·········(0)·52825144·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/a7/828e55d4c3861d7ada569ceba51a29fda80f3b.debug7 -rw-r--r--···0·root·········(0)·root·········(0)·52825656·2025-02-09·22:54:37.000000·./usr/lib/debug/.build-id/f3/ee57298096caacdbabfa723d408d4ad1664884.debug
8 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/8 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/
9 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/doc/9 drwxr-xr-x···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/doc/
10 lrwxrwxrwx···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/doc/macaulay2-dbgsym·->·macaulay210 lrwxrwxrwx···0·root·········(0)·root·········(0)········0·2025-02-09·22:54:37.000000·./usr/share/doc/macaulay2-dbgsym·->·macaulay2
⊟
483 MB
./usr/lib/debug/.build-id/a7/828e55d4c3861d7ada569ceba51a29fda80f3b.debug vs.
./usr/lib/debug/.build-id/f3/ee57298096caacdbabfa723d408d4ad1664884.debug
File has been modified after NT_GNU_BUILD_ID has been applied. Files 97% similar despite different names
⊟
989 B
readelf --wide --file-header {}
error from `readelf --wide --file-header {}`: readelf: Error: Unable to find program interpreter name
    
Offset 6, 15 lines modifiedOffset 6, 15 lines modified
6 ··OS/ABI:····························UNIX·-·GNU6 ··OS/ABI:····························UNIX·-·GNU
7 ··ABI·Version:·······················07 ··ABI·Version:·······················0
8 ··Type:······························DYN·(Shared·object·file)8 ··Type:······························DYN·(Shared·object·file)
9 ··Machine:···························Advanced·Micro·Devices·X86-649 ··Machine:···························Advanced·Micro·Devices·X86-64
10 ··Version:···························0x110 ··Version:···························0x1
11 ··Entry·point·address:···············0xeea1011 ··Entry·point·address:···············0xeea10
12 ··Start·of·program·headers:··········64·(bytes·into·file)12 ··Start·of·program·headers:··········64·(bytes·into·file)
13 ··Start·of·section·headers:··········52822456·(bytes·into·file)13 ··Start·of·section·headers:··········52822968·(bytes·into·file)
14 ··Flags:·····························0x014 ··Flags:·····························0x0
15 ··Size·of·this·header:···············64·(bytes)15 ··Size·of·this·header:···············64·(bytes)
16 ··Size·of·program·headers:···········56·(bytes)16 ··Size·of·program·headers:···········56·(bytes)
17 ··Number·of·program·headers:·········1517 ··Number·of·program·headers:·········15
18 ··Size·of·section·headers:···········64·(bytes)18 ··Size·of·section·headers:···········64·(bytes)
19 ··Number·of·section·headers:·········4219 ··Number·of·section·headers:·········42
20 ··Section·header·string·table·index:·4120 ··Section·header·string·table·index:·41
⊟
2.0 KB
readelf --wide --program-header {}
error from `readelf --wide --program-header {}`: readelf: Error: Unable to find program interpreter name
    
Offset 5, 23 lines modifiedOffset 5, 23 lines modified
  
5 Program·Headers:5 Program·Headers:
6 ··Type···········Offset···VirtAddr···········PhysAddr···········FileSiz··MemSiz···Flg·Align6 ··Type···········Offset···VirtAddr···········PhysAddr···········FileSiz··MemSiz···Flg·Align
7 ··PHDR···········0x000040·0x0000000000000040·0x0000000000000040·0x000348·0x000348·R···0x87 ··PHDR···········0x000040·0x0000000000000040·0x0000000000000040·0x000348·0x000348·R···0x8
8 ··INTERP·········0x0003cc·0x00000000000003cc·0x00000000000003cc·0x000000·0x00001c·R···0x18 ··INTERP·········0x0003cc·0x00000000000003cc·0x00000000000003cc·0x000000·0x00001c·R···0x1
9 ··LOAD···········0x000000·0x0000000000000000·0x0000000000000000·0x0003cc·0x0579a8·R···0x10009 ··LOAD···········0x000000·0x0000000000000000·0x0000000000000000·0x0003cc·0x0579a8·R···0x1000
10 ··LOAD···········0x000000·0x0000000000058000·0x0000000000058000·0x000000·0x67aef1·R·E·0x100010 ··LOAD···········0x000000·0x0000000000058000·0x0000000000058000·0x000000·0x67aef1·R·E·0x1000
11 ··LOAD···········0x001000·0x00000000006d3000·0x00000000006d3000·0x16e8ec·0x16e8ec·R···0x100011 ··LOAD···········0x001000·0x00000000006d3000·0x00000000006d3000·0x16e8cc·0x16e8cc·R···0x1000
12 ··LOAD···········0x0000c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x408c50·RW··0x100012 ··LOAD···········0x0000c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x408c50·RW··0x1000
13 ··DYNAMIC········0x1700c0·0x0000000000857700·0x0000000000857700·0x000000·0x0003c0·RW··0x813 ··DYNAMIC········0x1700c0·0x0000000000857700·0x0000000000857700·0x000000·0x0003c0·RW··0x8
14 ··NOTE···········0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x814 ··NOTE···········0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x8
15 ··NOTE···········0x0003a8·0x00000000000003a8·0x00000000000003a8·0x000024·0x000024·R···0x415 ··NOTE···········0x0003a8·0x00000000000003a8·0x00000000000003a8·0x000024·0x000024·R···0x4
16 ··NOTE···········0x16f8cc·0x00000000008418cc·0x00000000008418cc·0x000020·0x000020·R···0x416 ··NOTE···········0x16f8ac·0x00000000008418ac·0x00000000008418ac·0x000020·0x000020·R···0x4
17 ··TLS············0x1700c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x00166c·R···0x1017 ··TLS············0x1700c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x00166c·R···0x10
18 ··GNU_PROPERTY···0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x818 ··GNU_PROPERTY···0x000388·0x0000000000000388·0x0000000000000388·0x000020·0x000020·R···0x8
19 ··GNU_EH_FRAME···0x001000·0x00000000006ffc1c·0x00000000006ffc1c·0x000000·0x020c84·R···0x419 ··GNU_EH_FRAME···0x001000·0x00000000006ffbfc·0x00000000006ffbfc·0x000000·0x020c84·R···0x4
20 ··GNU_STACK······0x000000·0x0000000000000000·0x0000000000000000·0x000000·0x000000·RW··0x1020 ··GNU_STACK······0x000000·0x0000000000000000·0x0000000000000000·0x000000·0x000000·RW··0x10
21 ··GNU_RELRO······0x1700c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x017f40·R···0x121 ··GNU_RELRO······0x1700c0·0x00000000008420c0·0x00000000008420c0·0x000000·0x017f40·R···0x1
  
22 ·Section·to·Segment·mapping:22 ·Section·to·Segment·mapping:
23 ··Segment·Sections...23 ··Segment·Sections...
24 ···00·····24 ···00·····
25 ···01·····.interp·25 ···01·····.interp·
⊟
5.55 KB
readelf --wide --sections {}
error from `readelf --wide --sections {}`: readelf: Error: Unable to find program interpreter name
    
Offset 1, 8 lines modifiedOffset 1, 8 lines modified
1 There·are·42·section·headers,·starting·at·offset·0x32601b8:1 There·are·42·section·headers,·starting·at·offset·0x32603b8:
  
2 Section·Headers:2 Section·Headers:
3 ··[Nr]·Name··············Type············Address··········Off····Size···ES·Flg·Lk·Inf·Al3 ··[Nr]·Name··············Type············Address··········Off····Size···ES·Flg·Lk·Inf·Al
4 ··[·0]···················NULL············0000000000000000·000000·000000·00······0···0··04 ··[·0]···················NULL············0000000000000000·000000·000000·00······0···0··0
5 ··[·1]·.note.gnu.property·NOTE············0000000000000388·000388·000020·00···A··0···0··85 ··[·1]·.note.gnu.property·NOTE············0000000000000388·000388·000020·00···A··0···0··8
6 ··[·2]·.note.gnu.build-id·NOTE············00000000000003a8·0003a8·000024·00···A··0···0··46 ··[·2]·.note.gnu.build-id·NOTE············00000000000003a8·0003a8·000024·00···A··0···0··4
7 ··[·3]·.interp···········NOBITS··········00000000000003cc·0003cc·00001c·00···A··0···0··17 ··[·3]·.interp···········NOBITS··········00000000000003cc·0003cc·00001c·00···A··0···0··1
Offset 14, 38 lines modifiedOffset 14, 38 lines modified
14 ··[·9]·.rela.dyn·········NOBITS··········0000000000012378·0003cc·03e910·18···A··5···0··814 ··[·9]·.rela.dyn·········NOBITS··········0000000000012378·0003cc·03e910·18···A··5···0··8
15 ··[10]·.rela.plt·········NOBITS··········0000000000050c88·0003cc·006d20·18···A··5··26··815 ··[10]·.rela.plt·········NOBITS··········0000000000050c88·0003cc·006d20·18···A··5··26··8
16 ··[11]·.init·············NOBITS··········0000000000058000·001000·000017·00··AX··0···0··416 ··[11]·.init·············NOBITS··········0000000000058000·001000·000017·00··AX··0···0··4
17 ··[12]·.plt··············NOBITS··········0000000000058020·001000·0048d0·10··AX··0···0·1617 ··[12]·.plt··············NOBITS··········0000000000058020·001000·0048d0·10··AX··0···0·16
18 ··[13]·.plt.got··········NOBITS··········000000000005c8f0·001000·000020·08··AX··0···0··818 ··[13]·.plt.got··········NOBITS··········000000000005c8f0·001000·000020·08··AX··0···0··8
19 ··[14]·.text·············NOBITS··········000000000005c940·001000·6765a5·00··AX··0···0·6419 ··[14]·.text·············NOBITS··········000000000005c940·001000·6765a5·00··AX··0···0·64
20 ··[15]·.fini·············NOBITS··········00000000006d2ee8·001000·000009·00··AX··0···0··420 ··[15]·.fini·············NOBITS··········00000000006d2ee8·001000·000009·00··AX··0···0··4
21 ··[16]·.rodata···········NOBITS··········00000000006d3000·001000·02cc1c·00···A··0···0·3221 ··[16]·.rodata···········NOBITS··········00000000006d3000·001000·02cbfc·00···A··0···0·32
22 ··[17]·.eh_frame_hdr·····NOBITS··········00000000006ffc1c·001000·020c84·00···A··0···0··422 ··[17]·.eh_frame_hdr·····NOBITS··········00000000006ffbfc·001000·020c84·00···A··0···0··4
23 ··[18]·.eh_frame·········NOBITS··········00000000007208a0·001000·0db320·00···A··0···0··823 ··[18]·.eh_frame·········NOBITS··········0000000000720880·001000·0db320·00···A··0···0··8
24 ··[19]·.gcc_except_table·NOBITS··········00000000007fbbc0·001000·045d0c·00···A··0···0··424 ··[19]·.gcc_except_table·NOBITS··········00000000007fbba0·001000·045d0c·00···A··0···0··4
25 ··[20]·.note.ABI-tag·····NOTE············00000000008418cc·16f8cc·000020·00···A··0···0··425 ··[20]·.note.ABI-tag·····NOTE············00000000008418ac·16f8ac·000020·00···A··0···0··4
26 ··[21]·.tbss·············NOBITS··········00000000008420c0·1700c0·00166c·00·WAT··0···0·1626 ··[21]·.tbss·············NOBITS··········00000000008420c0·1700c0·00166c·00·WAT··0···0·16
27 ··[22]·.init_array·······NOBITS··········00000000008420c0·1700c0·000378·08··WA··0···0··827 ··[22]·.init_array·······NOBITS··········00000000008420c0·1700c0·000378·08··WA··0···0··8
28 ··[23]·.fini_array·······NOBITS··········0000000000842438·1700c0·000008·08··WA··0···0··828 ··[23]·.fini_array·······NOBITS··········0000000000842438·1700c0·000008·08··WA··0···0··8
29 ··[24]·.data.rel.ro······NOBITS··········0000000000842440·1700c0·0152c0·00··WA··0···0·3229 ··[24]·.data.rel.ro······NOBITS··········0000000000842440·1700c0·0152c0·00··WA··0···0·32
30 ··[25]·.dynamic··········NOBITS··········0000000000857700·1700c0·0003c0·10··WA··6···0··830 ··[25]·.dynamic··········NOBITS··········0000000000857700·1700c0·0003c0·10··WA··6···0··8
31 ··[26]·.got··············NOBITS··········0000000000857ac0·1700c0·002538·08··WA··0···0··831 ··[26]·.got··············NOBITS··········0000000000857ac0·1700c0·002538·08··WA··0···0··8
32 ··[27]·.data·············NOBITS··········000000000085a000·1700c0·000b60·00··WA··0···0·3232 ··[27]·.data·············NOBITS··········000000000085a000·1700c0·000b60·00··WA··0···0·32
33 ··[28]·.bss··············NOBITS··········000000000085ab80·1700c0·3f0190·00··WA··0···0·6433 ··[28]·.bss··············NOBITS··········000000000085ab80·1700c0·3f0190·00··WA··0···0·64
34 ··[29]·.comment··········PROGBITS········0000000000000000·16f8ec·00001f·01··MS··0···0··134 ··[29]·.comment··········PROGBITS········0000000000000000·16f8cc·00001f·01··MS··0···0··1
35 ··[30]·.debug_aranges····PROGBITS········0000000000000000·16f910·0098c6·00···C··0···0··835 ··[30]·.debug_aranges····PROGBITS········0000000000000000·16f8f0·0098c6·00···C··0···0··8
36 ··[31]·.debug_info·······PROGBITS········0000000000000000·1791d8·1ae627d·00···C··0···0··836 ··[31]·.debug_info·······PROGBITS········0000000000000000·1791b8·1ae6485·00···C··0···0··8
37 ··[32]·.debug_abbrev·····PROGBITS········0000000000000000·1c5f458·03ab23·00···C··0···0··837 ··[32]·.debug_abbrev·····PROGBITS········0000000000000000·1c5f640·03ab23·00···C··0···0··8
38 ··[33]·.debug_line·······PROGBITS········0000000000000000·1c99f80·296931·00···C··0···0··838 ··[33]·.debug_line·······PROGBITS········0000000000000000·1c9a168·296931·00···C··0···0··8
39 ··[34]·.debug_str········PROGBITS········0000000000000000·1f308b8·8f77bc·01·MSC··0···0··839 ··[34]·.debug_str········PROGBITS········0000000000000000·1f30aa0·8f77c3·01·MSC··0···0··8
40 ··[35]·.debug_line_str···PROGBITS········0000000000000000·2828078·0035d6·01·MSC··0···0··840 ··[35]·.debug_line_str···PROGBITS········0000000000000000·2828268·0035d6·01·MSC··0···0··8
41 ··[36]·.debug_loclists···PROGBITS········0000000000000000·282b650·5a6e3c·00···C··0···0··841 ··[36]·.debug_loclists···PROGBITS········0000000000000000·282b840·5a6e35·00···C··0···0··8
42 ··[37]·.debug_macro······PROGBITS········0000000000000000·2dd2490·0e0742·00···C··0···0··842 ··[37]·.debug_macro······PROGBITS········0000000000000000·2dd2678·0e0759·00···C··0···0··8
43 ··[38]·.debug_rnglists···PROGBITS········0000000000000000·2eb2bd8·13ec26·00···C··0···0··843 ··[38]·.debug_rnglists···PROGBITS········0000000000000000·2eb2dd8·13ec26·00···C··0···0··8
44 ··[39]·.symtab···········SYMTAB··········0000000000000000·2ff1800·08f4d8·18·····40·8680··844 ··[39]·.symtab···········SYMTAB··········0000000000000000·2ff1a00·08f4d8·18·····40·8680··8
45 ··[40]·.strtab···········STRTAB··········0000000000000000·3080cd8·1df32b·00······0···0··145 ··[40]·.strtab···········STRTAB··········0000000000000000·3080ed8·1df32b·00······0···0··1
46 ··[41]·.shstrtab·········STRTAB··········0000000000000000·3260003·0001b3·00······0···0··146 ··[41]·.shstrtab·········STRTAB··········0000000000000000·3260203·0001b3·00······0···0··1
47 Key·to·Flags:47 Key·to·Flags:
48 ··W·(write),·A·(alloc),·X·(execute),·M·(merge),·S·(strings),·I·(info),48 ··W·(write),·A·(alloc),·X·(execute),·M·(merge),·S·(strings),·I·(info),
49 ··L·(link·order),·O·(extra·OS·processing·required),·G·(group),·T·(TLS),49 ··L·(link·order),·O·(extra·OS·processing·required),·G·(group),·T·(TLS),
50 ··C·(compressed),·x·(unknown),·o·(OS·specific),·E·(exclude),50 ··C·(compressed),·x·(unknown),·o·(OS·specific),·E·(exclude),
51 ··R·(retain),·D·(mbind),·l·(large),·p·(processor·specific)51 ··R·(retain),·D·(mbind),·l·(large),·p·(processor·specific)
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623 KB
readelf --wide --symbols {}
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915 B
readelf --wide --notes {}
error from `readelf --wide --notes {}`: readelf: Error: Unable to find program interpreter name
    
Offset 1, 12 lines modifiedOffset 1, 12 lines modified
  
1 Displaying·notes·found·in:·.note.gnu.property1 Displaying·notes·found·in:·.note.gnu.property
2 ··Owner················Data·size·»  Description2 ··Owner················Data·size·»  Description
3 ··GNU··················0x00000010»  NT_GNU_PROPERTY_TYPE_0»    ······Properties:·x86·ISA·needed:·x86-64-baseline3 ··GNU··················0x00000010»  NT_GNU_PROPERTY_TYPE_0»    ······Properties:·x86·ISA·needed:·x86-64-baseline
  
4 Displaying·notes·found·in:·.note.gnu.build-id4 Displaying·notes·found·in:·.note.gnu.build-id
5 ··Owner················Data·size·»  Description5 ··Owner················Data·size·»  Description
6 ··GNU··················0x00000014»  NT_GNU_BUILD_ID·(unique·build·ID·bitstring)»   ····Build·ID:·a7828e55d4c3861d7ada569ceba51a29fda80f3b6 ··GNU··················0x00000014»  NT_GNU_BUILD_ID·(unique·build·ID·bitstring)»   ····Build·ID:·f3ee57298096caacdbabfa723d408d4ad1664884
  
7 Displaying·notes·found·in:·.note.ABI-tag7 Displaying·notes·found·in:·.note.ABI-tag
8 ··Owner················Data·size·»  Description8 ··Owner················Data·size·»  Description
9 ··GNU··················0x00000010»  NT_GNU_ABI_TAG·(ABI·version·tag)»     ····OS:·Linux,·ABI:·3.2.09 ··GNU··················0x00000010»  NT_GNU_ABI_TAG·(ABI·version·tag)»     ····OS:·Linux,·ABI:·3.2.0
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269 MB
readelf --wide --debug-dump=info {}
Max HTML report size reached
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1.29 KB
readelf --wide --debug-dump=macro {}
error from `readelf --wide --debug-dump=macro {}`: readelf: Error: Unable to find program interpreter name
    
Offset 11712, 15 lines modifiedOffset 11712, 15 lines modified
11712 ·DW_MACRO_define_strp·-·lineno·:·310·macro·:·PACKAGE_BUGREPORT·"https://github.com/Macaulay2/M2/issues"11712 ·DW_MACRO_define_strp·-·lineno·:·310·macro·:·PACKAGE_BUGREPORT·"https://github.com/Macaulay2/M2/issues"
11713 ·DW_MACRO_define_strp·-·lineno·:·313·macro·:·PACKAGE_NAME·"Macaulay2"11713 ·DW_MACRO_define_strp·-·lineno·:·313·macro·:·PACKAGE_NAME·"Macaulay2"
11714 ·DW_MACRO_define_strp·-·lineno·:·316·macro·:·PACKAGE_STRING·"Macaulay2·1.24.11"11714 ·DW_MACRO_define_strp·-·lineno·:·316·macro·:·PACKAGE_STRING·"Macaulay2·1.24.11"
11715 ·DW_MACRO_define_strp·-·lineno·:·319·macro·:·PACKAGE_TARNAME·"Macaulay2"11715 ·DW_MACRO_define_strp·-·lineno·:·319·macro·:·PACKAGE_TARNAME·"Macaulay2"
11716 ·DW_MACRO_define_strp·-·lineno·:·322·macro·:·PACKAGE_URL·"https://macaulay2.com/"11716 ·DW_MACRO_define_strp·-·lineno·:·322·macro·:·PACKAGE_URL·"https://macaulay2.com/"
11717 ·DW_MACRO_define_strp·-·lineno·:·325·macro·:·PACKAGE_VERSION·"1.24.11"11717 ·DW_MACRO_define_strp·-·lineno·:·325·macro·:·PACKAGE_VERSION·"1.24.11"
11718 ·DW_MACRO_define_strp·-·lineno·:·328·macro·:·PROFILING·011718 ·DW_MACRO_define_strp·-·lineno·:·328·macro·:·PROFILING·0
11719 ·DW_MACRO_define_strp·-·lineno·:·331·macro·:·REL·"6.1.0-37-cloud-amd64"11719 ·DW_MACRO_define_strp·-·lineno·:·331·macro·:·REL·"6.12.22+bpo-amd64"
11720 ·DW_MACRO_define_strp·-·lineno·:·334·macro·:·SIZEOF_INT_P·811720 ·DW_MACRO_define_strp·-·lineno·:·334·macro·:·SIZEOF_INT_P·8
11721 ·DW_MACRO_define_strp·-·lineno·:·337·macro·:·SIZEOF_LONG·811721 ·DW_MACRO_define_strp·-·lineno·:·337·macro·:·SIZEOF_LONG·8
11722 ·DW_MACRO_define_strp·-·lineno·:·350·macro·:·STDC_HEADERS·111722 ·DW_MACRO_define_strp·-·lineno·:·350·macro·:·STDC_HEADERS·1
11723 ·DW_MACRO_define_strp·-·lineno·:·356·macro·:·WITH_NEWLINE_CR·011723 ·DW_MACRO_define_strp·-·lineno·:·356·macro·:·WITH_NEWLINE_CR·0
11724 ·DW_MACRO_define_strp·-·lineno·:·359·macro·:·WITH_NEWLINE_CRLF·011724 ·DW_MACRO_define_strp·-·lineno·:·359·macro·:·WITH_NEWLINE_CRLF·0
11725 ·DW_MACRO_define_strp·-·lineno·:·362·macro·:·WITH_PYTHON·111725 ·DW_MACRO_define_strp·-·lineno·:·362·macro·:·WITH_PYTHON·1
11726 ·DW_MACRO_define_strp·-·lineno·:·365·macro·:·WITH_TBB·111726 ·DW_MACRO_define_strp·-·lineno·:·365·macro·:·WITH_TBB·1
⊟
325 KB
readelf --wide --debug-dump=loc {}
Max HTML report size reached
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222 KB
strings --all --bytes=8 {}
    
Offset 1, 6477 lines modifiedOffset 1, 6502 lines modified
1 GCC:·(Debian·14.2.0-19)·14.2.01 GCC:·(Debian·14.2.0-19)·14.2.0
2 ph8o82\02 ph8o82\0
3 s};-<N8/3 s};-<N8/
4 ZJzgjN/n4 ZJzgjN/n
5 t}<gg]_/5 t}<gg]_/
6 <9/:UsV896 <9/:UsV89
7 \28482868185837 \2848286818583
8 A=n0FvR/# 
9 Z%=&i=y!+ 
10 RM#4.P» / 
11 zWBewqDq 
12 oGb+"S,_ 
13 o^`3]$e[ 
14 3jp4g»  T1Q 
15 FH<6U9V| 
16 dpF»    wQGc; 
17 5<YFi:9v5 
18 +(n4iG'zU 
19 <C»     JJNz, 
20 _O]S:\\I 
21 <Fv=u{lV 
22 ,hR%<%oF 
23 u+%_l!+K 
24 K[1}i+N_ 
25 oC-UUx-U 
26 c4»     >?}GE 
27 tViVc*Fb 
28 d3334pR`a)M 
29 JA7L:F+h\ 
30 I^eBKzlp2 
31 $I9)98C> 
32 7%31^$(F( 
33 /^CaB&$L 
34 6::?q3wJ 
35 4wIJ-&{_ 
36 hRNaH3o@ 
37 No[La*@y 
38 BJsDk(&. 
39 fjY5__mO% 
40 O5?XD@vM9 
41 Ei9R~[7# 
42 YHq{QC)» 
43 "@$:?eht 
44 rMk»    pGxD 
45 hmw5:5x0 
46 +!'Js$&E 
47 G)m=|<<D 
48 wA-4s%2lO 
49 45K%gZIF 
50 ^V/X.QfFen 
51 )0[b8[b8[b8[b8[b8[b] 
52 -]»     R?$-O 
53 q}Ir%E.= 
54 m[-ccmc[- 
55 ?<t|ha^%ii 
56 XR?`2"|r 
57 Gn/4+,Hie 
58 _{u[:@Ot 
59 ]xgaxgaxgaxga 
60 I8p*yK>& 
61 ({LG}O0D 
62 N@9:@ex= 
63 "qW6K6~` 
64 q1pHfvWa 
65 f{/3((f8x 
66 S4j^Rju)M1', 
67 Ui\%Dk4) 
68 ]{|mny=2lq0 
69 u"':dRl4 
70 O8b&9"+o 
71 9i^lZe·* 
72 (,W}xFfK 
73 szRnrb~6 
74 sC·'CV»  ` 
75 o5Eu}FqA 
76 ^QHyaGw· 
77 ·Q0y3}b` 
78 \*CP'VEb 
79 yX9Qj\Z&S 
80 }D>'m|lB# 
81 b>;x,&>2 
82 fW]qhe#K 
83 6961>1*n 
84 >b}R.>tw<b 
85 b.>{$?W1 
86 }R.>h»  NG 
87 hsCsACDP 
88 ;OemhdhC 
89 x;#?,s!X 
90 v#dVya9g 
91 X6UnAe·( 
92 H+1>O@v$ 
93 .uzh((]c 
94 \"sh>E]M 
95 "A._B$B2 
96 5Rkd!2a> 
97 /t+Uh*=Q 
98 v-&pF:Wh 
99 ;)|E&?|O 
100 eb=@E8>' 
101 H\A&`O<1{f 
102 rKNItB`Z@o»      
103 PvE?SoS9 
104 ,»      :%zqtE 
105 lGQNayi;;} 
106 %*5<bjD^)!" 
107 v(x@{p*v 
108 $$YK|SE·@DK 
109 Fd:AGC-`,i:e 
110 2U»     XcA-` 
111 i)~4]rEr 
112 SXB%Y,?< 
113 |AO»    ~»   eL 
114 g~}LNY@k 
115 s{)$~aoH 
116 ka8%Jo6!W 
117 dv?z;SRx 
118 )]R#*@qN2 
119 l<'{~vTe> 
120 C%av[..*S 
121 IXGyg]+m8 }FkDQf%Z1f%Zqf}j
 9 OjtWu\jt7u
 10 GO2·Ly&G9]
 11 Zs/&)5"4
 12 f9=LPN>!
 13 Xv0|9E+j
 14 5fBC2Gd2!
Max diff block lines reached; 223891/227451 bytes (98.43%) of diff not shown.
⊟
213 MB
readelf --wide --decompress --string-dump=.debug_str {}
Max HTML report size reached
Generated by diffoscope 297